P-Adic Functional Analysis and P-Adic Automorphic Forms, By

p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES

COURSE BY RICHARD TAYLOR

Introduction Richard Taylor taught a course on p-adic families of automorphic forms at Stanford in Winter 2020. Except for the last two lectures, these are “live-TeXed” notes from the course, contributed and maintained by various participants in the course. Conventions are as follows: Each lecture gets its own “chapter,” and appears in the table of contents with the date. The last two chapters were written by the lecturer himself, as course meetings were cancelled due to COVID-19. Of course, these notes are not a faithful representation of the course, either in the mathematics itself or in the quotes, jokes, and philosophical musings; in particular, the errors are our fault. By the same token, any virtues in the notes are to be credited to the lecturer and not the scribe(s).1 Please email suggestions to [email protected] or [email protected]. Contributors • Dan Dore • Daniel Kim • Aaron Landesman • Lynnelle Ye

1This introduction has been adapted from Akhil Mathew’s introduction to his notes, with his permission. 1 2 COURSE BY RICHARD TAYLOR

Contents Introduction 1 1. 1/7/20: Course overview 3 2. 1/9/20: Introduction to p-adic functional analysis 6 3. 1/14/20: Properties of locally convex topological vector spaces 10 4. 1/16/20: Properties of complete LCTVSs 13 5. 1/21/20: Structure and examples of Banach spaces 17 6. 1/23/20: Properties of dual spaces 21 7. 1/28/20: More on dual spaces 25 8. 1/30/20: Finishing dual spaces, compact maps, and tensor products 29 9. 2/4/20: More on tensor products, nuclear spaces, and spaces of compact type 32 10. 2/6/20: Analytic functions and their roots 37 11. 2/11/20: Locally convex topological L-algebras 41 12. 2/13/20: Modules over Fr´echet-Stein and Banach algebras 46 13. 2/18/20: Characteristic power series 51 14. 2/20/20: More properties of characteristic power series 55 15. 2/25/20: Eigenspaces, and an introduction to locally analytic p-adic manifolds 60 16. 2/27/20: Functions and distributions on locally analytic manifolds and groups 65 17. 3/3/20: Distributions and representations for locally analytic groups 70 18. 3/5/20: Locally analytic representations and more on compact operators 74 19. 3/10/20: Automorphic forms on deﬁnite quaternion algebras 77 20. 3/12/20: The eigencurve 83 References 93 p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 3

1. 1/7/20: Course overview Q0 ˆ Let p be prime. Let A = v Qv = (Z ⊗ Q) × R. 1.1. Modular curves. Let’s start with two examples of the spaces whose cohomologies we will be interested in. n (1) The modular curve for GL2 of level p is n ˆ p n × Y (p ) = GL2(Q)\ GL2(A)/ GL2(Z ) × U(p ) × (SO2 R>0) where ˆ p = lim /m and Z ←−p-m Z Z n n U(p ) = ker(GL2(Zp) → GL2(Z/p Z)). ` This can be written in the form i Γi\(C − R) where the Γi are congruence subgroups in SL2(Z). (2) Let D be a quaternion algebra of center Q ramified at a prime r 6= p and ∞. We have ∼ × × n D ⊗ R = H, and (D ⊗ R) /R is compact. The Shimura variety for D of level p is n × ∞ × ˆp × n YD(p ) = D \(D ⊗ A ) /(OD) × U(p ) ∞ ˆ ∼ n where A = Z ⊗Z Q, noting that D ⊗ Qp = M2×2(Qp) so the subgroup U(p ) makes sense. n n Y (p ) is a 1-dimensional C-manifold or 2-dimensional R-manifold; YD(p ) is 0-dimensional. i n i n We are interested in the cohomologies H (Y (p ),A) and H (YD(p ),A) where A is Qp, Zp, s Z/p Z, or Qp/Zp. 0 1 0 n For GL2, only H and H are nontrivial and H is easy to understand because Y (p ) is the complement of finitely many points in a proper curve. So we’ll mainly be concerned 1 n 1 n 0 n with H (Y (p ),A) or Hc (Y (p ),A). For D, we only need to consider H (YD(p ),A) = n Map(YD(p ),A). These are finite A-modules. 1.2. Limits of cohomologies. If we take the limit lim H1 (Y (pn),A), the result has an −→n (c) n m action by GL2(Qp), since any g ∈ GL2(Qp) takes U(p ) into U(p ) for n m. This limit also has an action by ±1 T0 = Zp[Tq,Sq | q 6= r, p prime], n n where Tq,Sq are Hecke operators coming from the maps Y (p , q) ⇒ Y (p ). Finally, there is also an action by GQ = Gal(Q/Q). 0 n × ∞ × ˆp × Similarly, lim H (YD(p ),A) = LC(D \(D ⊗ ) /(O ) ,A) (the locally constant func- −→n A D tions) has an action by GL2(Qp) × T0. As representations, these limits are smooth (all stabilizers are open) and admissible (the fixed points of any open subgroup are a finite A-module). We will refer to both of them, where n n A = Zp, as “H”. We can recover the cohomology at finite level p by taking U(p )-invariants of H. 1 ∞ 0 ∞ We will write T(c) or TD for the image of T0 in H(c)(Y (p ), Qp/Zp) or H (YD(p ), Qp/Zp). We can complete p-adically to get Hˆ 1 (Y (p∞), ) = lim H1 (Y (p∞), /ps ) (c) Zp ←− (c) Z Z s Hˆ 0(Y (p∞), ) = lim H0(Y (p∞), /ps ) = Map (Y (p∞), ) D Zp ←− D Z Z cts D Zp s 4 COURSE BY RICHARD TAYLOR with the p-adic topology. These have actions by GL2(Qp) × T0 × GQ, where the action by GL2(Qp) is continuous and “unitary”, and the GQ is for the GL2-case only. We will call these ˆ ˆ ˆ ˆ U(pn) 1 n H. HQp = H ⊗Zp Qp is a p-adic Banach space. We again can recover H = H(c)(Y (p ), Zp) 0 n ˆ sm or H (YD(p ), Zp), and H = H (by which we mean the smooth vectors, that is, those with open stabilizer in GL2(Qp)). We can instead take the projective limit over the trace maps ··· Y (pn) → · · · → Y (p2) → Y (p) to get H˜ 1 := lim H1 (Y (pn), ) (c) ←− (c) Zp n H˜ 0 := lim H0(Y (pn), ). ←− D Zp n

These have actions of GL2(Qp) × T(c) × GQ and GL2(Qp) × TD respectively. We will call them H˜ . Then H˜ has an action by GL ( ) = lim [GL ( /ps )] Zp 2 Zp ←− Zp 2 Z Z J K ∼ and is ﬁnite over Zp GL2(Zp) . Note that Zp Zp = Zp T . We have approximately J K J K J K ˜ ˆ H = Homcts(H, Zp) ˆ ˆ (for GL2 we should put Hc instead of H) with the weak topology on the RHS. This is because blah m s blah m s H (Y (p ), Z/p Z) and Hc (Y (p ), Z/p Z) are dual under Poincare duality and the limits go as expected. Q 1.3. Localizations. T has only ﬁnitely many maximal ideals and can be written as m max Tm. Each Tm is a complete noetherian local Zp-algebra. For each m we have a unique continuous semisimple representation

rm : GQ → GL2(k(m)) such that (1) 1 0 r (c) ∼ , m 0 −1

(2) rm is ramiﬁed only at p and r, and (3) tr(rm(Frobq)) = Tq, det(rm(Frobq)) = qSq for q 6= p, r. Furthermore, 2 2 (4) in the D case, if ϕ lifts Frobr, we have r(tr(rm(ϕ))) = (1 + r) det(rm(ϕ)). In the GL2-case, rm is unramiﬁed at r.

Theorem 1.3.1 (Khare-Wintenberger). If r : GQ → GL2(Fp) satisﬁes (1), (2), and (4), and ∼ is irreducible, then there exists m and k(m) ,→ Fp such that r = rm.

We call m Eisenstein if rm is reducible. Assume m is not Eisenstein. Then there is rm : GQ → GL2(Tm), a continuous representation satisfying (1)-(4), which is the universal deformation of rm satisfying (1), (2), and (4). That is to say, if R is any complete noetherian local Zp-algebra and r : GQ → GL2(R) satisﬁes (1), (2), and (4), then there is a unique Tm → R taking rm to r. Let Vm be the underlying Tm[GQ]-module of rm. In the GL2-case, we can write ∼ + Hm = Vm ⊗Tm Hm ˆ ∼ ˆ + Hm = Vm ⊗Tm Hm p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 5

˜ ∼ ˜ + Hm = Vm ⊗Tm Hm for H+ such that H+ ∼= Hom (V ,H ). In the D case deﬁne H+ := H (similarly with m m Tm[GQ] m m m m ˆ ˜ + H, H). Then Hm etc. have actions by Tm × GL2(Qp). + Let θ : Tm → Qp be a continuous homomorphism. Hm [θ] is usually 0. For special θ, we ˆ + may get an irreducible smooth representation of GL2(Qp). On the other hand Hm[θ] (that is, ˆ + the subspace of x ∈ Hm such that T (x) = θ(T )x for all T ∈ Tm) is always nonzero and is a unitary Banach representation of GL ( ). Both eigenspaces depend only on (θ ◦ r )| , 2 Qp m GQp + ˆ + ˜ but Hm [θ] does not determine it, and Hm [θ] does, as does Hm ⊗Tm,θ Qp. ˆ + la ˆ + We may now take the subspace (Hm ) of Hm , consisting of the locally analytic vectors: ˆ + ˆ + 1 x ∈ Hm such that GL2(Qp) → Hm , g 7→ gx, is analytic in some neighborhood of 2. This ˆ + la g + has an action by the Lie algebra g = gl2 such that ((Hm ) ) = Hm. ˆ + la 1.4. Eigenvarieties. We may now take the Jacquet module of (Hm ) . That is, let 1 ∗ N = ⊂ GL ; 0 1 2

for Π a smooth representation its Jacquet module is JN (Π) = ΠN(Qp), the coinvariants of N, la a representation of T (Qp). Emerton deﬁned an analogue JN which works for locally analytic la ˆ + la ∼ × × representations; then JN ((Hm ) ) is a locally analytic representation of T (Qp) = Qp × Qp . ˇ × ∨ × ∨ Let T be the space of all locally analytic characters of T (Qp), that is, (Qp ) × (Qp ) . We × can write Qp in the form Z p × (ﬁnite) × (1 + pZp), × ∨ hence write (Qp ) in the form

Gm × (finite) × (open disc of radius 1 about 1). la ˆ + la ∨ This is a rigid analytic space, and JN ((Hm ) ) gives rise to a coherent sheaf F on T . ∨ This F still has an action by Tm; let T be the image of Tm in End(F ) over T . Taking Spa(T ) gives the eigenvariety E , which has a finite map to T ∨. Composing with the map ∨ ∨ from T to W = T (Zp) that forgets the values of the character on Qp gives the map E → W , which has discrete fibers (W is called “weight space”). We also have a map E → (Spf(Tm))η, where (Spf(Tm))η is also a rigid space.

(1) ∨ (Spf(Tm))η T

W ∨ Fixing determinants, E , W are 1-dimensional, (Spf(Tm))η is 2-dimensional, and T is 3-dimensional. Zp T η is the open unit disc. E → (Spf(Tm))η has finite fibers and its image is called the infiniteJ K fern; it is a sort of space-filling curve. The image of E → (Spf(Tm))η can be characterized in terms of (ϕ, Γ)-modules. Roughly, points on E correspond to representations of GQp whose associated rank-2 (ϕ, Γ)-module is × given by an extension of rank-1 (ϕ, Γ)-modules (that is, locally analytic characters of Qp ). 6 COURSE BY RICHARD TAYLOR

We will construct/examine these spaces in more detail throughout the quarter. We will start with the p-adic functional analysis necessary to make all of this rigorous.

2. 1/9/20: Introduction to p-adic functional analysis

Fix some notation: let L/Qp be a finite extension, O = OL its ring of integers, O/(π) = F × its finite residue field, λ = (π) its maximal ideal. Letting v : L Z be the discrete valuation, we define the absolute value on L to be |α| = (#F)−v(α). A topological vector space over L is a vector space V over L with a topology such that the addition map +: V × V → V and the scalar multiplication map L × V → V are continuous. An appropriately general class of topological vector spaces consists of locally convex topological vector spaces. We may define these either in terms of lattices or in terms of seminorms.

2.1. Lattices. A subset Λ ⊆ V is called a lattice if it is an O-submodule which spans V over L. Equivalently, for any x ∈ V , there is some α ∈ L× with αx ∈ Λ. × Note that if Λ is a lattice, then αΛ is a lattice for any α ∈ L . If Λ1, Λ2 are lattices, then Λ1 ∩ Λ2 is a lattice. Two lattices Λ, Λ0 are considered equivalent, written Λ ∼ Λ0, if there exist α, β ∈ L× such 0 0 0 0 0 that αΛ ⊆ Λ ⊆ bΛ. Note that if Λ1 ∼ Λ1 and Λ2 ∼ Λ2, then Λ1 ∩ Λ2 ∼ Λ1 ∼ Λ2. Given a collection of lattices L , we may define a topology on V . This topology JL is the coarsest topology such that all elements of L are open and V is a topological vector space. We define 0 × L = {α1Λ1 ∩ · · · ∩ αrΛr : αi ∈ L , Λi ∈ L}. This is the closure of L under scalar multiplication and finite intersection. Note that 0 JL = JL 0 , and that replacing the lattices in L by equivalent lattices does not change L . Using this, we can describe JL more explicitly: 0 JL = JL 0 = {U ⊆ V | ∀x ∈ U, ∃Λ ∈ L such that x + Λ ⊆ U}. In other words, we take L 0 as a fundamental system of neighborhoods of 0. To show this claim, it suffices to show that the set described above gives V the structure of a topological vector space. Continuity of scalar multiplication is equivalent to the statement that for any x ∈ V, α ∈ L×, Λ ∈ L , there is an open neighborhood of (α, x) in L × V which maps into αx + Λ. We may find β ∈ L× such that βx ∈ Λ and β ∈ αO. Then we have: αx + Λ ⊇ (α + βO)(x + α−1Λ) = αx + Oβx + βα−1Λ + Λ. This shows continuity of scalar multiplication. The case of addition is similar and easier. Note that if we define Lfto be the set of lattices equivalent to something in L , we have J = J . L Lf

2.2. Seminorms. A second approach to describing locally convex topological vector spaces, resembling the archimedean case more closely, is to use seminorms.A seminorm k · k on V is a map k · k: V → R≥0 such that: (1). kαxk = |α|kxk for all α ∈ L, x ∈ V . (2). kx + yk ≤ max(kxk, kyk) for all x, y ∈ V . p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 7

We call k · k a norm if in addition kxk = 0 implies x = 0. We say that two seminorms are 0 × 0 equivalent, written k · k ∼ k · k , if there exist C1,C2 ∈ R>0 such that C1k · k ≤ k · k ≤ C2k · k. 0 0 If k · k, k · k are seminorms and c ∈ R>0, then Ck · k and max(k · k, k · k ) are seminorms as well. Let V be a set of seminorms. We use this to deﬁne the topology JV on V as the weakest topology which makes V a topological vector space such that each k · k ∈ V is continuous. This topology has a basis of open sets of the form

x + {y ∈ V : kyki ≤ i, i = 1, . . . , r} where x ∈ V , k · k1,..., k · kr ∈ V , and 1, . . . , r ∈ R>0. If we deﬁne to be the set of seminorms equivalent to something in , we have J = J . Ve V V Ve 2.3. Translating between lattices and seminorms. These two approaches are equivalent, by the following constructions. Given a lattice Λ, we deﬁne a seminorm k · kΛ: ( 0 Lx ⊆ Λ kxkΛ = inf |α| = × α∈L× min{|α|: α ∈ L , x ∈ αΛ} Lx 6⊆ Λ. x∈αΛ

(There is a general theory for ﬁelds like Cp as well, but then you have to be a bit more careful since you can’t say min.) To show that k · kΛ is a seminorm, we check the ultrametric inequality kx + ykΛ ≤ max(kxkΛ, kykΛ). If Lx ⊆ Λ, i.e. kxkΛ = 0, then y ∈ αΛ if and only if x + y ∈ αΛ, and so kx + yk = kyk. If Lx 6⊆ Λ, Ly 6⊆ Λ, then there is some α with |α| minimal such that x ∈ αΛ and β with |β| minimal such that y ∈ βΛ. Then ( αΛ |α| ≥ |β| x + y ∈ αΛ + βΛ = βΛ |β| ≥ |α|

Thus, kx + ykΛ ≤ max(|α|, |β|) = max(kxkΛ, kykΛ) as desired. We have the following compatibilities: 0 • Λ ⊆ Λ implies that k · kΛ0 ≥ k · kΛ. −1 •k·k αΛ = |α| k · kΛ. 0 • If Λ ∼ Λ , then k · kΛ ∼ k · kΛ0 . •k·k Λ∩Λ0 = max(k · kΛ, k · kΛ0 ). On the other hand, if k · k is a seminorm, we obtain a lattice Λ(k · k) via its unit ball: Λ(k · k) = {x ∈ V : kxk ≤ 1}. This is always a lattice (this is easy to check). We have the following compatibilities: •k·k 0 ≥ k · k implies that Λ(k · k0) ⊆ Λ(k · k). • αΛ(k · k) = Λ(|α|−1k · k). • Λ(max(k · k, k · k0) = Λ(k · k) ∩ Λ(k · k0). • If k · k ∼ k · k0, then Λ(k · k) ∼ Λ(k · k0). × • Λ(k · kΛ) = {x ∈ V : inf{|α|: α ∈ L , x ∈ αΛ} ≤ 1} = Λ. × • Note that x ∈ αΛ(k · k) if and only if kxk ≤ |α|, so kxkΛ(k·k) = inf{|α|: α ∈ L , kxk ≤ −1 |α|}. We have k · k ≤ k · kΛ(k·k) ≤ |π |k · k, so these norms are equivalent. • Given a set of lattices L , let V be the corresponding set of seminorms k · kΛ for Λ ∈ L . Then JL = JV . 8 COURSE BY RICHARD TAYLOR

2.4. Locally convex topological vector spaces. Now, we define: Definition 2.4.1. V is a locally convex topological vector space if it is a topological vector space over L whose topology can be defined by a set of lattices, or equivalently by a set of seminorms. Here are some immediate properties: • V is a Hausdorff topological space if and only if the intersection of all open lattices Λ is 0. • A seminorm on V is continuous if and only if k · k−1([0, )) is open for all > 0. V is called normable if the topology can be defined by a single norm. Such V are always Hausdorff (but the converse is not true). The appropriate notion of morphism between locally convex topological vector spaces is that of a continuous linear map. We denote L (V,W ) for the set of all such maps. Note that f : V → W is continuous if and only if the preimage of any open lattice is open, if and only if for any continuous seminorm k · k on W , the seminorm f −1k · k = kf(·)k is continuous on V . We let LCTVS be the L-linear category of locally convex topological vector spaces over L with continuous linear maps. This category has all small colimits: given a covariant functor F : J → LCTVS, we have: M lim F = F A/hι (x) − ι (F (f)(x)): f : A → B ∈ arr(J)i −→ A B A∈ob(J)

(here, ιA etc. denotes inclusion in the direct sum). The topology on the direct sum is the canonical one and we use the quotient topology. Similarly, we have all small limits: given a covariant functor F : J → LCTVS, we have Y lim F = {(x ) ∈ FA: ∀f : A → B ∈ J, F (f)(x ) = x }. ←− A A B A∈ob(J) The topology is defined as the subspace topology of the product topology, i.e. a (sub-)basis of open sets is given by sets of the form (Λ × Q FB) ∩ lim F for A ∈ ob(J), Λ ⊆ FA an B6=A ←− open lattice. In particular, we have products, coproducts, kernels, cokernels, quotients, and finite products. Some properties: • For both limits and colimits, the underlying vector space is the corresponding limit/colimit in the category of vector spaces. • Finite products are equal to finite coproducts. • M M M (Vi/Wi) = ( Vi)/( Wi). • Direct sums and products preserve the property of being Hausdorff. • V/W is Hausdorff if and only if W is closed in V . On the other hand, LCTVS is not an abelian category: the continuous identity map (L, {O}) → (L, {L}) is an epimorphism which is not a cokernel. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 9

2.5. More properties of LCTVSs. If we are dealing with finite-dimensional vector spaces, not much is happening: Lemma 2.5.1. If V is a finite dimensional Hausdorff LCTVS then V is isomorphic to L⊕d with the topology defined by kxk∞ = maxi=1,...,d(|xi|). In particular, V is normable. (Note that the same is true for finite-dimensional vector spaces over an archimedean field: the Euclidean topology is the unique locally convex topology on Rn). Proof. Suppose that L is a collection of open lattices making L⊕d a Hausdorff LCTVS. Then for Λ ∈ L , define NΛ = ∩α∈L× αΛ ⊆ V . This is a L-vector subspace. Also, we have

NΛ1∩Λ2 = NΛ1 ∩ NΛ2 . 0 Choose Λ such that NΛ has minimal (finite) dimension. Then for all Λ ∈ L , the inclusion 0 NΛ∩Λ0 ⊆ NΛ must be an equality. Thus NΛ = NΛ∩Λ0 ⊆ NΛ0 for all open lattices Λ ∈ L . 0 Thus, NΛ ⊆ ∩Λ0∈L Λ = (0), as (V, L ) is Hausdorff; therefore NΛ = 0. Now, if Λ0 ∈ L is an open lattice, since V is finite dimensional, we may find some α ∈ L× 0 ⊕d 0 0 such that αei ∈ Λ for all i, i.e. that αO ⊆ Λ . This implies that all Λ ∈ L are open for the k · k∞ topology. We have to show that O⊕d is open in the topology defined by L . It will suffice to show that Λ ⊆ αO⊕d for some α ∈ L× with Λ as above. n−1 Suppose that this is not the case. Then for each n, there exists some xn ∈ Λ with π xn 6∈ ⊕d −(n−1) −n O , i.e. kxnk∞ ≥ |π| . Without loss of generality, we may take kxnk∞ = |π| . Then n ⊕d π xn ∈ O , which is a compact metric space with respect to the metric d(x, y) = kx − yk∞. n ⊕d × Therefore, by passing to a subsequence, we may assume π xn → y ∈ O . For any β ∈ L , n n we therefore have π βxn → βy. Since π βxn ∈ Λ for n 0, we see βy ∈ Λ. Thus, Ly ⊆ Λ, which is a contradiction. Lemma 2.5.2. For a locally convex topological vector space V , the following are equivalent: (1) V is metrizable (i.e. the topology can be defined by a metric). (2) The topology on V can be defined by a translation-invariant metric. (3) V is Hausdorff and the topology on V can be defined by countably many lattices (or equivalently countably many seminorms). Proof. • (2) implies (1) trivially. • (1) implies (3): If the topology on V is defined by a metric d, consider the open neighborhoods of 0 given by Ωn = {x | d(x, 0) ≤ 1/n} for n = 1, 2,.... Since V is a LCTVS, we can find an open lattice Λn ⊆ Ωn for each n. Since d defines the topology,

the Λn give a basis of neighborhoods of 0, so we may take J = J{Λn}. Note that ∩nΛn = {0}, so V is Hausdorff. • (3) implies (2): Suppose the topology on V is defined by a countable family of seminorms k · k1, k · k2,.... Then we define

kx − ykn d(x, y) = sup n . n 2 (1 + kx − ykn) r 1 – d is well-deﬁned and ≤ 1 always, as the function r 7→ 1+r = 1 − 1+r is a monotonically increasing function from [0, ∞) to [0, 1). – We clearly have d(x, y) = d(y, x). – d(x, y) = 0 if and only if kx − ykn = 0 for all n if and only if x = y (since V is Hausdorﬀ). 10 COURSE BY RICHARD TAYLOR

– kx − ykn ky − zkn d(x, z) ≤ sup max n , n = max(d(x, y), d(y, z)). n 2 (1 + kx − ykn) 2 (1 + ky − zkn) Why do we want this extra generality, instead of just using Banach spaces? Many natural spaces, e.g. locally analytic functions on Zp, are not Banach and not even Frechet.

3. 1/14/20: Properties of locally convex topological vector spaces Last time, we discussed when locally convex topological vector spaces are metrizable. Here are some additional facts about metrizable LCTVS’s: Lemma 3.0.1. (1) A normable LCTVS is metrizable. (2) A Hausdorﬀ ﬁnite dimensional LCTVS is metrizable. (3) A countable (inverse) limit of metrizable LCTVS’s is again metrizable. Proof. For (3), we note that the inverse limit has a topology generated by pulling back the seminorms on each Vi.

3.1. “Convergent” products. Suppose that (Vi, k · ki)i∈I ) are normed LCTVS’s. There are additional important constructions of normed LCTVS’s out of these, given by n Y o ⊥i∈I,0(Vi, k · k) = (xi) ∈ vi : ∀ > 0, kxiki < for all but ﬁnitely many i and n Y o ⊥i∈I (Vi, k · ki) = (xi) ∈ vi : kxiki is bounded independently of i .

We can think of ⊥0 as the set of sequences (xi) ∈ ⊥ with kxiki → 0. These constructions depend on the choice of norms, not just the topology. Note that these admit a continuous Q map to i Vi. We deﬁne a norm on these spaces by the supremum k(xi)k∞ = supi∈I kxiki. We claim that n ⊥0 is a closed subspace in ⊥ with respect to this norm topology. Indeed, suppose (xi ) is n a sequence of elements of ⊥0 converging to some (xi) ∈ ⊥, i.e. that supi∈I kxi − xik → 0. n Then for any > 0, we can choose some n such that kxi − xik < for all i. For all but n ﬁnitely many i, kxi k < , so by the ultrametric inequality kxik < for such i. Now we give some examples: Example 3.1.1. Let Ω be a compact topological space, and consider the space C(Ω,L) of

continuous functions from Ω to L. We equip this with a norm by kfk = supx∈Ω |f(x)|. This is a basic example of a normed LCTVS over L. Example 3.1.2. Deﬁne the ring of convergent power series

LhT i = {f(T ) = f0 + f1T + · · · ∈ L[[T ]]: |fi| → 0 as i → ∞} ∞ This is isomorphic to ⊥i=0,0(L, | · |). We can interpret LhT i as analytic functions on the “unit ball” OL = ∪K/L,[K:L]<∞OK . Indeed, given a ∈ OL, f(T ) = f0 + f1T + · · · ∈ LhT i, then we can define f(a) ∈ OL(a) ⊆ OL by 2 f(a) = f0 + f1a + f2a + ··· i i Note that this infinite sum makes sense by the fact that |fia | = |fi||a| ≤ |fi| → 0 (so by the ultrametric inequality, the difference between any two partial sums also goes to 0). p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 11

Lemma 3.1.3. For f ∈ LhT i, we have kfk = sup |f(t)|. t∈OL Proof. First of all, we have sup |f(t)| ≤ kfk. t∈OL Indeed, applying the ultrametric inequality tells us that i |f(t)| = max |fi||t| ≤ max |fi| = kfk i i To see equality, we first assume without loss of generality (by scaling f by an element of L×) × that kfk = 1, i.e. that fi ∈ O for all i and fi ∈ O for some i. We need to prove that there is some t ∈ such that |f(t)| = 1, i.e. that |f(t)| ∈ ×. OL OL Since we have assumed that fi ∈ O for all i, we can take the reduction f = f (mod π) ∈ F[T ]. (Note that it is indeed a polynomial, as |fi| < 1 for all but finitely many i). Since × fi ∈ O for at least one i, we see that f is a nonzero polynomial. Thus, there is some t ∈ F such that f(t) 6= 0. Then, lifting t to some t ∈ OL, we have that f(t) = f(t) 6= 0, so |f(t)| = 1. 3.2. Bounded subsets. Definition 3.2.1. We say that B ⊆ V is bounded if any continuous semi-norm on V is bounded on B. Lemma 3.2.2. Let N be a collection of continuous semi-norms generating the topology of V . Then the following are equivalent for a subset B ⊆ V : (1) B is bounded. (2) For any k · k ∈ N , k · k is bounded on B. (3) For any open lattice Λ, there is some a ∈ L× such that B ⊆ aΛ. Proof. • Since the semi-norms in N generate the topology of V , any continuous seminorm on V is bounded by the maximum of some finite set of semi-norms inside N . So (1) and (2) are equivalent. • Assume that B is bounded. If Λ is an open lattice, then k · kΛ is a continuous × semi-norm, so there is some a ∈ L such that kbkΛ ≤ |a| for b ∈ B. Unraveling definitions, this says that B ⊆ aΛ. Thus (1) implies (3). • Assume that (3) holds and let k · k be a continuous semi-norm. Then its unit ball Λ(k · k) is an open lattice, so there is some a ∈ L× with B ⊆ aΛ(k · k), i.e. kBk ≤ |a|. So (3) implies (1).

Lemma 3.2.3. (1) Suppose that B ⊆ V is bounded. Then hBiO , the O-module generated by B, is bounded. (2) If B ⊆ V is bounded, then its closure B is bounded. (3) If f : V → W is continuous linear and B ⊆ V is bounded, then f(B) is bounded. (4) If Vi is Hausdorﬀ for all i, then: L • B ⊆ i Vi is bounded if and only if priB is bounded for all i and priB = (0) for all but ﬁnitely many i. Q • B ⊆ i Vi is bounded if and only if priB is bounded for all i. 12 COURSE BY RICHARD TAYLOR

Proof. (1) This follows from the ultrametric inequality: kax + byk ≤ max(kxk, kyk) for a, b ∈ O, x, y ∈ V . (2) If k · k is continuous, then kBk ⊆ kBk, which is bounded. (3) If M ⊆ W is an open lattice, then f −1M ⊆ V is an open lattice, so B ⊆ af −1M for some a ∈ L× and thus f(B) ⊆ aM. (4) Note that in ⊕iVi, the topology is generated by open lattices of the form ⊕iΛi where Λi ⊆ Vi is an open lattice. Thus, the condition that B is bounded is equivalent to × saying that for any collection (Λi)i, there is some a ∈ L with priB ⊆ aΛi for each i. Certainly the condition in the lemma is sufficient. Conversely, suppose that priB is non-zero for infinitely many i, and let i1, i2,... be an infinite sequence of distinct such i. For all n, there exists some bn ∈ B such that

bn,in 6= 0. Then since Vin is Hausdorff, there exists Λin ⊆ Vin an open lattice such that πnb 6∈ Λ . Consider the open lattice Λ = Q∞ Λ × Q V ⊆ L V . Since B is n,in in n=1 in j6=in j i m m bounded, there exists an m such that π B ⊆ Λ, and therefore π bn,in ∈ Λin : taking n ≥ m gives a contradiction. Q Q In i Vi, the topology is generated by open lattices of the form Λi × j6=i Vi with Λi ⊆ Vi an open lattice. So B is bounded if for each choice of i, Λi, we can choose × a ∈ L such that priB ⊆ aΛi, i.e. that priB is bounded. Lemma 3.2.4. If V is metrizable and Λ ⊆ V is a lattice, then the following are equivalent: • Λ ⊆ V is open. • For all bounded subsets B ⊆ V , there exists some a ∈ L× such that B ⊆ aΛ. • For all bounded closed O-submodules B ⊆ V , there is some a ∈ L× such that B ⊆ aΛ. Proof. The second two points are equivalent due to the fact that replacing a bounded subset B by the closure of the O-submodule which it generates preserves the property of being bounded. Now, since V is metrizable, there exists a countable sequence Λ1 ⊇ Λ2 ⊇ Λ3 ··· of open lattices generating the topology on V (we can assume they are nested as we can always replace open lattices with finite intersections thereof). Suppose there is some Λ satisfying the condition as in the lemma but which is not open. n Then for all n, there exists some xn ∈ π Λn with xn 6∈ Λ. −n We claim the sequence {π xn} is bounded: we need to show that for any m, there is × −n −n some a ∈ L such that π xn ∈ aΛm for all n. If n ≥ m, we have π xn ∈ Λn ⊆ Λm. Then × −i we just need to choose a ∈ L such that π xi ∈ aΛm for i = 1, . . . , m − 1. × −n Thus, there is some a ∈ L such that π xn ∈ aΛ for each n. Therefore, there exists n−m some m such that xn ∈ π Λ for all n. Thus, for n ≥ m, we have xn ∈ Λ, which is a contradiction. Thus, we see that for a metrizable LCTVS, the topology can be described by understanding bounded sets instead of open sets. Since many LCTVS’s of interest are not metrizable, we identify a more general class of spaces with this property: Definition 3.2.5. A LCTVS V is bornological if for any lattice Λ ⊆ V , Λ is open if and only if for any bounded B ⊆ V we may find a ∈ L× with B ⊆ aΛ. We just showed that any metrizable LCTVS is bornological. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 13

Lemma 3.2.6. If V is bornological and f : V → W is linear, then f is continuous if and only if it takes bounded sets to bounded sets. Proof. We already have seen that a continuous map preserves boundedness. Conversely, assume that f takes bounded sets to bounded sets. Suppose that M ⊆ W is an open lattice: we want to show that the lattice f −1M is open. Let B ⊆ V be any bounded set. Since f preserves bounded sets and M is open, there is some a ∈ L× such that f(B) ⊆ aM, and therefore B ⊆ af −1(M). Since V is bornological, −1 we see that f M is open. Lemma 3.2.7. The following are equivalent for an LCTVS V : (1) V is normable. (2) The topology on V is generated by a single open lattice Λ such that ∩a∈L× aΛ = (0). (3) V is Hausdorff and V contains a bounded open lattice. In this case, the topology is generated by any bounded open lattice. Proof. • Let V be normable. Then V is Hausdorff, and the unit ball Λ(k · k) is a bounded open lattice, so (1) implies (3). • Suppose that V is Hausdorff and contains a bounded open lattice Λ. Let Λ0 be any open lattice. Since Λ is bounded, there is some a ∈ L× such that Λ ⊆ aΛ0, i.e. that a−1Λ ⊆ Λ0. Thus, Λ0 is open in the topology generated by Λ, so Λ generates the topology of V . Moreover, let 0 6= γ ∈ V be arbitrary. Since V is Hausdorff, there × is some a ∈ L with γ 6∈ aΛ (as Λ generates the topology). Thus, ∩a∈L× aΛ = (0). Thus, we see that (3) implies (2). • Finally, assume that the topology on V is generated by an open lattice Λ satisfying ∩a∈L× aΛ = (0). Then the topology on V is generated by the seminorm k · kΛ. This × seminorm is a norm, since if kxkΛ = 0 then by definition x ∈ aΛ for all a ∈ L , and therefore x = 0. Thus (2) implies (1). 3.3. Completeness. Just as in archimedean functional analysis, an essential role will be played by the property of completeness. Since we care about potentially non-metrizable (i.e. non-first-countable) LCTVS’s, we need to generalize the usual notion of Cauchy sequences to Cauchy nets. Let I be a partially ordered set which is directed: i.e. for all i, j ∈ I there is some k ∈ I with k ≥ i and k ≥ j. In this setting, we will say “for all i 0 ...” to mean “there exists some j such that for all i ≥ j . . .”. A Cauchy net (xi)i∈I is a collection of elements of V indexed by a directed set I such that for any open lattice Λ, xi − xj ∈ Λ whenever i, j 0. We say that an x ∈ V is a limit of (xi)i∈I if for any open lattice Λ, x − xi ∈ Λ for i 0.

Deﬁnition 3.3.1. A LCTVS V is complete if every Cauchy net (xi)i∈I in V has a unique limit in V .

4. 1/16/20: Properties of complete LCTVSs 4.1. General facts about completeness. Lemma 4.1.1. If a LCTVS V is ﬁnite-dimensional and Hausdorﬀ, V is automatically complete. 14 COURSE BY RICHARD TAYLOR

Proof. We have already seen that V ' L⊕n is normable with topology given by the norm

k(xi)k∞ = supi=1,...,n |xi|. Since V is normable, we can characterize completeness in terms of Cauchy sequences (rather than Cauchy nets). If x(m) is a Cauchy sequence in V , then for (m) (m) (m) each i = 1, . . . , n, xi is Cauchy as well, as |xi − xj | ≤ kxi − xjk. By completeness of L, (m) n the Cauchy sequence xi converges to a limit xi ∈ L. Let x = (xi)i=1 ∈ V . Since there are (m) only ﬁnitely many coordinates, this implies that x → x in the supremum norm k · k∞. Lemma 4.1.2. Let V be a Hausdorﬀ LCTVS and let W ⊆ V be a linear subspace with the subspace topology. If W is complete, then W is closed.

Proof. If x ∈ W , then for any open lattice Λ ⊆ V , there is an xΛ ∈ W ∩ (x + Λ). The set of 0 0 open lattices form a directed set, with Λ ≥ Λ if Λ ⊆ Λ , so the family (xΛ) is a net. In fact,

it is a Cauchy net: whenever Λ1, Λ2 ≥ Λ, then xΛ1 − xΛ2 ∈ Λ. As W is complete, there is a y ∈ W with xn → y. However, since xn → x as well, the fact that V is Hausdorﬀ implies that x = y. Lemma 4.1.3. If W ⊆ V is a closed subspace and V is complete, then so is W .

Proof. Left as an exercise. L Q Lemma 4.1.4. If (Vi)i∈I is a collection of complete LCTVS’s, then both i∈I Vi and i∈I Vi are complete.

Q µ Q Proof. • i∈I Vi is complete: Suppose that (x ), µ ∈ U is a Cauchy net in i Vi. Then µ µ (by e.g. continuity of the projection maps), xi is a Cauchy net for all i, so xi → xi µ for some xi ∈ Vi. Letting x = (xi)i∈I ∈ V , we claim that x → x: Let J be a finite subset of I, and let Λj ⊆ Vj be an open lattice for each j ∈ J. Then Q Q Q a basis of open lattices in i Vi consists of those of the form Λ = j∈J Λj × i∈I−J Vi, so we must show that for any such Λ, xµ − x ∈ Λ for µ 0. This says exactly that µ xj − xj ∈ Λj for µ 0, independently of j. This is true because J is finite. L µ µ • i∈I Vi is complete: Let (x )µ∈U be a Cauchy net. Each xi is a Cauchy net in Vi, so there is a limit xi ∈ Vi for each i. We claim that x = (xi)i∈I is in ⊕Vi and that xµ → x. Indeed, suppose that there is an infinite sequence i1, i2, i3,... of distinct indices such that xi 6= 0 for all n. Thus, there is some open lattice Λi ⊆ Vi with xi 6∈ Λi n L L n L n n n for each n. Consider the open lattice Λ = n Λin ⊕ j6=i Vj ⊆ Vi. n L Suppose that Λi ⊆ Vi is an open lattice for all i. A basis of open lattices in Vi L i is given by those of the form Λ = i Λi. µ L µ ν µ ν If (x ) is a Cauchy net in Vi, then for µ, ν Λ 0, x − x ∈ Λ, i.e. xi − xi ∈ Λi for all i. Corollary 4.1.5. Small (inverse) limits of complete LCTVS’s are again complete. Proof. This follows since lim V ⊆ Q V is a closed subspace. ←− i i i

Lemma 4.1.6. If (Vi, k · ki) are complete normed spaces, then the convergent products ⊥i(Vi, k · ki) and ⊥i,0(Vi, k · ki) are complete.

Proof. The argument is similar to the above. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 15

Given any LCTVS V , we define its completion V = lim V/Λ b −→ Λ⊆V with Λ ranging over open lattices. The inverse limit is taken in the category of O-modules, but × it has the structure of a vector space over L: for a ∈ L , we have a(xΛ + Λ) = (axa−1Λ + Λ). For any open lattice Λ, we let Λb be the kernel of Vb → V/Λ. This is an open lattice in Vb. The following properties hold: • The set of Λb for Λ ranging through all open lattices generates the inverse limit topology. • The natural map V/b Λb → V/Λ is an isomorphism. • The map V → Vb sending x to (x + Λ) is continuous. µ µ • Vb is complete: if x = (xΛ + Λ) is a Cauchy net in Vb, then for µ, ν Λ 0, we have µ ν µ xΛ − xΛ ∈ Λ. Thus, (xΛ + Λ) is a constant element of V/Λ for µ 0: denote this µ value by xΛ + Λ. Then if we define x = (xΛ + Λ), we have x → x. Lemma 4.1.7. If f : V → W is a continuous linear map, and if W is complete, then f factors uniquely through a continuous linear map fb: Vb → W . Proof. Let (x + Λ) ∈ V . Then we define f(x + Λ) = lim f(x ). To see that this makes Λ b b Λ ←−n n sense, we must see that (f(xn)) ∈ W is a Cauchy net. Indeed, given any open lattice M ⊆ W , −1 there is some open lattice Λ ⊆ f M. If Λ1, Λ2 ⊆ Λ, then xΛ1 − xΛ2 ∈ Λ and therefore

f(xΛ1 ) − f(xΛ2 ) ∈ M. The claims about continuity, uniqueness, etc. are immediate. 4.2. Banach and Fréchet spaces. Definition 4.2.1. • A complete normable space is called a Banach space. • A complete metrizable space is called a Fréchetspace. Lemma 4.2.2. (1) A countable (inverse) limit of Fréchetspaces is a Fréchetspace. (2) The convergent products ⊥ or ⊥0 of normed Banach spaces is a Banach space. (3) The quotient of a Banach (resp. Fréchet)space by a closed subspace is Banach (resp. Fréchet). Proof. (1) We have shown that a limit of complete spaces is complete. Then it is metrizable because the norms on the limit can be defined by pulling back norms from the individual pieces. Details left as an exercise. (2) Again completeness follows from previous work, and the topology is defined by one norm. Details left as an exercise. (3) Let Λ1 ⊇ Λ2 ⊇ Λ3 ⊇ · · · be a countable sequence of open lattices generating the topology on V . Their images in V/W generate the quotient topology, so V/W is metrizable. Thus, we can check completeness by looking at Cauchy sequences. Suppose that xn + W is a Cauchy sequence in V/W , i.e. for any i, for n, m i 0, xn − xm + W ⊆ Λi + W . By passing to a subsequence, we may assume without loss of generality that xn+1 − xn ∈ Λn + W . Choose yn ∈ Λn with xn+1 − xn ∈ yn+1 + W . 0 0 0 We define x1 = x1, x2 = x1 + y2, and so on: we have xn = x1 + y2 + y3 + ··· + yn. 0 0 By construction, we have xn + W = xn + W for all n. Also, (xn) is a Cauchy 16 COURSE BY RICHARD TAYLOR

0 0 sequence in V , as xn+1 − xn = yn+1 ∈ Λn. Thus, it has some limit x ∈ V , and 0 (xn + W ) = (xn + W ) → x + W in V/W . Lemma 4.2.3. If V is a Fréchetspace and M ⊆ V is a closed lattice, then M is open. −n Proof. Since M is a lattice, we have V = ∪n∈Zπ M. Suppose that M is not open. Then M contains no open set, i.e. M is nowhere dense. The same is true for each π−nM (as multiplication by π is an isomorphism). Since V is a complete metric space, the Baire category theorem implies that it cannot be a countable union of closed nowhere dense sets, so we have a contradiction. This property is often useful, so we define more generally: Definition 4.2.4. A LCTVS V is called barrelled if any closed lattice in V is also open. Lemma 4.2.5. Suppose that V,W are Fréchetspaces and that f : V → W is a linear function, not assumed to be continuous. (1) (The closed graph theorem): f is continuous if and only if Γ(f) := {(x, fx) ∈ V ⊕ W : x ∈ V } ⊆ V ⊕ W is a closed subset. (2) If f is a continuous bijection, then f is a homeomorphism. (3) (The open mapping theorem) If f is continuous and surjective, then f is open. Proof. The proof of the closed graph theorem (1) is omitted: see [6] or adapt the well-known proof from functional analysis over R. To deduce part (2), apply the closed graph theorem to f −1: continuity of f means that Γ(f) is closed in V ⊕ W , and Γ(f −1) maps bijectively to Γ(f) under the isomorphism W ⊕ V → V ⊕ W . To deduce the open mapping theorem (3), apply (2) to the induced map V/kerf → W , which must therefore be a homeomorphism. As f is the composition of this induced map with the open map V → V/kerf, we see that f is open.

Lemma 4.2.6. If (Vi, k · ki) are normed spaces, (W, k · k) is a Banach space, fi : Vi → W is continuous linear for all i, and if there exists some C ∈ R>0 such that kfi(x)k ≤ Ckxki for all i and for all x ∈ Vi, then there exists a unique continuous linear map f : ⊥i,0(Vi, k·ki) → (W, k·k) whose restriction to each Vi is fi. P Proof. f is deﬁned by sending (xi) to i fi(xi). Since kxiki → 0, kfi(xi)k ≤ Ckxiki → 0, so this sum converges. We have the following strong structure theorem for Banach spaces over L: Lemma 4.2.7. Let V be a Banach space. Then there is some index set I such that V ' ⊥i∈I,0(L, | · |) as topological vector spaces.

If xi ∈ V maps to ei ∈ ⊥i∈I,0(L, | · |) under some such isomorphism, we say that xi is a Banach basis of V . Note that kxik is bounded. Any element y ∈ V can be written uniquely P as i aixi where ai ∈ L, and |ai| → 0 ’as i → ∞’, i.e. for any > 0, |ai| < for all but ﬁnitely many i. Proof. Let k · k0 be a norm on V which deﬁnes the topology. Set kxk = min γ γ∈|L|={0}∪(#F)Z γ≥kxk0 p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 17

Note that kxk0 ≤ kxk ≤ (#F)kxk0. k · k is a norm which is equivalent to k · k0, so we may assume without loss of generality that kV k = |L|. Let Λ = Λ(k · k) be the corresponding unit ball, and consider Λ = Λ/πΛ. This vector space has a basis xi over F: choose some xi ∈ Λ with kxik = 1 mapping to xi. By the above lemma, there is a well-deﬁned continuous linear map ⊥i∈I,0(L, | · |) → V by sending ei to xi, and we claim that this is an isometry:

Consider (ai) ∈ ⊥i,0(L, | · |): we have k(ai)k = maxi |ai| = |ai0 | for some i0. Then we have X X ai aixi = ai0 xi =: ai0 α. ai0 ai P ai We have ∈ O, so α = xi ∈ Λ − πΛ, as xi is a basis for Λ/πΛ. Since Λ is the ai0 ai0 unit ball for k · k and k · k is valued in |L×| = {0} ∪ |π|Z, we thus have kαk = 1, so P k i aixik = |ai0 | = k(ai)k. Now, it suffices to show that ⊥i,0(L, | · |) has dense image inside of V : since it is also complete, this implies the image is equal to V . Let y0 ∈ V be arbitrary. We can write y0 = ay 0 with a ∈ L and |a| = ky k, so y ∈ Λ − πΛ. Then, since the xi span Λ/πΛ, we may find 0 P 0 0 some x = i ai xi with y = x + πy1 for some y1 ∈ Λ. Continuing in this manner, we can n P n Pn j j n+1 find x = i ai xi for each n with y = j=0 π x + π yn+1 and yn+1 ∈ Λ. Thus, we have Pn j j 0 k j=0 π x − yk → 0, so y and therefore y is in the closure of the image of ⊥i,0(L, | · |). Note that a given norm k · k on V might not exactly correspond to the supremum norm k(ai)i∈I k∞ = maxi |ai| under the isomorphism above; however, the two norms will be equivalent. (Indeed, the equivalent norm k · k0 defined in the above proof will coincide with the supremum norm).

5. 1/21/20: Structure and examples of Banach spaces 5.1. Structure of Banach spaces. We saw last time that any Banach space V has a Banach basis {xi}i∈I , i.e. a set of elements of V with kxik bounded such that any x ∈ V P may be written uniquely as x = i aixi. Any such set deﬁnes a topological isomorphism ⊥I,0(L, | · |) → V .

Lemma 5.1.1. If {xi}i∈I and {yj}j∈J are Banach bases of V , then the cardinality of I is equal to the cardinality of J. Proof. If I is finite, then V has finite dimension #I and therefore #J = #I. Thus we may assume that both I and J are infinite. P ∞ For i ∈ I, we write xi = j∈J ai,jyj. For each i, let Ji = ∪n=1{j ∈ J : |aij| ≥ 1/n}. This is a countable subset of J such that aij = 0 for j 6∈ Ji. Now, for all i ∈ I, xi is in the closed

subset ⊥j∈∪iJi,0(L, | · |) ⊆ ⊥J,0(L, | · |) = V . Since {xi} is dense in V , this inclusion must be an equality. Thus J = ∪i∈I Ji, and therefore #J ≤ #I · ℵ0 = #I. Lemma 5.1.2. If V is a Banach space, then the cardinality of any Banach basis is countably infinite if and only if V contains a dense subspace of countably infinite dimension. L Proof. Write V = ⊥I,0(L, | · |). Then if #I is countably infinite, i∈I L ⊆ V is a dense subspace of countably infinite dimension. Conversely, let V0 ⊆ V be a dense subspace with countable basis x1, x2,.... Then we have P xn = i∈I aniei with ani → 0. As above, the set In = {i ∈ I : ani 6= 0} is countable, and V0 18 COURSE BY RICHARD TAYLOR

is contained in the closed subspace ⊥∪nIn,0(L, | · |) ⊆ ⊥I,0(L, | · |) = V . Since V0 is dense, this is an equality and thus I = ∪nIn is a countable set. Lemma 5.1.3. If V is a Banach space and W ⊆ V is a closed subspace, then V ' W ⊕V/W . Note: Here, as everywhere in this course, ‘subspace’ means “linear subspace with the subspace topology”. This is a stronger condition than being the image of a continuous injection.

Proof. Since V/W is Banach, we have an isomorphism V/W ' ⊥i∈I,0(L, | · |) corresponding to a Banach basis {xi}. Let π : V → V/W be the quotient map. We may choose xei ∈ V with kx k bounded such that π(x ) = x (as kx k = inf −1 kyk is bounded). ei ei i i y∈π ({xi}) Then, we define a map s: V/W ' ⊥i∈I,0(L, | · |) by sending xi to xei; we have π ◦ s = idV/W . Now, sending (y, z) to y + s(z) defines a continuous homomorphism W ⊕ V/W → V with continuous inverse v 7→ (v − sπv, πv). 5.2. Examples of Banach spaces. Many important examples of Banach (and Fréchet, locally convex, etc.) spaces are realized as various spaces of functions. Example 5.2.1. Let Ω be a compact topological space, and define C(Ω,L) to be the L- algebra of continuous functions from Ω to L. This admits the supremum norm kfk =

supx∈Ω(|f(x)|) = maxx∈Ω(|f(x)|), which is well-defined due to compactness of Ω. We may easily see that it is a norm (e.g. deducing the ultrametric inequality from the pointwise ultrametric inequality |(f + g)(x)| ≤ max{|f(x)|, |g(x)|}). We claim that C(Ω,L), topologized with this norm, is complete, and therefore a Banach space. Indeed, let (fi) be a Cauchy net. Then for all > 0, for all i, j 0, kfi − fjk < . This says exactly that |fi(x) − fj(x)| < for all x ∈ Ω, so we may define f(x) = limi fi(x). We need to check that fi → f in C(Ω,L). Given x, , we can choose j x, such that |fj(x) − f(x)| < . But we also have |fi(x) − fj(x)| < for i, j 0, not depending on x. Then by letting j be sufficiently large, we conclude that for all > 0, we can find i 0 such that for all x ∈ Ω, |fi(x) − f(x)| < , as desired. −1 n Finally, we need to check that f is continuous, that is, we need to show that f (a + π OL) n is open for all a ∈ L and n ∈ Z. For i 0, for all x, |fi(x) − f(x)| < |π| . This implies n n −1 n that fi(x) ∈ a + π OL if and only if f(x) ∈ a + π OL. Hence f (a + π OL) is equal to −1 n fi (a + π OL), which is open. Therefore f is continuous.

Consider the compact topological space Zp. Then we have: x x(x−1)(x−2)···(x−n+1) Lemma 5.2.2 (Mahler). The functions n = n for n = 0, 1, 2,... is a Banach basis for C(Zp,L). x Proof. As they are polynomials in x, the n are certainly continuous. For x ∈ Z>0, we know x x x that n ∈ Z ⊆ Zp. Thus by continuity, n ∈ Zp for all x ∈ Zp, i.e. k n k ≤ 1. Now, there x exists a unique continuous map ⊥Z≥0,0(L, | · |) → C(Zp,L) sending en to n . We will see that it is an isomorphism. Consider the “linear diﬀerence” map D : C(Zp,L) → C(Zp,L) given by D(f)(x) = f(x + 1) − f(x). We have kD(f)k ≤ kfk, so D is continuous. By an elementary induction, we see that n X n (Dnf)(x) = f(x + n − i)(−1)i i i=0 p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 19

We claim that kDnfk → 0 as n → ∞. To see this, apply the above formula with n = pk for some k: k p −1 k k X p k (Dp f)(x) = f(x + pk) − f(x) + f(x + pk − i)(−1)i + ((−1)p + 1)f(x) i i=1 Since pk → 0 as k → ∞ and f is continuous, we see that f(x + pk) − f(x) → 0 as k → ∞. This is uniform in x by compactness of Zp (and Dini’s theorem). In addition, note that the pk pk binomial coeﬃcients i are divisible by p. Also, the number ((−1) + 1) (which is 0 unless p = 2, in which case it is 2) is divisible by p. Thus, for k large, we have kDpk fk ≤ |p|kfk < kfk Since kDgk ≤ kgk for any g, the above inequality is true for any m ≥ pk as well. Thus, we see that kDmfk → 0 as m → ∞. n Now, we may deﬁne a map C(Zp,L) → ⊥Z≥0,0(L, | · |) by sending f to (D f)n(0). Note nx mx that (D n )(0) = 1 and (D n )(0) = 0 for m 6= n, so the composite map ⊥Z≥0,0(L, | · |) →

⊥Z≥0,0(L, | · |) is the identity and therefore C(Zp,L) → ⊥Z>0,0(L, | · |) is surjective. n It is also injective, since (D f)(0) = 0 implies f(n) = 0 for all n ∈ Z≥0 inductively, because n Pn in (D f)(0) = i=0 f(x + n − i)(−1) i ). Then by continuity, f = 0 identically. Since a bijective continuous map between Banach spaces is an isomorphism, we get the desired result.

Fix some r ∈ R>0. We define the closed unit disc of radius r in L as D(r) = {t ∈ L: |t| ≤ r}. The ring of analytic functions on this disc is defined as ∞ X n n O(D(r)) = {f = cnT ∈ L[[T ]]: |cn|r → 0 as n → ∞}. n=0 P n Note that for t ∈ D(r) and f ∈ O(D(r)), the sum f(t) = cnt converges in L(t) ⊆ L. We P n define a norm on O(D(r)) for f = n cnT ∈ O(D(r)) by n kfkr = max |cn|r . n∈Z≥0 We may easily check that this is a norm (e.g. deducing the ultrametric inequality from the ultrametric inequality |cn + dn| ≤ max(|cn|, |dn|)). We claim that O(D(r)) is a Banach space. P n Indeed, let (fi) be a Cauchy net. Writing fi = n fi,nT , we see that (fi,n) is a Cauchy net P n for each n, so there is some fn ∈ L with fi,n → fn for each n. Now define f(T ) = n fnT . We must check that f = limi fi and that f ∈ O(D(r)). −n Indeed, given > 0, for all i, j 0, we have |fi,n − fj,n| < r uniformly in n. Also, for −n any fixed n, |fj,n − fn| < r for j n, 0. So by taking j large enough in the uniform-in-n −n −n bound |fi,n − fj,n| < r , we see that |fi,n − fn| < r for i 0 uniformly in n. Thus −n f = limi fi. Moreover, as for any fixed i, |fi,n| < r for n ,i, taking i large enough we −n see that |fn| < r for n 0. We can see immediately that O(D(1)) = LhT i has Banach basis 1,T,T 2,.... More generally, we calculate that a Banach basis for O(D(r)) is given by 1, πm(1)T, πm(2)T 2,... l log r m with m(n) = n log |π| . There is another important characterization of the norm k · kr given above: 20 COURSE BY RICHARD TAYLOR

Q Lemma 5.2.3. If f ∈ O(D(r)) then kfkr = supt∈D(r) |f(t)|. If moreover r ∈ p , then kfkr = maxt∈D(r) |f(t)|. Note that |f(t)| makes sense for f(t) ∈ L, as | · | on L has a unique extension to an absolute value on L. Proof. We immediately deduce from the ultrametric inequality that

X n n |f(t)| = cnt ≤ max |cn|r = kfkr. n n To see the converse, we ﬁrst assume that r ∈ pQ. This assumption lets us reduce to the case r = 1: we replace L by a ﬁnite extension L0/L such that there is some α ∈ L0 with |α| = r, and replace f(T ) by g(T ) = f(αT ). By construction, we see that g ∈ O(D(1)), kgk1 = kfkr, and supt∈D(1) |g(t)| = sups∈D(r) |f(s)|. Also, by dividing by an appropriate element of L, we may assume that kfk1 = 1, i.e. that f ∈ O T . Thus, we may consider the mod-π reduction f ∈ F[T ]. Since kfk1 = 1, we see J K that f 6= 0. Thus, there exists some α ∈ F with f(α) 6= 0. Lifting α to α ∈ OL, we see that f(α) ∈ ×, i.e. that |f(α)| = 1. OL Q Q For r 6∈ p , note that, as p is dense in R>0, kfkr = sup r0

kfkr = sup sup |f(t)| = sup |f(t)| = sup |f(t)| r0

• B = {{b}: b ∈ V }. Then the topology LB(V,W ) is called the weak topology. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 21

• B = {B ⊆ V : B is bounded}. Then the topology LB(V,W ) is called the strong topology.

6. 1/23/20: Properties of dual spaces

Let V,W be LCTVSs. Last time, we deﬁned the LCTVS LB(V,W ), where B is a collection of bounded subsets of V , as the space of linear functionals from V to W where, for each

B ∈ B and continuous seminorm k · k on W , we include the seminorm kfkB = supx∈B kf(x)k. A basis of neighborhoods of the identity is Λ(B,M) where B ∈ B and M ⊂ W is an open lattice. The two most important examples (though others do appear sometimes) are when B is the collection of singleton points, giving us Ls(V,W ), the “weak topology” on L (V,W ); and when B is the collection of all bounded subsets, giving us Lb(V,W ), the “strong topology” on L (V,W ). ∨ ∨ ∨ We write V = L (V,L). To specify the topology, we will write Vs ,Vb . 6.1. Some preliminary observations.

0 id 0 Lemma 6.1.1. If B ⊂ B then LB0 (V,W ) → LB(V,W ) is continuous, that is, the B topology is ﬁner than the B topology. In particular, the strong topology is stronger than the weak topology.

Lemma 6.1.2. If W is Hausdorff and ∪B∈BB is dense in V , then LB(V,W ) is Hausdorff (so Ls, Lb are Hausdorff).

Proof. Let 0 6= f ∈ L (V,W ). Since ∪B∈BB is dense, there must be some b ∈ B ∈ B with f(b) 6= 0. Since W is Hausdorﬀ, there exists some open lattice M ⊆ W with f(b) 6∈ M. Therefore, f is not contained in the open lattice Λ(B,M). Lemma 6.1.3. If f : V → W is continuous linear, deﬁne the dual map f ∨ : W ∨ → V ∨ by g 7→ g ◦ f. ∨ ∨ ∨ If B is a collection of bounded subsets of V , then f : WfB → VB is continuous. In ∨ ∨ ∨ ∨ ∨ ∨ particular, f : Ws → Vs and f : Wb → Vb are continuous, since for example the latter factors as ∨ ∨ ∨ Wb → W{fB|B⊂V bounded} → Vb . Proof. If B ⊂ V is bounded then (f ∨)−1Λ(B, O) = {g ∈ W ∨ | (g ◦ f)(B) ⊂ O} = Λ(f(B), O). 6.2. The Hahn-Banach theorem and some consequences.

Theorem 6.2.1 (Hahn-Banach). Suppose W0 ⊂ V0 are vector spaces and that k · k is a ˜ seminorm on V0. If f : W0 → L and |f| is bounded by k · k on W0, then there is f : V0 → L ˜ ˜ such that |f|W0 = f, |f| ≤ k · k.

Proof. Consider the poset of pairs (U, fU ) where W0 ⊂ U ⊂ V0, fU : U → L satisﬁes 0 0 0 fU |W0 = f, and |fU | ≤ k · k on U, and we say that (U, fU ) ≥ (U , fU ) if U ⊂ U and fU |U 0 = fU 0 . By Zorn’s lemma, there is a maximal (U, fU ), so WLOG (W0, f) is maximal. We need to prove that W0 = V . 22 COURSE BY RICHARD TAYLOR

Suppose not. Let x ∈ V0 \ W0. Let U = W0 + Lx. To get a contradiction, we need to ﬁnd b ∈ L such that for all y ∈ W0 and a ∈ L, |f(y) + ab| ≤ ky + axk.

Actually it suffices to find b such that |f(y) + b| ≤ ky + xk for all y ∈ W0, since this implies the above. So for y ∈ W0, let B(y) = {b ∈ L | |f(y) + b| ≤ ky + xk}. This contains −f(y), so is nonempty, and is closed and compact. It suffices to show that any finite intersection of B(y)s is nonempty, since then by compactness we would also have

∩y∈W0 B(y) 6= ∅. In fact we claim that B(y1) ∩ B(y2) = B(y1) if kx + y1k ≤ kx + y2k, because

B(y1) ∩ B(y2) = {b ∈ L | |f(y1) + b| ≤ ky1 + xk, |f(y2) + b| ≤ ky2 + xk}

and if b ∈ B(y1), then

|f(y2) + b| ≤ max(|f(y1) − f(y2)|, |f(y1) − b|)

≤ max(ky1 − y2k, ky1 + xk)

≤ max(ky1 + xk, ky2 + xk, ky1 + xk) = ky2 + xk.

So we are done. Corollary 6.2.2. If V is a Hausdorff LCTVS and 0 6= x ∈ V , there is f ∈ V ∨ with f(x) = 1. Proof. There is a continuous seminorm k · k on V with kxk ≥ 1. Let f : Lx → L, x 7→ 1. ˜ ˜ ˜ Then there is f : V → L with f(x) = 1 and |f| ≤ k · k. Then f is also continuous. Corollary 6.2.3. If W ⊂ V is a subspace of a LCTVS (with the subspace topology) and ∨ ˜ ∨ ˜ f ∈ W , then there is f ∈ V such that f|W = f. Proof. Since f is continuous, |f| is bounded by some seminorm k · k. Then f has an extension ˜ ˜ f bounded by k · k. So f is also continuous. Corollary 6.2.4. If M ⊂ V is a closed O-submodule and x ∈ V \ M, then there is f ∈ V ∨ with fM ⊂ O and f(x) ∈/ O. Proof. There is an open lattice Λ ⊂ V with (x + Λ) ∩ M = ∅, that is, x∈ / Λ + M. So WLOG −1 M is an open lattice. Let f : Lx → L, x 7→ π . Then |f| ≤ k · kM on Lx, so we can find an ˜ ˜ −1 ˜ ˜ extension f : V → L such that f(x) = π and |f| ≤ k · kM , so that |f| ≤ 1 on M. Lemma 6.2.5. If V is a Hausdorff LCTVS and W ⊂ V is finite dimensional, then V ∼= W ⊕ V/W .

∨ Proof. Let x1, . . . , xn be a basis of W . Let f1, . . . , fn be the dual basis of W (note that since W is ﬁnite-dimensional, the continuous dual and linear dual are the same thing). Find ˜ ∨ fi ∈ V extending fi. Let F : V → W (⊂ V ) X ˜ x 7→ fi(x)xi. i p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 23

2 This is continuous and satisﬁes F |W = idW and F = F . So we have an isomorphism ker F −→∼ V/W . Then we have maps V ∼= W ⊕ ker F x 7→ (F x, x − F x) x + y ← (x, y) [ which are mutually inverse, hence isomorphisms. 6.3. Double duals.

∨ ∨ Lemma 6.3.1. If V is a Hausdorﬀ LCTVS then there is a continuous bijection δ : V → (Vs )s given by δ(x)(f) = f(x).

∨ ∨ ∨ ∨ We deﬁne Vs := (Vs )s , that is, V with the topology of (Vs )s . A generating set of open lattices is given by f −1O for f ∈ V ∨; such sets are always open in V , so this topology is weaker than the given topology on V . Thus, Vs called the weak topology (or weak*-topology) on V . ∨ Proof. First, we see that δ(x) is continuous on Vs , since δ(x)−1(O) = {f ∈ V ∨ | f(x) ∈ O} = Λ({x}, O) ∨ is open in Vs . We also see that δ itself is continuous, since δ−1Λ({f}, O) = {x ∈ V | f(x) ∈ O} = f −1O is open. Next, δ is injective because if 0 6= x ∈ V , there is f ∈ V ∨ with f(x) 6= 0 by Hahn-Banach, so δ(x)(f) 6= 0. ∨ ∨ Finally, we check that δ is surjective. Let d ∈ (Vs ) . By deﬁnition it is a continuous ∨ function d : Vs → L. We have −1 d O ⊃ Λ({x1, . . . , xn}, O)

for some x1, . . . , xn ∈ V . WLOG x1, . . . , xn are linearly independent. Let W = Lx1⊕· · ·⊕Lxn. ∼ ∨ ∼ ∨ ∨ ∨ ∨ By Lemma 6.2.5, we have V = W ⊕U, hence V = W ⊕U , where U = {f ∈ V | f|W = 0}. ∨ Again give W the basis f1, . . . , fn where fi(xj) = δij. m Multiplying the previous condition on the xis by π , we have m m π O ⊃ dΛ({x1, . . . , xn}, π O) for all m, hence ! X m ∨ m d π Ofi + U ⊂ π O i ∨ m ∨ ∨ for all m, hence dU ⊂ π O for all m, hence dU = {0}. But for any f ∈ Vs , we have P ∨ f − i f(xi)fi ∈ U , so X X d(f) = f(xi)d(fi) = f d(fi)xi i P for all f. This can be rewritten as d = δ ( i d(fi)xi), so d has a preimage, as desired. ∨ ∨ Corollary 6.3.2. V → (Vb )s has dense image (and likewise if we replace the strong topology with any topology which reﬁnes the weak topology). . 24 COURSE BY RICHARD TAYLOR

∨ ∨ Proof. Suppose not. Then by Hahn-Banach we can ﬁnd a continuous 0 6= f :(Vb )s → L ∨ ∨ ∨ such that f is zero on the closure of the image of V , that is, a 0 6= f ∈ ((Vb )s )s that is zero ∨ ∨ ∨ ∨ ∨ on the image of V . But ((Vb )s )s = Vb , so we have 0 6= f ∈ Vb such that f(V ) = {0}, which is a contradiction. ∨ ∨ Lemma 6.3.3. If V is Hausdorﬀ and bornological, δ : V → (Vb )b gives a topological ∨ ∨ isomorphism from V to its image. If furthermore V is complete then δ(V ) is closed in (Vb )b . ∨ ∨ Note that if V is Banach and dim V = ∞, V → (Vb )b is not surjective. (However, there are interesting cases where it is.)

6.4. Duals of direct sums and products.

L ∨ ∼ Q ∨ L ∨ ∼ Q ∨ Lemma 6.4.1. (1) ( i Vi)s −→ i(Vi)s and ( i Vi)b −→ i(Vi)b . L ∨ ∼ Q ∨ L ∨ ∼ Q ∨ (2) i(Vi)s −→ ( i Vi)s and i(Vi)b −→ ( i Vi)b . Proof. (1) The fact that these are bijections comes from the universal property of ⊕. To get that they are homeomorphisms, first reduce to the case that Vi is Hausdorff for all i by letting Wi ⊂ Vi be the closure of {0} and noting that if the statement is true when everything is Hausdorff, then

∨ ∨ ∨ M ∼ M M ∼ M ∼ Y ∨ ∼ Y ∨ Vi = Vi/ Wi = Vi/Wi −→ (Vi/Wi) = Vi .

Now assuming everything is Hausdorﬀ, it is easy to check that the natural map is L continuous. We now check that the map is open. If B ⊂ Vi is bounded, pri B is bounded for all i and {0} for almost all i. So ( ) X Y Λ(B, O) = (fi) | ∀b ∈ B, fi(bi) ∈ O ⊃ Λ(pri B, O). i i ∨ Q Since Λ(pri B, O) is Vi for almost all i, i Λ(pri B, O) is open, and thus Λ(B, O) is too. (2) The map in the opposite direction is !∨ Y M ∨ Vi → Vi i

f 7→ f|Vi . L ∨ To check that the image really lands in Vi , note that there is some ﬁnite I0 such that Y Y −1 Λi × Vi ⊂ f O,

i∈I0 i/∈I0 Q Q so f i/∈I Vi ⊆ O. As i6∈I Vi is an L-subspace of the product, this implies that Q 0 0 f V ⊆ ∩ × a = 0. i6∈I0 i a∈L O This map has the desired properties (composites with the map in the other direction are the identity, and both are continuous) by a similar argument as before. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 25

7. 1/28/20: More on dual spaces ∨ ∨ Last time we defined V = L (V,L), or specifically VB where B is a collection of bounded ∨ ∨ subsets of V . We defined Vs and Vb , and found that ∨ M Y ∨ Vi = Vi,s/b s/b ∨ Y M ∨ Vi = Vi,s/b. s/b ∨ ∨ Also, we saw that if V is Hausdorff, V → (Vs )s is a continuous bijection. 7.1. Dualizing norms. Lemma 7.1.1. Suppose V is a normable LCTVS with norm k · k. Then |f(x)| kfk∨ = sup x∈V kxk x6=0 ∨ ∨ defines a norm on V which generates the strong topology Vb . Similarly, if W is normed, one can define a norm on L (V,W ) that gives the strong topology Lb(V,W ). Proof. First note that the supremum exists because |f| is a continuous seminorm on V . For the purpose of the proof, define a new seminorm kfk0 = sup |f(x)|. v∈V kxk≤1 We can see that this is equivalent to the one in the lemma as follows. Since we can scale any element x by an element of L× to lie in the annulus |π| < kxk ≤ 1, and scaling x by L× does |f(x)| not change the quantity kxk , we have |f(x)| kfk∨ = sup . x∈V kxk |π|

Λ(B, O) ⊃ Λ(aΛ(k · k), O) = a−1Λ(Λk · k, O) = a−1Λ(k · k0).

Lemma 7.1.2. ∨ ∼ ∨ ∨ (⊥0(Vi, k · ki))b = ⊥ Vi,b, k · ki .

Proof. The desired isomorphism is θ : f 7→ (f|Vi ). Clearly θ is linear. We have

∨ |f(x)| ∨ kf|Vi ki = sup ≤ (kfk∞) x∈Vi kxki |π|≤kxk≤1

∨ for all i, so kθ(f)k∞ ≤ kfk∞. On the other hand

∨ −1 −1 kfk∞ ≤ |π| sup |f(x)| ≤ |π| sup sup |f(xi)| kxk∞≤1 i kxiki≤1

−1 f(xi) −1 ∨ −1 ≤ |π| sup sup = |π| sup kf|Vi ki = |π| kθ(f)k∞. i |π|≤kxiki≤1 kxiki i

∨ In conclusion the seminorms kθ(·)k∞ and k · k∞ are equivalent, so θ is injective and a homeomorphism onto its image. ∨ It remains to show that θ is surjective. If (fi) is an element of the RHS with kfik ≤ C Pi for all i, and (xi) ∈ ⊥0(Vi, k · ki), then |fi(xi)| ≤ Ckxiki goes to 0 as i → ∞, so fi(xi) P i converges. So we may deﬁne f(x) = fi(xi). Then f is linear, f|Vi = fi, and f is continuous because

|f(x)| ≤ sup |fi(xi)| ≤ C sup kxiki = Ck(xi)k∞. i i Note that we can similarly show that if W is Banach, then

∨ ∼ ⊥i(Lb(Vi,W ), k · ki ) = Lb(⊥0(Vi, k · ki),W ). Lemma 7.1.3. We have an injection

∨ ∨ ∨ ⊥0(Vi,b, k · ki ) ,→ (⊥(Vi, k · ki))b and the image is closed (but not in general surjective).

Proof. Similar to before. The map is X (fi) 7→ (xi) 7→ fi(xi) . P One checks that fi(xi) converges and that this map is a homeomorphism onto its image. ∨ The image is closed because Vi,b is complete for all i, which we will prove in a moment. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 27

7.2. Stuﬀ about strong duals. Lemma 7.2.1. If V is Hausdorﬀ and bornological (e.g. metrizable), then

∨ ∨ δ : V → (Vb )b is a homeomorphism onto its image, and if V is complete then its image is closed.

∨ ∨ ∨ ∨ Proof. δ factors as V → (Vs )s ⊂ (Vb ) and the first map is a bijection, so it is well-defined ∨ and injective. To show that δ is continuous, we need to show that if B ⊂ Vb is bounded, then δ−1Λ(B, O) is open in V . We can rewrite \ δ−1Λ(B, O) = {x ∈ V | ∀f ∈ B, f(x) ∈ O} = f −1O. f∈B Since V is bornological, it suffices to show that if C ⊂ V is bounded, then there is a ∈ L× T −1 with aC ⊂ f∈B f O. But we can find b ∈ L such that bB ⊂ Λ(C, O), so \ \ f −1O ⊃ f −1O ⊃ bC. f∈B f∈Λ(C,b−1O) To show that δ is open, let Λ ⊂ V be an open lattice. It is enough to show that (1) Λ = δ−1Λ(Λ(Λ, O), O) ∨ and (2) Λ(Λ, O) ⊂ Vb is bounded. For (1), Λ ⊂ δ−1Λ(Λ(Λ, O), O) is clear, and for the other inclusion, if x ∈ δ−1Λ(Λ(Λ, O), O), then for all f ∈ Λ(Λ, O) we have f(x) ∈ O, so x ∈ Λ. (As in, if x were not in Λ, Hahn-Banach would give f ∈ V ∨ such that f(Λ) ⊂ O but f(x) ∈/ O.) For (2), if B ⊂ V is bounded, we need to show that there is a ∈ L× such that aΛ(Λ, O) ⊂ Λ(B, O). But if b ∈ L× is such that bB ⊂ Λ, then b−1Λ(B, O) = Λ(bB, O) ⊃ Λ(Λ, O). ∨ ∨ Lemma 7.2.2. If every closed bounded O-submodule of V is compact then V (Vb )b . Proof. We don’t have time for this. See Proposition 15.3 of Schneider.

Note that the above condition usually doesn’t apply to Banach spaces. If V = ⊥Z>0,0(L, |·|), ∨ ∼ then Vb = ⊥Z>0 (L, | · |) contains ⊥Z>0,0(L, | · |) as a proper closed subset. Then if we take a continuous

0 6= f : ⊥Z>0 (L, | · |)/⊥Z>0,0(L, | · |) → L, δ ∨ ∨ then f∈ / im(V → (Vb ) ), because f(ei) = 0 for all i, and if f = δ(x) then δ(x)(ei) = xi = 0 for all i so x = 0.

∨ ∨ Remark 7.2.3. If V is any inﬁnite-dimensional Banach space, then V → (Vb ) is not surjective. Indeed, choosing a countable Banach sub-basis, we may split V = W ⊕ W 0 with

W ' ⊥Z>0,0(L, | · |), and use the preceding example. 28 COURSE BY RICHARD TAYLOR

Example 7.2.4. Recall that we have ∼ C(Zp,L) =⊥Z≥0,0 (L, | · |) x via n 7→ en. From this we have a map ∨ ∼ A : C(Zp,L)b −→ O T ⊗O L ∨ J K P∞ n as follows. Let µ ∈ C(Zp,L)b . We want to write µ as a power series n=0 anT with an ∈ L, |an| bounded. We should have ∞ X x A(µ) = µ T n = µ ((1 + T )x) , n n=0 at least formally. If t ∈ L with |t| < 1 then we can honestly write A(µ)(t) = µ ((1 + t)x). ∨ r The strong topology on O T ⊂ C(Zp,L) is the p-adic topology, generated by π O T . J K ∨ J K The weak topology on O T ⊂ C(Zp,L) is the (π, T )-adic topology: for a sum of the form P x J K cn n , cn ∈ L, |cn| → 0, the associated open set ( N ) X x X n X Λ cn , O = anT ∈ O T | ancn ∈ O n J K n=1

(where N is such that cn ∈ O for n > N) contains nX n −1 o big anT ∈ O T | an ∈ cn O for n = 0,...,N ⊃ (π, T ) . J K 7.3. Stuff about completeness. When is LB(V,W ) complete? Definition 7.3.1. We call a subset H ⊂ L (V,W ) equicontinuous if for all open lattices M ⊂ W , there is an open lattice Λ ⊂ V with Λ ⊂ f −1M for all f ∈ H. Lemma 7.3.2 (Banach-Steinhaus). If V is a Fréchetspace (or, more generally, barrelled— that is, every closed lattice is open) then any bounded subset of Ls/b(V,W ) is equicontinuous. (Note that any bounded subset in Lb(V,W ) is bounded in Ls(V,W ), so we can drop the /b.) Proof. Let H be the bounded subset in question. Suppose M ⊂ W is an open lattice. Then T −1 f∈H f M is a closed O-submodule. Since V is barrelled, it is sufficient to prove that T −1 × f∈H f M is a lattice. If x ∈ V then H ⊂ aΛ({x},M) for some a ∈ L . That is, if f ∈ H −1 −1 −1 T −1 then f(a x) = a f(x) ∈ M, so a x ∈ f∈H f M.

Lemma 7.3.3. Suppose W is complete and B contains all singletons and H ⊂ LB(V,W ) is closed. Under either of the following conditions, H is complete. (1) V is bornological and B = b. (2) H is equicontinuous. ∨ Corollary 7.3.4. (1) If V is bornological then Vb is complete. (2) If V,W are Fr´echetthen Lb(V,W ) is Fr´echet. (3) if V is normable and W is Banach then Lb(V,W ) is Banach.

Proof of Lemma. Let (fi) be a Cauchy net in H. Then for all x ∈ V ,(fi(x)) is a Cauchy net in W . (Reason: if M ⊂ W is any open lattice then Λ({x},M) is an open lattice in LB(V,W ), so for i, j x,M 0, we have fi − fj ∈ Λ({x},M), meaning fi(x) − fj(x) ∈ M.) So fi(x) converges to a limit f(x) ∈ W , giving us a linear map f : V → W . It suﬃces to check that f is continuous; then we can conclude that fi → f in LB(V,W ). p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 29

To show this, let B ∈ B and M ⊂ W be an open lattice. Then for i, j B,M 0 we have (fi − fj)(B) ⊂ M, and for j x 0 we have fj(x) − f(x) ∈ M, so by choosing j large enough depending on x ∈ B, we see that for i B,M 0 we have (fi − f)(B) ⊂ M. We will ﬁnish this next time.

8. 1/30/20: Finishing dual spaces, compact maps, and tensor products 8.1. Finishing up dual spaces. We were in the middle of proving the following lemma.

Lemma 8.1.1. Suppose W is complete, B is a collection of bounded subsets of V containing all singletons, and H ⊂ LB(V,W ) is closed. Under either of the following conditions, H is complete. (1) V is bornological and B = b. (2) H is equicontinuous (e.g. if V is barrelled and H is bounded).

Proof. We took a Cauchy net (fi) in H, found that fi(x) converged to some f(x) ∈ W for all x ∈ V , and thus obtained a linear map f : V → W . It suffices to check that f is continuous: then if B ∈ B and M ⊂ W is an open lattice, we have (f − fi)(B) ⊂ M for i B,M 0, so (f − fi) ∈ Λ(B,M), so fi → f in LB(V,W ). (1) It suffices to prove that if B ⊂ V is bounded, then f(B) is bounded. Given M ⊂ W , × we need to find a ∈ L such that f(B) ⊂ aM. Choose i such that (f − fi)(B) ⊂ M. × We can find a ∈ L with |a| ≥ 1 and fi(B) ⊂ aM, so f(B) ⊂ M + aM = aM. Note that we really did need to use the strong topology for this argument to work: to check continuity of a map out of a bornological space, we need to verify that f(B) is bounded for all open sets B. (2) Given M ⊂ W an open lattice, there is Λ ⊂ V an open lattice with fi(Λ) ⊂ M for all i. For all x ∈ V we have (f − fi)(x) ∈ M for i 0. Since M is closed, we conclude that if x ∈ Λ then f(x) ∈ M, that is, f(Λ) ⊂ M.

∨ This implies, for example, that if V is bornological then Vb is complete, and if V,W are Fr´echet/Banach then Lb(V,W ) is Fr´echet/Banach.

∼ ∨ ∨ Lemma 8.1.2. If V is barrelled and Hausdorﬀ (e.g. Fr´echet),then V −→ (Vs )b .

∨ ∨ Proof. We have seen that δ : V → (Vs )b is a linear bijection. To see that it is open, let ∨ ∨ Λ ⊂ V be an open lattice. Then Λ(Λ, O) ⊂ Vs is bounded, since it is bounded in Vb . Then ∨ ∨ Λ(Λ(Λ, O), O) is an open lattice in (Vs )b . But we have seen that Λ(Λ(Λ, O), O) is precisely the image of Λ under δ. ∨ −1 To see that δ is continuous, suppose B ⊂ Vs is bounded. We need to show that δ Λ(B, O) is open in V . By deﬁnition this is \ {x ∈ V | ∀f ∈ B, f(x) ∈ O} = f −1O. f∈B

T −1 But the Banach-Steinhaus theorem tells us that B is equicontinuous, so f∈B f O is indeed open. 30 COURSE BY RICHARD TAYLOR

8.2. Compact and completely continuous maps. Now, we isolate a class of continuous linear maps which have particularly well-behaved spectral theory. Definition 8.2.1. We call f ∈ L (V,W ) compact if there is an open lattice Λ ⊂ V with fΛ compact in W . Let C (V,W ) = {f ∈ L (V,W ) | f compact}. This is an L-subspace of L (V,W ). Lemma 8.2.2. (1) If f : V → W is compact and B ⊂ V is bounded, then fB is compact. f g (2) If V −→ W −→ U are continuous and linear, U is Hausdorff, and either f or g is compact, then g ◦ f is compact. Proof. (1) Let Λ ∈ V be an open lattice such that fΛ is compact. Then B ⊂ aΛ for some a ∈ L×, so fB ⊂ af(Λ). (2) First suppose that f is compact, and let Λ ⊆ V be an open lattice such that fΛ is compact. Then gfΛ = gfΛ. Since U is Hausdorff and fΛ is compact, then gfΛ is compact and closed, so gfΛ is compact. Now suppose instead that g is compact. Let M ⊆ W be an open lattice such that gM is compact. By continuity of f, we may pick an open lattice Λ ⊆ V with fΛ ⊆ M. Thus gfΛ ⊆ gM, so it is compact. Note that if f ∈ L (V,W ) with W Hausdorff and im f is finite-dimensional, then f is compact. This is because if M ⊂ W is an open lattice, then M ∩ im f is bounded, and so if Λ ⊂ V is an open lattice with fΛ ⊂ M, then fΛ ⊂ M ∩ im f is bounded; so fΛ is compact.

Lemma 8.2.3. Let V be normed, W complete, and Λ0 ⊂ V the unit ball. Then (1) f ∈ L (V,W ) is compact if and only if for any open lattice M ⊂ W ,

#(fΛ0/M ∩ fΛ0) < ∞.

(2) C (V,W ) ⊂ Lb(V,W ) is closed.

Proof. (1) In this case f is compact if and only if fΛ0 is compact (as Λ0 is an open bounded lattice). If fΛ0 is compact, then for M ⊂ W an open lattice, fΛ0/fΛ0 ∩ M is also compact. But it is additionally discrete (since for any M ⊆ W an open lattice, the cosets of fΛ0 ∩ M in fΛ0 form a disjoint open cover), hence finite. In the other direction, assume the finiteness statement and consider the map fΛ → lim fΛ /fΛ ∩ M. 0 ←− 0 0 M This is a bijection because of the completeness of W : if we have a compatible sequence (xM +(fΛ0 ∩M)), then (xM ) is a Cauchy net, so has a limit x. It is a homeomorphism by definition of the open sets on both sides. Also we have

fΛ0/fΛ0 ∩ M = (fΛ0 + M)/M = (fΛ0 + M)/M = fΛ0/M ∩ (fΛ0).

We conclude that fΛ0 is proﬁnite, hence compact. (2) Let f ∈ L (V,W ) \ C (V,W ). Then there is M ⊂ W an open lattice such that

#(fΛ0/fΛ0 ∩ M) = #((fΛ0 + M)/M) = ∞. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 31

For any g ∈ Λ(Λ0,M), we have (f + g)Λ0 ⊂ fΛ0 + M, so

((f + g)Λ0 + M)/M = (fΛ0 + M)/M

is infinite, so g is not compact. That is, (f + Λ(Λ0,M)) ∩ C (V,W ) = ∅. Definition 8.2.4. We say that f ∈ L (V,W ) has finite rank if dim im f < ∞. Let CC (V,W ) be the closure of the space of finite rank elements of Lb(V,W ). We call elements of CC (V,W ) “completely continuous”. Lemma 8.2.5. If W is Hausdorff, C (V,W ) ⊂ CC (V,W ). Corollary 8.2.6. If W is complete and V is normed, then C (V,W ) contains all finite rank maps V → W and is closed, so C (V,W ) = CC (V,W ). Proof of Lemma 8.2.5. Let f ∈ C (V,W ). Given B ⊂ V bounded and M ⊂ W an open lattice, we need to find g ∈ L (V,W ) of finite rank with f − g ∈ Λ(B,M). Since fB is compact, (fB + M)/M = (fB + M)/M is finite. Choose e1, . . . , ed ∈ f(B) Pd such that i=1 Lei + M ⊃ f(B) + M. WLOG the eis are linearly independent mod M, and −1 Pd we can scale them so that ei ∈ π M − M for all i. Let W1 = i=1 Lei. ∨ Let f1, . . . , fd be the dual basis of W1 associated to e1, . . . , ed. Then |fi(x)| ≤ |π|kxkM for ˜ ∨ ˜ all i. By Hahn-Banach, we may choose fi ∈ W such that |fi| ≤ |π|k · kM and ( 1 if i = j f˜(e ) = i j 0 otherwise. Let F : W → W X ˜ y 7→ fi(y)ei.

so if y ∈ M then F (y) ∈ M. This means that if y ∈ W1 + M, then F (y) − y ∈ M (because if y = y1 + y2 with y1 ∈ W1 and y2 ∈ M, then F (y) = y1 + F (y2) = y + F (y2) − y2, but F (y2) − y2 ∈ M). In particular this applies if y ∈ fB + M. So (f − F ◦ f)(B) = (id −F )(fB) ⊂ M,

that is, f − F ◦ f ∈ Λ(B,M). But im(F ◦ f) ⊂ W1, so F ◦ f is ﬁnite rank, and is our desired ﬁnite rank approximating map. Lemma 8.2.7. Suppose V,W are Banach spaces and f ∈ L (V,W ). Then TFAE: (1) f is compact ∨ ∨ ∨ (2) f ∈ L (Wb ,Vb ) is compact ∨∨ ∨ ∨ ∨ ∨ ∨ ∨ (3) f :(Vb )b → (Wb )b factors through the closed inclusion W ⊂ (Wb )b . Proof. We don’t have time for this. See Lemma 16.4 of Schneider. 32 COURSE BY RICHARD TAYLOR

8.3. Tensor products. There are (at least) two distinct notions of tensor products of LCTVSs, corresponding to the following two notions of continuity for bilinear maps. Definition 8.3.1. A bilinear map β : V × W → U is called (1) separately continuous if for all v ∈ V , the map W → U given by w 7→ β(v, w) is continuous, and similarly for all w ∈ W , the map V → U given by v 7→ β(v, w) is continuous. (2) (jointly) continuous if β is continuous with respect to the product topology on V × W . Definition 8.3.2. (1) We can define a topology on V ⊗ W (the “inductive topology”) by declaring a lattice Λ ⊂ V ⊗ W to be open if for all v ∈ V , {w ∈ W | v ⊗ w ∈ Λ} is open in W , and for all w ∈ W , {v ∈ V | v ⊗ w ∈ Λ} is open in V . We will write V ⊗ι W for V ⊗ W with this topology. ⊗ Note that V × W −→ V ⊗ι W is separately continuous, and universal among separately continuous bilinear maps on V × W : if β : V × W → U is separately continuous, the induced map V ⊗ι W → U is continuous. (2) We can define a different topology on V ⊗ W by taking Λ ⊗O M,→ V ⊗L W , for open lattices Λ ⊂ V and M ⊂ W , as a generating set of open lattices in V ⊗ W . This is called the “projective topology”. We will write V ⊗π W for V ⊗ W with this topology. ⊗ Similarly to the previous situation, V × W −→ V ⊗π W is (jointly) continuous, and if β : V × W → U is continuous, then the induced map V ⊗π W → U is continuous.

Lemma 8.3.3. (1) If V , W are Hausdorﬀ then V ⊗ι W , V ⊗π W are Hausdorﬀ. (2) The projective topology on V ⊗ W is coarser than the inductive topology (that is, id V ⊗ι W −→ V ⊗π W is continuous). 0 0 0 0 (3) If f ∈ L (V,V ), g ∈ L (W, W ), then f ⊗ g : V ⊗ι W → V ⊗ι W and f ⊗ g : 0 0 V ⊗π W → V ⊗π W are continuous. 0 0 0 0 (4) If V ⊂ V , W ⊂ W are subspaces, then V ⊗π W ,→ V ⊗π W is a subspace. Proof. Exercise.

Lemma 8.3.4. If V and W are normable with norms k · kV and k · kW , then V ⊗π W is normable with the norm ( m ) X kzk = inf max (kxikV kyikW ) | z = xi ⊗ yi . i i=1

Proof. WLOG kV kV = |L|, kW kW = |L|. Let Λ0 = Λ(k · kV ), M0 = Λ(k · kW ) be the open unit balls in V and W . Then our proposed norm also satisﬁes kV ⊗ W k = |L|. We claim that Λ(k · k) = Λ0 ⊗O M0, so that the topologies match. It is clear that Pm Λ(k · k) ⊃ Λ0 ⊗O M0. In the other direction, if kzk ≤ 1, then we can write z = i=1 xi ⊗ yi so that kxikV kyikW ≤ 1 for all i. WLOG we can rescale the xi, yi so that kxikV = 1; then xi ∈ Λ0 for all i. This also implies that kyik ≤ 1 for all i, so yi ∈ M0 for all i, and we have z ∈ Λ0 ⊗O M0 as desired. 9. 2/4/20: More on tensor products, nuclear spaces, and spaces of compact type 9.1. More on tensor products. Last time, we deﬁned the conditions of separately continuous and plain/jointly continuous for bilinear maps β : V × W → U. We called the resulting p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 33

topologies on V ⊗W the injective topology (written V ⊗ι W ) and projective topology (written V ⊗π W ). Lemma 9.1.1. If β : V × W → U is bilinear, then TFAE: (1) β is continuous. (2) For all open lattices N ⊂ U, there are open lattices Λ ∈ V , M ⊂ W with β(Λ,M) ⊂ N. (3) For all continuous seminorms k · k on U, there are continuous seminorms k · kV on V and k · kW on W with

kβ(x, y)k ≤ kxkV kykW for all x ∈ V , y ∈ W . Proof. (2) implies (1): let N ⊂ U be any open lattice, and choose Λ, M as in (2). For any x ∈ V and y ∈ W , for a, b ∈ L, we have β(x + aΛ, y + bM) = β(x, y) + β(Λ, ay) + β(bx, M) + abβ(Λ,M) ⊂ β(x, y) + N if ay ∈ M, bx ∈ Λ, and ab ∈ O; but we can choose a, b with these properties. So β is continuous. (3) implies (2): given N, let k · kN be a seminorm associated to N, and choose k · kV , k · kW as in (3). Then take Λ = Λ(k · kV ), M = Λ(k · kW ). (1) implies (3): since kβk is continuous, there are continuous seminorms k · kV , k · kW on V,W such that if kxkV ≤ and kykW ≤ , then kβ(x, y)k ≤ 1. Rescaling them gives k · kV , k · kW satisfying the desired inequality. Lemma 9.1.2. If V , W are Fréchet(or in fact if one is Fréchetand the other metrizable) then ∼ (1) V ⊗π W = V ⊗ι W . (2) if β : V × W → U is bilinear, β being separately continuous implies that it is continuous. (Note that this implies (1).) Proof. For (2), it suffices to prove that if N ⊂ U is an open lattice, there are Λ ⊂ V and M ⊂ W open lattices with β(Λ,M) ⊂ N. Suppose otherwise. Then since V and W are metrizable, we can find (xn, yn) ∈ V × W with (xn, yn) → 0 but β(xn, yn) ∈/ N. Note that the map

W → Ls(V,U) y 7→ β(−, y) is continuous: for x ∈ V and N 0 ⊂ U an open lattice, the preimage of Λ({x},N 0) under this map is {y ∈ W | β(x, y) ∈ N 0}, which is open since β is separately continuous. On the other hand {yn} ⊂ W is bounded, so {β(−, yn)} ⊂ Ls(V,U) is bounded. Therefore by Banach-Steinhaus, {β(−, yn)} is equicontinuous. Therefore there is an open lattice Λ ⊂ V such that β(Λ, yn) ⊂ N for all n. But xn ∈ Λ for n 0, so β(xn, yn) ∈ N for n 0, contradiction.

Deﬁnition 9.1.3. Deﬁne V ⊗b πW to be the completion of V ⊗π W .

Lemma 9.1.4. If V , W are Fr´echet/Banach,then V ⊗b πW is also Fr´echet/Banach. 34 COURSE BY RICHARD TAYLOR

Lemma 9.1.5. Let V , W be Hausdorff. Note that the map ∨ Vb ⊗ W → Lb(V,W ) f ⊗ y 7→ (x 7→ f(x)y) ∨ factors through θ : Vb ⊗π W → CC (V,W ). Then θ is a homeomorphism onto its image, which is dense in CC (V,W ). Proof. Schneider Lemmas 18.1 and 18.9. Corollary 9.1.6. If V is Hausdorff and bornological and W is complete, we have an isomor- ∨ ∼ phism Vb ⊗b πW −→ CC (V,W ). Lemma 9.1.7. If V is Hausdorff and bornological and W is complete, and either any closed bounded O-submodule of V is compact, or any closed bounded O-submodule of W is compact, ∨ ∼ then we have an isomorphism Vb ⊗b πW −→ Lb(V,W ). Proof. Schneider Corollary 18.8.

9.2. Nuclear spaces. Let Λ ⊂ V be an open lattice, k · kΛ the corresponding seminorm on V . Then we can take the completion V → V ∧ = V ∧ , Λ k·kΛ

giving a Banach space. Furthermore, if M0 is the open unit ball in some other Banach space −1 W , and f : V → W is continuous linear, then f M0 is an open lattice in V , and f is also −1 continuous in the topology on V generated by f M0. So any such f factors through a ∧ completion V → V −1 → W of this form. f M0 We call V ∧ = V ∧ “a Banach completion of V ”. If Λ0 ⊂ Λ, then we can factor V → V ∧ Λ k·kΛ Λ ∧ ∧ as V → VΛ0 → VΛ . Deﬁnition 9.2.1. We call V nuclear if for all open lattices Λ ⊂ V , there is another open 0 ∧ ∧ lattice Λ ⊂ Λ such that VΛ0 → VΛ is compact. Lemma 9.2.2. V is nuclear if and only if for all f : V → W where f is continuous linear and W is Banach, f is compact.

Proof. Forward direction: assume V is nuclear. Let M0 ⊂ W be a bounded open lattice (i.e. −1 ∧ ∧ a unit ball). Let Λ ⊂ f M be an open lattice such that V → V −1 is compact. Then 0 Λ f M0 ∧ ∧ f : V → W factors as V → V → V −1 → W , hence is compact. Λ f M0 ∧ Backward direction: let Λ ⊂ V be an open lattice. Then g : V → VΛ is compact, so there is an open lattice Λ0 ⊂ V such that gΛ0 is compact. By intersecting with Λ, we can assume 0 ∧ ∧ ∧ 0 that Λ ⊂ Λ. Then V → VΛ factors through VΛ0 → VΛ , which is compact since it takes Λ to gΛ0.

Lemma 9.2.3. (1) Vs (i.e. V with the weak-∗ topology) is nuclear. (2) Subspaces of nuclear spaces (with the subspace topology) are nuclear. (3) Quotients of nuclear spaces by closed nuclear subspaces are nuclear. (4) Products of nuclear spaces are nuclear. (5) Countable direct sums of nuclear spaces are nuclear. (6) If V and W are nuclear, V ⊗π W is nuclear. ι1 ι2 (7) If we have a chain V1 ,→ V2 ,→ V3 ,→ · · · where each ιj is a homeomorphism onto its image and V is nuclear for all i, then lim V is nuclear. i −→ i p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 35

Remark 9.2.4. An inﬁnite-dimensional Banach space is never nuclear, since a Banach space V is nuclear if and only if id : V → V is compact.

π3 π2 π1 Lemma 9.2.5. (1) If we have a chain · · · → V3 → V2 → V1 where Vi is Hausdorff and π is compact for all i, lim V is nuclear. i ←− i (2) If we have a chain V ,→ι1 V ,→ι2 V ,→ι3 ··· with ι compact for all j, then lim V i 2 3 j −→j j is nuclear. We will not prove these; they are all in Schneider. 9.3. LCTVSs of compact type. Definition 9.3.1. A LCTVS V is said to be “of compact type” if it is a limit of a chain of the form ι1 ι2 V1 ,→ V2 ,→ V3 ,→ · · · with Vn Banach and ιn compact for all n. Lemma 9.3.2. (1) If V is of compact type, then V is complete, bornological, and barrelled. ∨ ∼ ∨ ∨ (2) If V is of compact type, then Vb is nuclear and Fréchetand V −→ (Vb )b . ∨ ∼ ∨ ∨ (3) If W is nuclear and Fréchet,then Wb is compact type and W −→ (Wb )b . (4) There is an anti-equivalence of categories {nuclear Fréchetspaces} ↔ {LCTVSs of compact type} ∨ W 7→ Wb ∨ Vb ← V. [ ι1 ι2 Lemma 9.3.3. (1) If we have V1 ,→ V2 ,→ V3 ,→ · · · with each Vn a Hausdorff LCTVS and ι compact, then lim V is of compact type. n −→ n (2) A countable direct sum of LCTVSs of compact type is of compact type. (3) If V is compact type and W ⊂ V is closed, then (a) W and V/W are of compact type. ∨ ∨ (b) (V/W )b → Vb is a homeomorphism onto a closed subspace. ∨ ∼ ∨ ∨ (c) Wb = Vb /(V/W )b . So that we don’t go too long without any proofs, we will prove a technical lemma that is important for proving many of the above statements. Lemma 9.3.4. Let V be a LCTVS of compact type, so that it is a limit V = lim V as above. −→ n (1) An O-submodule Λ ⊂ V is an open lattice if and only there are bounded open lattices Λn ⊂ Vn for all n such that Λ contains

Λ1 + Λ2 + Λ3 + ··· ,

where by Λi we mean the closure of Λi in V , or equivalently in Vi+1. (2) B ⊂ V is bounded if and only if B ⊂ Vn is bounded for some n. In particular, closed bounded subsets in V are compact.

Proof. (1) Forward direction: if Λ is an open lattice then Λ ∩ Vn is an open lattice, which contains a bounded open lattice Λn, which satisﬁes the desired condition. Backward direction: Λ ⊂ V is open if and only if Λ ∩ Vn is open for all n. But since L Λ contains Λi,Λ ∩ Vn contains Λn, and is indeed open. 36 COURSE BY RICHARD TAYLOR

(2) Backward direction: easy. Forward direction: suppose the claim fails for some bounded B ⊂ V . We will

recursively ﬁnd x1, x2,... ∈ B such that xi ∈ Vni for n1 < n2 < ··· , and Λni ⊂ Vni a bounded open lattice, such that

2 m−1 x1, πx2, π x3, . . . , π xm ∈/ Λn1 + ··· + Λnm . n Then since B is bounded, there should be n such that π xi ∈ Λn1 + Λn2 + ··· for all n i. But π xn+1 ∈/ Λn1 + Λn2 + ··· + Λnm for any m, so we would have a contradiction. 1−m Suppose x1, . . . , xm−1 have been chosen. B is not contained in π (Λn1 + ··· + 1−m Λnm−1 ), since otherwise it would be bounded in Vnm−1 . Let xm ∈ B \ π (Λn1 + ··· + m−1 Λnm−1 ). We have xm ∈ Vnm for some nm > nm−1, and π xm ∈/ Λn1 + ··· + Λnm−1 .

There is an open lattice M ⊂ Vnm+1 such that m−2 m−1 x1 + M, . . . , π xm−1 + M, π xm + M

do not meet Λn1 + ··· + Λnm−1 . But we can ﬁnd a bounded open lattice Λnm ⊂ Vnm

with Λnm ⊂ M and in fact Λnm ⊂ M, so indeed 2 m−1 x1, πx2, π x3, . . . , π xm ∈/ Λn1 + ··· + Λnm .

9.4. Examples. If r1 > r2, the restriction map O(D(r1)) → O(D(r2)) on the space of analytic functions on the closed disc of radius r1 is compact. This is because ( ) X n −n im Λ(k · kr1 ) = cnT | |cn| ≤ r1 ∀n n Q is in bijection with n≥0 O via ( ) Y X n −n θ : O → cnT | |cn| ≤ r1 ∀n n≥0 n X m(n) n (an) 7→ π anT n

m(n) −n where m(n) ∈ Z is minimal such that |π| ≤ r1 , that is, n log r m(n) ≥ − 1 . log |π| We can check that θ is continuous as follows. The open set ( ) X n −n −n cnT | |cn| ≤ r1 ∀n, |cn| ≤ r2 ∀n ⊂ O(D(r2)) n can be rewritten as ( ) X n −n −n cnT | |cn| ≤ r1 ∀n, |cn| ≤ r2 ∀n = 1,...,N n p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 37

−n −n for some N depending on (since r1 > r2, we have r1 < r2 for n suﬃciently large). The inverse image of the latter under θ is ( ) Y (an) ∈ O | |an| < βn, n = 1,...,N n≥0 Q for some βn = βn(), which is open. Since n≥0 O is compact, θ is actually a homeomorphism, and im θ is compact.

+ Deﬁnition 9.4.1. (1) O (D(r)) = lim O(D(r1)), called the space of overconvergent −→r1>r analytic functions, is a LCTVS of compact type. ◦ (2) O(D (r)) = lim O(D(r1)), the space of analytic functions on an open disc, is a ←−r1

10. 2/6/20: Analytic functions and their roots

Last time, we defined spaces of compact type: limits of chains V1 ,→ V2 ,→ V3 ,→ · · · where the Vis are Banach and the inclusions are compact. We also defined nuclear Fréchet spaces: those such that for any open lattice Λ ⊂ V , there is a smaller open lattice Λ0 ⊂ Λ ⊂ V such that V ∧ → V ∧ is compact. We found that these are anti-equivalent notions via taking k·kΛ0 k·kΛ strong duals. We also defined +(D(r)) = lim (D(r )), O −→ O 1 r1>r the space of overconvergent analytic functions on the disc of radius r, and saw that this was of compact type. (These are sometimes useful in rigid geometry, for instance in de Rham cohomology, where overconvergent forms somehow work better than convergent forms.) And we defined (D◦(r)) = lim (D(r )), O ←− O 1 r1

P m 10.1. Dividing analytic functions. We call f = m fmT ∈ O(D(r)) n-distinguished if n m n kfkr = |fn|r and |fm|r < |fn|r for all m > n. If f =6 0 then it is n-distinguished for some n, since the terms go to zero.

Lemma 10.1.1 (Weierstrass division). If f ∈ O(D(1)) is n-distinguished and g ∈ O(D(1)), then we can write g = qf + r for a unique q ∈ O(D(1)) and r ∈ L[T ] with deg(r) < n; moreover

kgk1 = max(kqk1kfk1, krk1). 38 COURSE BY RICHARD TAYLOR

Proof. First suppose that indeed g = qf + r with q ∈ O(D(1)) and r ∈ L[T ] with deg(r) < n; we will check that we must then have kgk1 = max(kqk1kfk1, krk1). We may assume WLOG that kfk1 = 1 and max(kqk1kfk1, krk1) = 1. Then q, f, r ∈ O T , and we can reduce them J K mod π to get q, f, r ∈ F[T ]. We certainly have kgk1 ≤ 1. If it were the case that kgk1 < 1, then we would have g = 0 = qf + r. But since max(kqk1kfk1, krk1) = 1, we know that one of r or q is nonzero, that r has degree < n, and that f is nonzero and has degree n, so it is impossible for qf + r to cancel out to 0. Now we will ﬁnd the desired q and r. First, we ﬁnd q ∈ O(D(1)) and r ∈ L[T ] with deg(r) < n such that kg − (qf + r)k1 ≤ |π|kgk1.

To do so, again assume WLOG that kfk1 = 1 and kgk1 = 1. Then we have deg f = n, and by usual polynomial division we can write g = qf + r ∈ F[T ] with deg r < n. Choose q, r lifting q, r such that r ∈ O[T ] with deg(r) < n. These satisfy the desired condition. Next we show that for all N ∈ Z>0, we can ﬁnd q ∈ O(D(1)) and r ∈ L[T ] with deg(r) < n such that N kg − (qf + r)k1 ≤ |π| kgk1. We do this by induction on N. Suppose we have found q, r satisfying this condition for N − 1, so we have N−1 kg − (qf + r)k1 ≤ |π| kgk1. Then applying our base case to g − (qf + r) in place of g, we can ﬁnd q0, r0 such that 0 0 N kg − (qf + r) − (q f + r )k1 ≤ |π|kg − (qf + r)k1 ≤ |π| kgk, from which we see that q + q0, r + r0 satisfy the condition for N. Now the map O(D(1)) × L[T ]deg

is an isometry (since kqf + rk1 = max(kqk1, krk1)), hence injective with closed image, since the space on the left hand side is complete. But we also just proved that it has dense image. So it is an isomorphism. 10.2. Factoring analytic functions. Lemma 10.2.1 (Weierstrass preparation). Suppose r ∈ pQ and f ∈ O(D(r)) is n- distinguished. Then f = gh for unique g, h such that g ∈ L[T ] is n-distinguished and satisfies × kgkr = 1, and h ∈ O(D(r)) . Proof. Since the lemma asserts that the factorization is unique, it suffices to prove it after replacing L by a finite extension (then we can use Galois descent). So we may assume WLOG that r ∈ |L|, and then use the change of variables T 7→ aT for some a ∈ L× with |a| = r to reduce to the case r = 1. Then as before, we can assume WLOG that kfk1 = 1. We have f = gh if and only if T n = h−1f + (T n − g), where T n − g is degree < n. Then uniqueness follows from uniqueness in Lemma 10.1.1. For existence, use Lemma 10.1.1 to write T n = qf + r where q ∈ O(D(1)) and r ∈ L[T ] with × deg(r) < n, and max(kqk1, krk1) = 1. It suffices to prove that q ∈ O(D(1)) , so that we can p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 39

−1 n set h = q and g = T − r. (Note that since krk1 ≤ 1, this means that g is n-distinguished and kgk1 = 1.) Reducing mod π, we get T n = qf + r in F[T ]. Since f is degree n and r is degree < n, we conclude that deg q = 0 but q 6= 0, which is to say that q = a − πq0 with a ∈ O×, 0 0 q ∈ O(D(1)), and kq k1 ≤ 1. Then ! πq0 πq0 2 q−1 = a−1 a + + + ··· a a

converges in O(D(1)), as desired. Corollary 10.2.2. If 0 6= f ∈ O(D(r)), then f has only ﬁnitely many zeros in D(r). In fact if f is n-distinguished it has at most n zeros in D(r).

Corollary 10.2.3. If r ≥ 1 then O(D(r)) → C(Zp,L) is injective and continuous. Corollary 10.2.4. If r ∈ pQ, O(D(r)) is noetherian, in fact a PID.

Proof. If I = (fi) is an ideal of O(D(r)), then I can be rewritten as (gi) where fi = gihi with × hi ∈ O(D(r)) and gi ∈ L[T ]. But then (gi) ⊂ L[T ] is generated by a single polynomial, so I is as well. Corollary 10.2.5. (1) If r ∈ pQ, f ∈ O(D(r)) is n-distinguished, and we write f = gh with h ∈ O(D(r))× and g ∈ L[T ] monic of degree n, then O(D(r))/(f) = O(D(r))/(g) = L[T ]/(g). (2) The maximal ideals of O(D(r)) are given by (g) where g ∈ L[T ] is irreducible and all the roots of g lie in D(r). For such g, we have isomorphisms ∼ L0 := L[T ]/(g) −→ O(D(r))/(g) ∼ L[T ]/(gn) −→ O(D(r))/(gn) 0 ∼ ∧ ∼ ∧ L x = L[T ](g) −→ O(D(r))(g). J K Proof. For example, to see that O(D(r))/(g) isomorphic to L[T ]/(g), we take the natural map L[T ]/(g) → O(D(r))/(g), and observe that it is an isomorphism because any element of deg

Proof. Assume WLOG that f(0) = 1. We will recursively deﬁne fn, gn as follows: f1 = f, n and fn = fn+1gn is the factorization from Lemma 10.2.1 where fn is viewed in O(D(p )), so n × that gn ∈ L[T ] and fn+1 ∈ O(D(p )) , except rescaled so that fn+1(0) = 1 and gn(0) = 1. n × Note that fn+1 is 0-distinguished (because everything in O(D(p )) is 0-distinguished), and n n−1 all roots of gn lie in D(p ) \ D(p ). 1 For the recursion to proceed, we need to show that fn+1 ∈ O(A ), or equivalently that n fn+1 ∈ O(D(R)) for any R ≥ p . Assuming inductively that fn ∈ O(D(R)), we have by 40 COURSE BY RICHARD TAYLOR

Lemma 10.1.1 that fn = qgn + r where q ∈ O(D(R)) and r ∈ L[T ] satisfies deg(r) < deg(gn). But we have fn+1gn = fn = qgn + r, implying (fn+1 − q)gn = r; since deg(r) < deg(gn), this means r = 0 and fn+1 = q ∈ O(D(R)) as desired. n−1 Now since fn is 0-distinguished in O(D(p )), we have for all r that (n−1)m |mth coefficient of fn|p ≤ 1 r m |mth coefficient of f |rm ≤ . n pn−1 1−n n−1 1 Then we have kfn − 1kr ≤ rp if r ≤ p , implying that fn → 1 in O(A ), hence Qn−1 i=1 gi → f. Also, since gi(0) = 1, the mth coefficient of gi can be written a sum of m-fold i−1 products of inverses of roots of gi; since the roots of gi all have size at least p , this means that for all r we have 1−im 1−i kgi − 1kr ≤ sup rp = rp , m≥1 which goes to 0 as i → ∞. P m 1 The Newton polygon of f = m amT ∈ O(A ), f(0) = 1, is drawn on the xy-plane as follows (see Figure 1). First we plot (m, − log |am|) for all m. In O(D(r)), f is i-distinguished if i is the maximal index maximizing

i log |ai|+i log r |ai|r = p .

− log |am| So if log r ≤ m for all m, then f is 0-distinguished in O(D(r)); but when log r passes − log |am| min m , then for i attaining this minimum we have − log |a | log |a | + i log r ≥ log |a | + i · i = 0, i i i and for the greatest such i and the corresponding r = r1, f is i-distinguished in O(D(r1)). log |ai| We conclude that f has exactly i zeros in D(r1), which must be of valuation exactly i , and none of lower valuation. We can continue; for a diﬀerent index j, we have

log |aj| + j log r ≥ log |ai| + i log r

− log |aj |+log |ai| − log |aj |+log |ai| when log r passes j−i , so for the greatest j > i minimizing j−i and the corresponding choice of r = r2, f is j-distinguished in O(D(r2)), and thus has j − i additional log |aj |−log |ai| zeros of valuation exactly j−i . And so on. In conclusion, if we take the lower convex hull of the points in our plot, for every line segment of slope α and horizontal length n in the convex hull, f has n zeros of valuation −α. This lower convex hull is the Newton polygon of f. See Figure 1 for an example where i = 4 and j = 7. 10.3. Locally analytic functions. Let h −h LAh = {f : Zp | L | f continuous, ∀a ∈ Z/p Z, f(a + t) ∈ O(D(p ))}. This is the space of locally analytic functions of radius of analyticity p−h. It can also be written as M O(D(p−h)). a∈Z/phZ On it we have the norm

kfkh = max kf(a + t)kp−h = max max |f(a + t)|. a∈Z/phZ a∈Z/phZ t∈D(p−h) p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 41

− log|am| r2

m 4 7

Figure 1. Newton polygon of a power series

0 For h > h, the natural inclusion LAh(Zp,L) → LAh0 (Zp,L) is compact, and so LA( ,L) = lim LA ( ,L) Zp −→ h Zp h

is a LCTVS of compact type. This is the space of locally analytic functions on Zp. In summary, we have the chain of inclusions

O(D(1)) = LA0(Zp,L) ,→ LAh(Zp,L) ,→ LA(Zp,L) ,→ C(Zp,L). ◦ Next time, we’ll see that LA(Zp,L) is dual to O(D (r)).

11. 2/11/20: Locally convex topological L-algebras 11.1. Duals of function spaces. We have been discussing various spaces of functions on Zp: • O(D(1)) consists of the analytic functions on the closed unit disc: these are given globally by a power series with radius of convergence (at least) 1. This is a Banach space. • For h ∈ Z>0, we have the space LAh(Zp,L) of locally analytic functions. This h is the space of continuous functions f : Zp → L such that for any a ∈ Z/p Z, f(a + t) ∈ O(D(p−h)). This is a Banach space, isomorphic to the direct sum over a ∈ Z/phZ of O(D(p−h)). • The space of locally analytic functions on Z is LA(Z ,L) := lim LA (Z ,L). It is a p p −→h h p LCTVS of compact type. • The space of continuous functions f : Zp → L with the supremum norm is a Banach space C(Zp,L). We have inclusions of each space in the space below it in the above list.

Lemma 11.1.1. For any h, the space LAh(Zp,L) has a Banach basis n x ! | n = 0, 1, 2,... . ph n

Proof. See e.g. [4, §1.7.2]. (It’s very computational.) 42 COURSE BY RICHARD TAYLOR

Corollary 11.1.2. The strong dual of LAh(Zp,L) is isomorphic to the space ( ∞ ) X n n cnT : |cn| ! is bounded ph n=0 and this isomorphism restricts to the isomorphism of the strong dual of C(Zp,L) with the space ( ∞ ) X n cnT : |cn| is bounded n=0 P n x defined by ( cnT ) n = cn. ∨ Note that the above space C(Zp,L) of power series with bounded coefficients may be identified with the space of bounded analytic functions on the open unit disc D0(1).

Corollary 11.1.3. The strong dual of LA(Zp,L) may be identiﬁed with the space

0 nX n n o O(D (1)) = cnT : |cn|r → 0 for all r < 1 in a manner compatible with the above identifications. Likewise, the strong dual of O(D(1)) may be compatibly identified with ( ) X n 1/(p−1) cnT : |cn||n!| is bounded ,→ O(D(|p| )). n Proof. We can prove all of the above statements by studying Banach bases and using the formula X n n X 1 n val (n!) = ∼ = p ph p ph p − 1 h≥0 h≥0 which gives 1 n n h ! ∼ |p| p (p−1) , ph j k so “|c |rn → 0 as n → ∞ for all r < 1” is equivalent to “|c | n ! is bounded for all n n ph h ≥ 1”. 11.2. Locally convex topological L-algebras. Definition 11.2.1. • By a locally convex topological L-algebra, abbreviated ‘LCTA’, we mean a LCTVS A together with the structure of an L-algebra (associative with identity) such that the multiplication map A × A → A is continuous. • A morphism of LCTAs is a morphism of L-algebras which is continuous. • A seminorm k · k on a LCTA A is algebraic if kabk ≤ Ckakkbk for all a, b ∈ A. Lemma 11.2.2. If A is an LCTA whose underlying LCTVS is normable with topology defined by a norm k · k, then k · k is algebraic. Proof. By continuity, there exists an > 0 such that if kak ≤ , kbk ≤ , then kabk ≤ 1. This implies kakkbk kabk ≤ 2|π|2 for all a, b ∈ A. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 43

Remark 11.2.3. If k · k is an algebraic seminorm and kabk ≤ Ckakkbk, then k · k is equivalent to the seminorm k · k0 = Ck · k which satisﬁes kabk0 ≤ kak0kbk0. Example 11.2.4. (1) If Ω is a compact topological space, then C(Ω,L) is a commutative Banach L-algebra via the pointwise multiplication map (f · g)(ω) = f(ω)g(ω). Indeed, we have:

kf · gk∞ = sup |f(ω)||g(ω)| ≤ kfk∞kgk∞. ω∈Ω (2) O(D(r)) is a commutative Banach L-algebra via multiplication of power series X n X m X X s fnT · gmT = fngmT . n m s n+m=s Indeed, we have:

X s n m X n X m sup fngm r ≤ sup sup |fn|r |gm|r ≤ k fnT krk gmT kr. s s n+m=s n+m=s n m

(3) If V is a Banach space over L with norm k · k, then the Banach space Lb(V,V ) is a Banach L-algebra via composition of linear operators. Recall that the norm on kf(x)k Lb(V,V ) is the operator norm kfk = sup06=x∈V kxk . Indeed, we have kf(g(x))k kf ◦ gk = sup 06=x∈V kxk kf(g(x))k = sup x∈V kxk g(x)6=0 kf(g(x))k kg(x)k = sup . x∈V kg(x)k kxk g(x)6=0 11.3. Topological modules over LCTAs. Now that we have algebras, we can also discuss modules. Deﬁnition 11.3.1. Suppose A is a LCTA. By a locally convex topological A-module (LCTM), we mean a LCTVS M together with a structure of a (left) A-module such that the scalar multiplication map A × M → M is continuous.

Remark 11.3.2. If A and M are normable with norms k · kA, k · kM , then the continuity of scalar multiplication A × M → M is equivalent to requiring that there exist C ∈ R>0 such that kamkM ≤ CkakAkmkM for all a ∈ A, m ∈ M. In the case of ﬁnitely generated modules over noetherian LCTAs, this structure is automatic.

Lemma 11.3.3. Suppose that A is a (left) noetherian Banach algebra. Then: (i) If M is a finitely generated Banach A-module, then any (abstract) A-submodule of M is closed. (ii) If M is a finitely generated Banach A-module and N is any Banach A-module, then any A-linear map f : M → N is continuous. If N is also finitely generated, then f is strict, i.e. M/kerf → Imf is an isomorphism. 44 COURSE BY RICHARD TAYLOR

(iii) Any finitely generated (abstract) A-module has a unique structure as a Banach A- module. (iv) The forgetful functor {finitely generated Banach A-modules} → {finitely generated A-modules} is an equivalence of categories. Proof. The proofs of all of these statements boil down to the Banach open mapping theorem. Without loss of generality, suppose that the norm on A satisfies kabk ≤ kakkbk for all a, b ∈ A. (i) Let N ⊆ M be an A-submodule, and let N be its closure. Since A is noetherian and M is finitely generated, N is a finitely generated A-module: choose generators ⊕n P x1, . . . , xn. This defines a surjection A → N sending (ai) to aixi. Note that any map from the free module A⊕n to a LCTM M is continuous. Let y1, . . . , yn be the image of the standard basis. Then by continuity of scalar multiplication on M, for any continuous seminorm k · kM on M, there is a C > 0 such that X aixi ≤ max kaixikM ≤ max Ckaik. M Now, consider the open unit ball ˚Λ := {a ∈ A: kak < 1} ⊆ A. This is an open subset of A which is closed under addition and multiplication. Thus, P ˚ Λxi ⊆ N is open. Thus, we may write X xi = yi + aijxj j ˚ with yi ∈ N and aij ∈ Λ. This can be rearranged to

x1 y1 . . (idn −(aij))  .  =  .  . xn yn Multiplying this by the matrix series 2 B = 1 + (aij) + (aij) + ··· , ˚ which converges as aij ∈ Λ, we ﬁnd that

x1 y1 . .  .  = B  .  , xn yn

that is, xi ∈ N for all i, so N = N is closed. (ii) As M is ﬁnitely generated, we may choose generators x1, . . . , xn ∈ M, and thus a continuous surjection π : A⊕n → M which is open by the open mapping theorem. ⊕n P Then the map f ◦ π : A → N sending (ai) to aif(xi) is continuous. Since π is open, this implies that f is continuous. Since N is ﬁnitely generated, the image of f is closed and Banach. Thus, the continuous bijection M/ ker f → im f is a homeomorphism by the open mapping theorem. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 45

(iii) Let M be a ﬁnitely generated A-module. Then choosing generators x1, . . . , xn, there ⊕n P is a continuous, thus open, surjection π : A → M sending (ai) to aixi. By part (i), the kernel ker π is closed, so the quotient space A⊕n/ ker π is Banach. Thus, π induces a bijection from the Banach A-module A⊕n/ ker π to M; this gives M the topology of a Banach A-module. Now, if M has any other Banach A-module topology, then π induces a continuous bijection A⊕n/ ker π → M. This is a homeomorphism by the open mapping theorem.

Remark 11.3.4. If A is a LCTA and k · k is an algebraic seminorm, then Abk·k is a Banach algebra. 11.4. Fréchet-Stein algebras. Definition 11.4.1 (Schneider-Teitelbaum). Suppose that A is a Fréchet algebra. We call A a Fréchet-Steinalgebra if there exist algebraic seminorms k · k1 ≤ k · k2 ≤ · · · defining the topology on A such that

(1) Abk·kn is noetherian for all n.

(2) The map Abk·kn → Abk·kn−1 is flat for all n: i.e. the functor M 7→ Abk·kn−1 ⊗ M is Abk·kn exact. Note that in particular, A = lim A . ←−n bk·kn Example 11.4.2. • Any noetherian Banach algebra is a Fréchet-Stein algebra (e.g. O(D(1))). • O(D0(r)) and O(A1) are Fréchet-Stein algebras. To see the latter, we use the following lemma:

Q Lemma 11.4.3. (1) If r1 < r2 are in p , then the inclusion map induces an isomorphism

O(\D(r2)) → O(D(r1)). k·kr1 0 (2) If r1 < r, then the inclusion map induces an isomorphism O(\D (r)) → O(D(r1)). k·kr1 Q (3) If r1 < r2 are in p , then the inclusion map O(D(r2)) → O(D(r1)) is ﬂat.

Proof. (1) The inclusion O(D(r2)) ,→ O(D(r1)) is an isometry when we use the r1-norm

on the left side. The induced map O(\D(r2)) → O(D(r1)) is thus an isometry as k·kr1 well. As both sides are Banach, it is thus injective with closed image. This image is moreover dense, as it includes L[T ] and L[T ] ⊆ O(D(r1)) is dense. Thus, it is an isomorphism. (2) The proof is the same as (1). (3) Suppose that M,→ N is an injection of ﬁnitely generated O(D(r2))-modules. We

need to prove that the map M ⊗O(D(r2)) O(D(r1)) → N ⊗O(D(r2)) O(D(r1)) is injective (exercise: it suffices to check flatness on finitely generated modules). Let x be in the kernel. We will show that Ann(x) is not contained in (g) for any maximal ideal (g) of the principal ideal domain O(D(r1)); thus, Ann(x) = (1), so x = 0. Choose some maximal ideal (g); we want to show that x is in the kernel of the localization

M ⊗ O(D(r1)) → M ⊗ O(D(r1))(g). 46 COURSE BY RICHARD TAYLOR

Note that we may take g ∈ L[T ] to be an irreducible monic polynomial whose roots in L all have valuation at most r1. By Krull’s intersection theorem, the map

M ⊗ O(D(r1))(g) → M ⊗ O(\D(r1))(g)

is injective. Moreover, we have seen that O(\D(r1))(g) ' Ld[T ](g). We have a commutative diagram:

M ⊗ O(D(r1)) N ⊗ O(D(r2))

M ⊗ O(D(r1))(g) N ⊗ O(D(r2))(g)

M ⊗ O(\D(r1))(g) N ⊗ O(\D(r2))(g)

Thus, to conclude that x is annihilated by an element not contained in (g), it suﬃces to see that the bottom map

M ⊗ O(\D(r1))(g) → N ⊗ O(\D(r2))(g) is injective. We claim that the map

O(D(r2)) → O(\D(r1))(g) ' Ld[T ](g)

is flat. Indeed, note that g ∈ L[T ] ⊆ O(D(r2)), and we may identify Ld[T ](g) as the completion of O(D(r2)) at the maximal ideal (g). 12. 2/13/20: Modules over Frechet-Stein´ and Banach algebras Last time we introduced the notion of a Fréchet-Stein algebra: a Fréchet algebra A with a defining set of algebraic seminorms

k−k1 ≤ k−k2 ≤ k−k3 ≤ · · · (implying A = lim A∧ ) such that ←−i k−ki (1) each Banach algebra A∧ is noetherian, and k−kn (2) the canonical map A∧ → A∧ is flat for all n. k−kn+1 k−kn Any noetherian Banach algebra is an example, as are O(D◦(r)) and O(A1). Now we will state some facts about modules over Fréchet–Stein algebras without proofs, due to lack of time. This will allow us to think about sheaves on e.g. O(A1) (note that this is not affinoid, but we can define coherent sheaves on each O(D(r))). 12.1. Coherent sheaves on Fréchet-Stein algebras.

Definition 12.1.1. By a coherent sheaf on a Fréchet-Stein algebra (A, {k−kn}), we mean a sequence of finitely generated A∧ -modules M together with isomorphisms k−kn n

∧ =∼ Ak−k ⊗A∧ Mn+1 −→ Mn. n k−kn+1 p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 47

These form an abelian category Coh(A, {k−kn}) where kernels, cokernels, and direct sums are taken termwise. (This follows from flatness of the connecting maps.) If M = {Mn} is a coherent sheaf, we define its global sections as Γ( ) = lim M . F ←− n Lemma 12.1.2. (1) The collection of A-modules isomorphic to global sections of objects of Coh(A, {k−kn}) is independent of the choice of the defining seminorms {k−kn}. We call such A-modules coadmissible. (2) The map Γ(M ) → Mn has dense image, and moreover ∧ =∼ Ak−kn ⊗A Γ(M ) −→ Mn. (3) The functor

Γ: Coh(A, {k−kn}) → {coadmissible A-modules} is an equivalence of categories. In particular, it is exact.

0 Proof. (1) The point is that if we have any other collection of seminorms k−kn then we have 0 ∧ ∧ k−k ≤ Ck−km for some m, so we can tensor from A to A 0 to get a ﬁnite module n k−km k−kn 0 Mn from Mm. This works the other way as well. (2) Surjectivity follows from abstract nonsense. Let us now show that the given map is injective. Write Γ(M ) = M and suppose that we have k X ai ⊗ xi 7→ 0 ∈ Mn i=1 for a ∈ A∧ and x ∈ M. We look at the map Ak → M sending (a ) 7→ P a x . The kernel i k−kn i i i i of the corresponding map (A∧ )k → M is ﬁnitely generated, and using surjectivity to lift k−kn n the generators, we can form a diagram Ar Ak M

r k A∧ A∧ M k−kn k−kn n where the bottom row is exact and the top row has composite equal to 0. Then we can tensor to get a map r k A∧ A∧ A∧ ⊗ M k−kn k−kn k−kn A

=∼ =∼ r k A∧ A∧ M . k−kn k−kn n

Then chasing the diagram, you show that the right vertical map is an isomorphism. Left-exactness of the global sections functor is obvious from the description as an inverse limit, but it is not so obvious that the derived functor vanishes. To see that Γ is exact, use the Mittag-Leﬄer condition; since each Mn is noetherian, the dense images of Mn+k → Mn must stabilize. 48 COURSE BY RICHARD TAYLOR

Lemma 12.1.3. (1) If f : M → N is an A-linear map of coadmissible A-modules, then ker f, im f, coker f are coadmissible. (2) If M1,M2 ⊆ N are coadmissible A-modules, so is M1 + M2. (3) If N is a coadmissible A-module and M ⊆ N is a finitely generated submodule, then it is coadmissible. (4) Any finitely generated A-module is coadmissible. (5) The category of coadmissible A-modules is an abelian subcategory of A-modules. So far we have been doing just algebra; we are forgetting about the topology. This is because if M is a coadmissible A-module, then M = lim M and each M is a uniquely a ←− n n Banach A∧ -module. So M becomes a Fréchet A-module in a natural way. Again, this k−kn topology is independent of the choice of defining seminorms {k−kn}. Lemma 12.1.4. Again, let A be a Fréchet–Steinalgebra. (1) A-linear maps between coadmissible A-modules are automatically continuous with respect to the canonical topology. (2) Coadmissible A-submodules of coadmissible modules are closed. (3) If M is coadmissible and N ⊆ M is a closed submodule, then N and M/N are coadmissible. The upshot is that coadmissible modules over Fréchet-Stein algebras behave exactly the same way as finitely generated modules over noetherian Banach algebras.

12.2. Banach-free modules. Now we will go back to Banach algebras, most of the time noetherian and commutative Banach algebras. But we will look at bigger modules. (Richard suspects that the following also extends to Fréchet-Stein algebras, but can’t find a reference and didn’t have time to check.) Definition 12.2.1. Let A be a Banach algebra. We say that a Banach A-module M is Banach-free (or potentially ON-able in the literature following Kevin Buzzard—see [3]) if ∼ M =⊥I,0 (A, k−k). We say that M is Banach-projective (or has property (Pr) in the literature) if M is a direct summand of a Banach-free Banach A-module. Lemma 12.2.2. A Banach A-module M is Banach-projective if and only if for every continuous A-linear surjection N1 N2 of Banach A-modules, every continuous A-linear map M → N2 lifts to a continuous A-linear map M → N1. Proof. For the forward direction, choose M 0 so that 0 ∼ M ⊕ M =⊥I,0 A.

0 Then the map f : M → N2 factors through M ⊕ M , so we may as well assume that M is Banach-free. Then there is a Banach basis {ei} ⊂ M, with images {f(ei)} ∈ N2. The set {f(ei)} is bounded, and the map N1 N2 is open by the open mapping theorem. It follows that we can ﬁnd lifts yi ∈ N1 of f(ei) ∈ N2 such that kyik is bounded. Then we can lift the map f to M → N1 by sending ei 7→ yi. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 49

For the reverse direction, given such an M, there is a lift

⊥I,0 A M

id M.

This diagram gives a splitting of ⊥I,0 A with M as a direct summand. From now on, in this lecture, we assume that A is a noetherian commutative Banach algebra. Lemma 12.2.3. Let A be a noetherian commutative Banach algebra. If M is a finitely generated A-module, then M is projective if and only if M is Banach-projective. Proof. If M is projective, then M ⊕ M 0 ∼= A⊕r as finitely generated A-modules. Then since both sides have canonical topologies that agree, M is also Banach-projective. Now suppose that M is Banach-projective. Since M is finitely generated, we have some surjection ⊕r A M of A-modules, which is necessarily continuous. This map splits as a map of Banach A-modules, so it splits as a map of A-modules. We now want to use finitely generated modules to approximate all modules.

Lemma 12.2.4. Suppose M ⊆⊥I,0 A is a finitely generated A-submodule. Then ⊕J (1) there exists a finite subset J ⊆ I such that the projection πJ : M,→ A of M onto the J-component is injective, (2) M is automatically closed, and (3) for all > 0, there exists a finite J ⊆ I such that

kπJ x − xk ≤ kxk for all x ∈ M. Proof. (1) Let M (1),...,M (r) generate M. We visualize this as a matrix with r rows and (j) (j) columns corresponding to I. Write M = mi . For each i ∈ I, let i∈I (1) (r) r xi = mi , . . . , mi ∈ A , r and let N ⊆ A be the submodule generated by the xi. Then because A is noetherian, N is a ﬁnitely generated module, so there exists a ﬁnite subset J ⊆ I such that N is generated by the xj for j ∈ J. We claim that this J works. If

X (k) πJ akm = 0,

Pr (k) then for all j ∈ J we have k=1 akmj = 0. This just means that the linear map Ar → A X (bk) 7→ akbk k 50 COURSE BY RICHARD TAYLOR

P (k) vanishes on xj for all j ∈ J, and hence on N. That means that k akmi = 0 for all i, so P (k) akm = 0. (2) Given any x∈ / M, we need to ﬁnd a U ⊆⊥I,0 A containing x but disjoint from M. Find a ﬁnite J ⊆ I such that πJ J (M, x) ,→⊥I,0 A −→ A J J is injective. Then πJ M is closed in A , so its complement is an open subset of A containing −1 J πJ (x) but disjoint from πJ M. It follows that the inverse image πJ (A \ πJ (M)) is a neighborhood of x disjoint from M. (3) Taking generators mi ∈ M, the map r A M is open by the open mapping theorem. This simply means that there exists δ > 0 such that for all m ∈ M with kmk ≤ δ, we can write r X m = aimi, kaik ≤ 1 for all i. i=1

Rescaling, we see that for all m ∈ M, there exist ai ∈ A such that X kmk m = a m , ka k ≤ . i i i δ|π| Given > 0, there exists a ﬁnite subset J ⊆ I such that

kπJ mi − mik ≤ δkmik P for all i. It follows that for all m, if we write m = aimi as above, then kmk kπJ m − mk ≤ sup kaikkπJ mi − mik ≤ δ|π| = kmk. i=1,...,r δ|π| This ﬁnishes the proof. Let M,N be Banach A-modules. Then there is a closed subspace

LA(M,N) ⊆ Lb(M,N)

of linear maps that are A-linear. We say that f ∈ LA(M,N) has finite rank if its image is a finitely generated A-module. We then define the space of completely continuous maps as

CC A(M,N) = closure of ﬁnite rank maps in LA(M,N). This is consistent with our previous notation in the case A = L.

Lemma 12.2.5. If N is Banach-free, then CC A(M,N) is also the closure of the set of f ∈ LA(M,N) such that im(f) is contained in a ﬁnite free A-module.

Proof. If f ∈ LA(M,N) has finite rank, then we need to see that f is in the closure of those g ∈ LA(M,N) such that im(g) is contained in a finite free A-module. We first write

N =⊥I,0 A, and then given > 0, there exists a ﬁnite subset J ⊆ I such that

kπJ x − xk < kxk p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 51

⊕J for all x ∈ im(f). Then im(πJ ◦ f) ⊆ A , and on the other hand kf(x)k kπ f(x) − f(x)k ≤ ≤ kxk. J kfk

So we have kπJ f − fk ≤ .

13. 2/18/20: Characteristic power series 13.1. More on complete continuity. As before, ﬁx A a noetherian commutative Banach algebra (where we will assume that kabk ≤ kakkbk), and M,N Banach A-modules (where we will assume that kamk ≤ kakkmk). Last time we deﬁned

closed CCA(M,N) ⊂ LA(M,W ) ⊂ Lb(M,N) where the first space is the “completely continuous” maps, the closure of those maps of finite rank (i.e. whose image is finitely generated).

Lemma 13.1.1. (1)

LA(⊥I,0(A, k · k),N) = ⊥I (N, k · kN )

LA(⊥I,0(A, k · k), ⊥J,0(A, k · k)) = ⊥I ⊥J,0(A, k · k).

(2)

closed CCA(⊥I,0(A, k · k), ⊥J,0(A, k · k)) = ⊥J,0⊥I (A, k · k) ⊂ ⊥I×J (A, k · k).

That is, if {ei}i∈I is a Banach basis for ⊥I,0(A, k · k), and f(ei) = (fij)j∈J , then

(1) f ∈ LA(··· ) if and only if kfijk is bounded and limj→∞ kfijk = 0 for all i, and (2) f ∈ CCA(··· ) if and only if kfijk is bounded and limj→∞ supi kfijk = 0. Proof. (1) is basically the same as in the case A = L, so we will not do it.

For (2), assume that kfijk is bounded and limj→∞ supi kfijk = 0. We need to check that

kf − πJ0 ◦ fk → 0

0 0 kgxk as J ⊂ J,#J < ∞ gets large. But since kgk = supx6=0 kxk , we have

kf − πJ0 ◦ fk ≤ sup sup kfijk = sup sup kfijk → 0 i j∈J\J0 j∈J\J0 i

as desired. For the other direction, as the given condition on f is closed, it suﬃces to prove that if f is ﬁnite rank, then

(fij) ∈ ⊥J,0⊥I (A, k · k). But for all > 0, there is J 0 ⊂ J with #J 0 < ∞ such that for all x ∈ im f we have kπJ0 x − xk < kxk, and therefore for all x ∈ M, we have kπJ0 f(x) − f(x)k < kfkkxk. Then 0 for j∈ / J , we have supi kfijk ≤ kfk. 52 COURSE BY RICHARD TAYLOR

13.2. Functions on the affine line over Banach algebras. We will see that if f is 1 completely continuous, then det(1 + T f) makes sense and lies in O(AA). (Here ( ∞ ) X n n O(D(r)A) = anT | an ∈ A, kankr → 0 as n → ∞ n=0 ( ∞ ) X ( 1 ) = lim (D(r) ) = a T n | a ∈ A, ∀r > 0, ka krn → 0 as n → ∞ , O AA ←− O A n n n r n=0 P n n and the latter has norms k anT kr = supn kankr .) Lemma 13.2.1. Suppose f(T ) ∈ A[T ] is a monic polynomial (or more generally the leading × 1 1 coefficient of f is in A ), and suppose g ∈ O(AA). Then there is a unique q ∈ O(AA) and r ∈ A[T ] with deg(r) < deg(f) such that g = qf + r. Proof. Existence: let deg(f) = d. By Euclidean division for polynomials, we may write n P n T = qnf + rn for qn, rn ∈ A[T ] with deg(rn) < deg(f). Let g = gnT . Then define P∞ P∞ q = n=0 gnqn and r = n=0 gnrn. We need these to converge in k · kr for all r ∈ R>0. But the coefficients of qn are polynomials in the coefficients of f of degree ≤ n, and n−d n−d−1 similarly for rn (maybe with degree ≤ n + 1). More precisely, qn = T + an−d−1T + n−d−2 an−d−2T + ··· where an−d−1 is linear in the coefficients of f, an−d−2 is quadratic in the coefficients of f, and so on. Consequently i n−d−i n−d n−d kqnkr ≤ sup kfk1r ≤ max(kfk1 , r ) i and n−d+1 d−1 krnkr ≤ kfk1 r . So n−d kgnqnkr ≤ kgnk max(kfk1, r) → 0

as n → ∞, and similarly for kgnrnkr. Uniqueness: if uniqueness fails, then we can ﬁnd qf = r with deg r < deg f and one of q and r nonzero. Then we can get a contradiction by arranging that q, f, r ∈ Λ(k · k) T and have nonzero reductions mod π. Since f is monic, we have deg(f (mod π)) = deg fJ, whichK gives a contradiction mod π. The upshot is that these spaces can be thought of as spaces of inﬁnite matrices with some condition on the rows and columns.

1 Corollary 13.2.2. If f, g ∈ A[T ] with f monic and g = qf for q ∈ O(AA), then q ∈ A[T ]. 13.3. The characteristic power series. Lemma 13.3.1. Let A be a commutative noetherian Banach algebra and M a Banach projective Banach A-module. Let ϕ ∈ CCA(M,M). We can associate to each ϕ a unique element ∞ X n 1 1 + tn(ϕ)T = det(1 − ϕT ) ∈ O(AA) n=1 such that 1 (1) det(1 − (·)T ): CCA(M,M) → O(AA) is continuous. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 53

(2) If im ϕ ⊂ N where N is a ﬁnite projective A-submodule of M, then

det(1 − ϕT ) = det(1 − ϕ|N T ) (note that the RHS is the algebraically deﬁned determinant). (3) If ϕi ∈ CCA(Mi,Mi), so that ϕ1 ⊕ ϕ2 ∈ CCA(M1 ⊕ M2,M1 ⊕ M2), then

det(1 − (ϕ1 ⊕ ϕ2)T ) = det(1 − ϕ1T ) det(1 − ϕ2T ). Proof. Uniqueness: for M Banach-free, this is implied by properties (1) and (2), because any completely continuous operator can be approximated by operators whose images lie inside finite free submodules. For M Banach-projective, choose N so that M ⊕ N is Banach-free and use that det(1 − (ϕ ⊕ 0)T ) = det(1 − ϕT ). Existence: again we will reduce to the Banach-free case. If we have already defined det(1 − (·)T ) in the Banach-free case, and ϕ ∈ CCA(M,M) with M ⊕ N Banach-free, we can just define det(1 − ϕT ) := det(1 − (ϕ ⊕ 0)T ). If M ⊕ N 0 is also Banach-free, then M ⊕ N ⊕ M ⊕ N 0 is Banach-free, and comparing the maps (ϕ, 0, 0, 0) and (0, 0, ϕ, 0) on this space, which are the same up to an automorphism, we see that this definition is independent of the choice of N. Now assume M is Banach-free, and write it in the form ⊥I,0(A, k · k). Then we know that if P ϕ(ei) = j∈I ϕijej, then kϕijk is bounded for all i, j (say by B > 1), and limj→∞ supi kϕijk = P∞ n 0. Then we will set det(1 − ϕT ) = 1 + n=1 tn(ϕ)T where n X X Y tn(ϕ) = (−1) sign(σ) ϕiσ(i). S⊂I σ permutation of S i∈S #S=n

We can ﬁnd J0 ⊂ J with #J0 < ∞ such that kϕijk ≤ 1 if j∈ / J0. For all > 0, we can furthermore ﬁnd J ⊂ J with #J < ∞ such that kϕ k ≤ if j∈ / J . Then if S J , we ij B#J0 ( have

Y #J0 ϕiσ(i) ≤ B · = , B#J0 i∈S

so tn(ϕ) converges. In fact, if #S > #J, we can strengthen this to

#S−#J Y #J0 #S−#J ϕiσ(i) ≤ B · ≤ , B#J0 i∈S n 1 so in fact ktn(ϕ)k ≤ C . We conclude that indeed det(1 − ϕT ) ∈ O(AA). (But note that so far, our deﬁnition seems to depend on the choice of Banach basis.) To show continuity, recall that we can estimate

ka1a2 ··· an − b1b2 ··· bnk

≤ max(ka1 − b1kka2k · · · kank, kb1kka2 − b2kka3k · · · kank,..., kb1k · · · kbn−1kkan − bnk) and therefore

0 0 n−#J−1 n k det(1 − ϕT ) − det(1 − ϕ T )kr ≤ kϕ − ϕ k sup( r ). n Now let’s show property (2). Suppose im(ϕ) ⊂ N where N is ﬁnite projective. We have

det(1 − ϕT ) = lim det(1 − πJ ◦ ϕT ) J⊂I #J<∞ 54 COURSE BY RICHARD TAYLOR since for all > 0 there is J < I with #J < ∞ and kπJ x − xk < kxk for all x ∈ N. This limit can also be written as lim det(1 − πJ ◦ ϕ|A⊕J T ) J since the coefficients of det(1 − πJ ◦ ϕT ) are X X Y sign(σ) (πJ ◦ ϕ)iσ(i) S:#S=n σ and (πJ ◦ ϕ)iσ(i) is 0 unless σ(i) ∈ J for all i ∈ S, that is, unless S ⊂ J. The limit can again be rewritten as lim det(1 − ϕ ◦ πJ |N T ) = det(1 − ϕ|N T ) J by comparing A⊕J and N using the maps ϕ A⊕J −→ N −→πJ A⊕J ϕ N −→πJ A⊕J −→ N and noting that ϕ ◦ πJ |N → ϕ|N as J → ∞. Note that from this we see by reducing to the algebraic case that det(1 − ϕT ) is independent of the choice of Banach basis. For property (3), we again reduce to the case where im(ϕ) ⊂ N and N is finite projective (i) (1) (i) over A as follows: if ϕ1 → ϕ1 and ϕ2 → ϕ2, and each ϕj has image contained in something (i) (i) finite projective, then ϕ1 ⊕ ϕ2 → ϕ1 ⊕ ϕ2.

Deﬁnition 13.3.2. If a ∈ A, then det(1 − aϕ) = det(1 − T ϕ) |T =a.

Lemma 13.3.3. (1) If ϕ, ψ ∈ CCA(M,M), then det(1 − ϕ) det(1 − ψ) = det(1 − (ϕ + ψ − ϕψ))

where ϕ + ψ − ϕψ ∈ CCA(M,M). (2) If ϕ ∈ CCA(M,N) and ψ ∈ LA(N,M), then ϕ ◦ ψ ∈ CCA(N,N), ψ ◦ ϕ ∈ CCA(M,M), and det(1 − ϕ ◦ ψT ) = det(1 − ψ ◦ ϕT ). (3) If N ⊂ M is closed and ϕ ∈ CCA(M,N), then det(1 − ϕT ) = det(1 − ϕ|N T ). Proof. For all of these, one uses the same strategy: reduce to the Banach-free case using M → M ⊕ M 0, reduce to the case where im ϕ is contained in a ﬁnite projective module, and apply the corresponding algebraic result. In the next lecture, we will address the following issues:

(1) If f ∈ A[T ] with f(0) = 0 and ϕ ∈ CCA(M,M), what is det(1 − f(ϕ)T )? (2) Do we have compatibility with base change ψ : A → B? Base change for Banach modules works as follows. Let ψ : A → B be a continuous algebra map (between commutative noetherian Banach algebras), and assume as usual that kaa0k ≤ kakka0k, kbb0k ≤ kbkkb0k, kψak ≤ kak.

Let M be a Banach module over A with kamk ≤ kakkmk. Then let B⊗b AM be the completion of B ⊗A M with respect to Λ(k · kB) ⊗Λ(k·kA) Λ(k · kM ). This has the universal property that if N is a Banach B-module and ϕ : M → N is continuous and A-linear, then ϕ factors uniquely through

M → B⊗b AM m 7→ 1 ⊗ m p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 55

so that the induced map B⊗b AM → N is continuous and B-linear. (Recall that the topology on B ⊗A M is given by kxk = inf sup kb kky k.) P i i x= bi⊗yi,bi∈B,yi∈M i

14. 2/20/20: More properties of characteristic power series 14.1. Review and base change. Let A be a noetherian commutative Banach algebra, M a Banach-projective Banach A-module, and ϕ ∈ CCA(MM). We saw last time that we can 1 continuously associate to ϕ a power series det(1 − T ϕ) ∈ O(AA) such that

det(1 − T (ϕ1 ⊕ ϕ2)) = det(1 − T ϕ1) det(1 − T ϕ2), and if im ϕ ⊂ N where N is a ﬁnite projective A-module then

det(1 − ϕT ) = det(1 − ϕ|N T ),

and if ϕ1 ∈ CCA(M,N) and ϕ2 ∈ LA(N,M) then

det(1 − T ϕ1 ◦ ϕ2) = det(1 − T ϕ2 ◦ ϕ1),

and if N ⊂ M is closed and ϕ ∈ CCA(M,N), then

det(1 − ϕT ) = det(1 − ϕ|N T ),

and if ϕ1, ϕ2 ∈ CC (M,M), then

det(1 − ϕ1T ) det(1 − ϕ2T ) = det(1 − (ϕ1 + ϕ2 − ϕ1ϕ2)T ).

Also, for a ∈ A, we write det(1 − aϕ) = det(1 − T ϕ)|T =a. Lemma 14.1.1. Let ψ : A → B be a continuous map of commutative noetherian Banach algebras and M a Banach-projective Banach A-module. Let ϕ ∈ CCA(M,M). Then (1) B⊗b AM is Banach projective over B (and if M is Banach-free then so is B⊗b AM). (2) 1 ⊗ ϕ ∈ CCA(B⊗bM,B⊗bM). (3) ψ(det(1 − T ϕ)) = det(1 − T (1 ⊗ ϕ)). Proof. Once we know (1), then (2) and (3) immediately follow, because the “matrix” of 1 ⊗ ϕ is the same as the “matrix” of ϕ. So we only need to check (1). Here, the Banach-projective case immediately reduces to the Banach-free case. So, choosing a Banach basis, we need to show that 0 B⊗ ⊥I,0 (A, k−k) →⊥I,0 (B, k−k); 1 ⊗ ei 7→ ei

is a homeomorphism. It is well-deﬁned and continuous because the elements 1 ⊗ ei are bounded; then one can show that it has dense image and is closed, then that it is surjective, and then that it is an isometry. 14.2. Functional calculus for characteristic power series. If f(T ) ∈ A[T ], f(0) = 0, what is det(1 − f(ϕ)T )? In the ﬁnite-dimensional case, if Y det(1 − ϕT ) = (1 − αiT ),

then the eigenvalues of f(ϕ) are f(αi), so Y det(1 − f(ϕ)T ) = (1 − f(αi)T ). 56 COURSE BY RICHARD TAYLOR

2 d So suppose that deg(f) = d and f(T ) = F1T + F2T + ··· + FdT . We can certainly deﬁne e Y Se (1 − f(Ai)T ) ∈ Z[F1,...,Fd,A1,...,Ae,T ] = Z[F1,...,Fd,E1,...,Ee,T ] i=1 where Ei is the degree i elementary symmetric function in the Ais (so E1 = A1 + ··· + Ae, etc.). Let e Y (1 − f(Ai)T ) = Gd,e(F1,...,Fd,E1,...,Ee,T ). i=1 Then we have

Gd,e+1(F1,...,Fd,E1,...,Ee, 0,T ) = Gd,e(F1,...,Fd,E1,...,Ee,T )

Gd+1,e(F1,...,Fd, 0,E1,...,Ee,T ) = Gd,e(F1,...,Fd,E1,...,Ee,T ) via the map

Z[E1,...,Ee+1] → Z[E1,...,Ee] Ei 7→ Ei for i ≤ e

Ee+1 7→ 0 induced by the map of ambient rings

Z[A1,...,Ae+1] → Z[A1,...,Ae] Ae+1 7→ 0. Now let ∞ X i (i) Gd,e(F ,E,T ) = 1 + T Gd,e(F ,E). i=1 (i) Qi ij Here Gd,e(F ,E) is a Z-linear combination of monomials of the form j=1 Fij Aj which m1 me happens to also be a Z-linear combination of expressions of the form Fi1 ··· Fii E1 ··· Ee . Qi ij (i) m1 me In order for j=1 Fij Aj to appear in Gd,e(F ,E) from the expansion of Fi1 ··· Fii E1 ··· Ee , we must have X X di ≥ ij = αmα ≥ i.

(i) This implies that Gd,e cannot involve Eα for α > di, so it stabilizes and we can write (i) (i) Gd,e = Gd,∞ for e ≥ di. r1 re Furthermore, if r1 ≥ · · · ≥ re, then A1 ··· Ae is a monomial of maximal A1-exponent r1−r2 r2−r3 re r1 re (i) in the expansion of E1 E2 ··· Ee . Since A1 ··· Ae can only appear in Gd,e(F ,E) if P r1 ≤ d, we conclude that d ≥ mα. Now assume that A is a Banach algebra, and f ∈ A[T ] with f(0) = 0 and deg(f) = d, 2 3 and g ∈ A[T ] with g(0) = 1 and deg(g) = e. Write g(T ) = 1 − g1T + g2T − g3T + ··· and 2 f(T ) = f1T + f2T + ··· , and let (i) (i) Gd,e(f, g) = Gd,e(f1, . . . , fd, g1, . . . , ge). Then we must have (i) i d −i kGd,∞(f, g)k ≤ kfk1kgkrr p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 57

m1 me for all r > 1, since in each term fi1 ··· fii g1 ··· ge , we have

m1 me −(m1+2m2+··· ) m1 2m2 kg1 ··· ge k = r (kg1kr) (kg2kr) ··· m1+m2+··· −i α −i d ≤ r sup kgαkr ≤ r kgkr. α Similarly, interpreting g(1) − g(2) as the polynomial with constant coeﬃcient 1 and jth (1) (2) nonconstant coeﬃcient given by gj − gj , we can bound (i) (1) (2) i −i (1) (2) d−1 (1) (2) kGd,∞(f, g − g )k ≤ kfk1r max(kg kr, kg kr) kg − g kr. 1 These bounds remain true for g ∈ O(AA). So we have ∞ X (i) i D(f, g) = 1 + Gd,∞(f, g)T ∈ A T i=1 J K but in fact skfk i kG(i) (f, g)ksi ≤ kgkd 1 d,∞ r r 1 for any r > 1 (since we assumed g ∈ O(AA) and g(0) = 1), and this goes to 0 as i → ∞ for 1 r > skfk1, so actually D(f, g) ∈ O(AA). 1 In conclusion, if f ∈ A[T ] with deg(f) = d and f(0) = 0, and g ∈ O(AA) with g(0) = 1, 1 then we have found D(f, g) ∈ O(AA) such that g 7→ D(f, g) is continuous (since i (1) (2) (1) (2) (1) (2) d−1 kfk1s kD(f, g − g )ks ≤ kg − g kr max(kg kr, kg kr) sup i r

for any r > 1, say r > kfk1s), and such that if g is a polynomial of degree e, then

D(f, g)(T ) = Gd,e(f, g, T ). Q Note that if A were an algebraically closed ﬁeld, we could write g(T ) in the form (1 − αiT ) Q where αi ∈ L and αi → 0 as i → ∞, and then we would have D(f, g)(T ) = (1 − f(αi)T ). Lemma 14.2.1. D(f, g(1)g(2)) = D(f, g(1))D(f, g(2)). Proof. By continuity, we can reduce to the case where g(1) and g(2) are polynomials. Then we can further reduce to the universal case where A = Z[F1,...,Fd,E1,...,Ee] and e e0 (1) Y (2) Y 0 g (T ) = (1 − AiT ), g (T ) = (1 − AiT ). i=1 i=1 Then since the claim is an equality of polynomials, we can replace A by Frac(A), and thus factorize e e0 (1) Y (2) Y 0 D(f, g )(T ) = (1 − f(Ai)T ),D(f, g )(T ) = (1 − f(Ai)T ), i=1 i=1 e e (1) (2) Y Y 0 D(f, g g )(T ) = (1 − f(Ai)T ) (1 − f(Ai)T ). i=1 i=1 58 COURSE BY RICHARD TAYLOR

Lemma 14.2.2. For A, M/A, ϕ ∈ CCA(M,M), and f ∈ A[T ] with f(0) = 0 as usual, we have f(ϕ) ∈ CCA(M,M) and det(1 − f(ϕ)T ) = D(f, det(1 − ϕT )). The proof is standard given what we already have.

14.3. Reciprocal polynomials. Lemma 14.3.1. Let f, g, h ∈ A[T ] have constant coefficient 0 and leading coefficient in A×, with f of degree d. Define f ∗(T ) = T df(1/T ). So if 2 d f(T ) = f0 + f1T + f2T + ··· + fdT then ∗ d d−1 f (T ) = f0T + f1T + ··· + fd. Then (1) g∗(T ) D 1 − , g(T ) = (1 − T )deg g. g∗(0) (2) f ∗(T ) f ∗(T ) D 1 − , g + hf | = D 1 − , g | . f ∗(0) T =1 f ∗(0) T =1

f ∗(T ) × (3) D 1 − f ∗(0) , g |T =1 ∈ A if and only if (f, g) = A[T ].

Q ∗ Proof. (1) We reduce to the universal case, where g(T ) = (1 − AiT ) and g (T ) = Q (T − Ai). Then

∗ Q !! g (T ) Y j(Ai − Aj) Y D 1 − , g(T ) = 1 − T 1 − = (1 − T ). g∗(0) ± Q A i j j i (2) This is because f ∗(T ) D 1 − , g | = f ∗R (f, g) f ∗(0) T =1 d d,deg g

where Rd,deg g(f, g) is the resolvent of f and g. Looking up how the resolvent works and ﬁlling in the details is an exercise. (3) Same as Part (2). Lemma 14.3.2. Let A be a commutative Banach algebra, f(T ) ∈ A[T ] such that f(0) = 1 and × 1 f ∗(T ) the leading coeﬃcient of f is in A , and g ∈ O(AA) with g(0) = 1. Then D 1 − f ∗(0) , g |T =1 × 1 is in A if and only if (f, g) = O(AA). 1 Proof. Let g = qf + r where r ∈ A[T ] with deg(r) < deg(f) and q ∈ O(AA). Then by Part (2) of Lemma 14.3.1 and continuity, we have f ∗(T ) f ∗(T ) D 1 − , g | = D 1 − , r | f ∗(0) T =1 f ∗(0) T =1 p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 59

and the RHS is in A× if and only if (f, r) = A[T ], by Part (3) of Lemma 14.3.1. If (f, r) = A[T ] 1 1 then f and r also generate O(AA) when viewed in that ring, in which (f, g) = (f, r) = O(AA). 1 1 Conversely, if (f, g) = O(AA), then we can ﬁnd λ, µ ∈ O(AA) with λf + µr = 1, and compute f ∗(T ) f ∗(T ) 1 = D 1 − , 1 = D 1 − , λf + µr | f ∗(0) f ∗(0) T =1 f ∗(T ) f ∗(T ) f ∗(T ) = D 1 − , µr | = D 1 − , µ | D 1 − , r | f ∗(0) T =1 f ∗(0) T =1 f ∗(0) T =1

f ∗(T ) × so indeed D 1 − f ∗(0) , g |T =1 ∈ A . 14.4. Adjugates.

Lemma 14.4.1. Let A, M/A, and ϕ ∈ CCA(M,M) be as before. Let ∞ X n det(1 − ϕT ) = 1 + tn(ϕ)T . n=1 Then the formal power series ∞ n X n X m P (T, ϕ) = T tn−m(ϕ)ϕ , n=0 m=0 Pn m where the coeﬃcient m=0 tn−m(ϕ)ϕ lies in A[ϕ] ⊂ LA(M,M), has the following properties: (1) If M is ﬁnite free over A, then P (T, ϕ) is the adjugate matrix of 1 − T ϕ. Pn m n (2) For all r ∈ R>0, k m=0 tn−m(ϕ)ϕ k r → 0 as n → ∞. (3) (1 − T ϕ)P (T, ϕ) = det(1 − T ϕ) id. Proof. (3) is a formal computation: ∞ n n−1 ! ∞ X n X m X m X n (1 − T ϕ)P (T, ϕ) = T tn−m(ϕ)ϕ − tn−1−m(ϕ)ϕ = T tn(ϕ) id . n=0 m=0 m=0 n=0 So this will remain true whenever we know it makes sense. This implies (1), since in that case everything is a polynomial. For (2), we reduce to when M is Banach-free, with basis ⊥I,0 (A, k · k). Then we have sup kϕijk = {rα} i

for a sequence r1 ≥ r2 ≥ r3 ≥ · · · → 0 (note that all but countably many must equal 0).

Choose jα so that rα = supi kϕijα k. We claim that

n X m tn−m(ϕ)ϕ ≤ max(r1 ··· rn−1kϕk, ktn(ϕ)k). m=0

Proof: Let Jm = {j1, . . . , jm}. We have πJm ◦ ϕ → ϕ as m → ∞. It suﬃces to prove the

inequality for πJm ◦ ϕ, so replace ϕ by πJm ◦ ϕ. Then n n ! X X t (ϕ)ϕm ≤ max kt (ϕ)k, kϕk t (ϕ)(ϕ ◦ π )m−1 . n−m n n−m Jm m=0 m=1 60 COURSE BY RICHARD TAYLOR

⊕Jm Since ϕ ◦ πJm ∈ EndA(A ), we have now reduced to the case of matrices that have ﬁnite dimensional domain as well as range. But if #I < ∞ then the adjugate matrix of (1 − T ϕ) is given by determinants of cofactors (Cramer’s rule), so the coeﬃcient of T n is a sum of terms that are products of n “ϕij”s, so indeed n n−1 X X t (ϕ)(ϕ ◦ π )m−1 = t (ϕ)(ϕ ◦ π )m ≤ r ··· r . n−m Jm n−1−m Jm 1 n−1 m=1 m=0

15. 2/25/20: Eigenspaces, and an introduction to locally analytic p-adic manifolds Let A be a commutative noetherian Banach algebra, M a Banach-projective Banach A-module, and ϕ ∈ CCA(M,M). Previously, we deﬁned 1 det(1 − T ϕ) ∈ O(AA). We also deﬁned the adjugate matrix P (T, ϕ) ∈ 1 , where A[ϕ] ∈ (M,M), and O AA[ϕ] LA showed that

(1 − T ϕ)P (T, ϕ) = det(1 − T ϕ) idM . 15.1. Roots of characteristic power series. Lemma 15.1.1. If det(1 − aϕ) ∈ A× for a given a ∈ A, then 1 − aϕ is an isomorphism.

−1 −1 Proof. The inverse of (1 − aϕ) is (1 − aϕ) = det(1 − aϕ) P (a, ϕ). Corollary 15.1.2. If P (T ) ∈ A[T ] with P (0) = 1 and leading coeﬃcient in A×, and if 1 ∗ (P (T ), det(1 − T ϕ)) = O(AA), then P (ϕ) is an isomorphism. ∗ deg P d Recall that by P (T ) we mean T P (1/T ), so if P (T ) = 1 + P1T + ··· + PdT then ∗ d P (T ) = Pd + Pd−1T + ··· + T . The ﬁnite-dimensional intuition for this is that if P (T ) and det(1 − T ϕ) are coprime, then the inverses of the eigenvalues of ϕ are not roots of P , so the eigenvalues of ϕ are not roots of P ∗, so P ∗(ϕ) should have no kernel.

P ∗(ϕ) Proof. Let ψ = 1 − P ∗(0) , a polynomial in ϕ with no constant term, hence an element of CCA(M,M). Then we have P ∗(T ) det(1 − T ψ) = D 1 − , det(1 − T ϕ) . P ∗(0) 1 × By Lemma 14.3.2 from last time, (P (T ), det(1 − T ϕ)) = O(AA) implies that det(1 − ψ) ∈ A , P ∗(ϕ) ∗ so 1 − ψ = P ∗(0) is invertible, so P (ϕ) is an isomorphism. Lemma 15.1.3. Let a ∈ A. Suppose d k det(1 − T ϕ)| = 0 dT T =a for k = 0, . . . , h − 1, but d h det(1 − T ϕ)| ∈ A×. dT T =a p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 61

(Note that the hth derivative being a unit makes this a stronger condition than saying that

det(1 − T ϕ) has a root at a of multiplicity h.) Then we can find e ∈ A[ϕ] ⊂ LA(M,M) such that • e2 = e, • eM is finite projective over A of rank h, • (1 − aϕ)heM = (0), × • (1 − aϕ) is an isomorphism on (1 − e)M and det(1 − aϕ|(1−e)M ) ∈ A , and • if h > 0 then a ∈ A×. The finite-dimensional intuition for this is that we are isolating the subspace of M of generalized ϕ-eigenvalue a, that is, finding the Jordan normal form of ϕ. Proof. We have already done the case h = 0, so assume h > 0. We will use the equation (1 − T ϕ)P (T, ϕ) = det(1 − T ϕ) id . Plugging in a, we get (1 − aϕ)P (a, ϕ) = 0. Differentiating both sides k times with respect to T , we get d k d k−1 d k (1 − T ϕ) P (T, ϕ) − kϕ P (T, ϕ) = det(1 − T ϕ) id . dT dT dT For 1 ≤ k < h, plugging in a gives d k d k−1 (1 − aϕ) P (T, ϕ)| = kϕ P (T, ϕ)| . dT T =a dT T =a This implies inductively that d k (1 − aϕ)k+1 P (T, ϕ)| = 0. dT T =a On the other hand, for k = h, we get d h d h−1 (1 − aϕ) P (T, ϕ)| − hϕ P (T, ϕ)| ∈ A× id . dT T =a dT T =a

Rescaling and assigning terms appropriately, this gives us ψ1, ψ2 ∈ A[ϕ] such that (1 − h aϕ)ψ1 + ψ2 = 1 and (1 − aϕ) ψ2 = 0. Furthermore, we can write h h (1 − aϕ) ψ1 + ψ2ψ3 = 1

for some ψ3 ∈ A[ϕ] which we can ﬁnd by writing ψ2 = (1 − aϕ)ψ1 − 1 and solving for ψ3 (or h by raising (1 − aϕ)ψ1 + ψ2 = 1 to the hth power). Let ψ2ψ3 = e. Then indeed (1 − aϕ) e = 0, and also h h (1 − e)e = (1 − aϕ) ψ1 ψ2ψ3 = 0, that is, e2 = e. This also means that (1 − e)2 = (1 − e), or h−1 h (1 − aϕ)((1 − aϕ) ψ1 )(1 − e) = 1 − e, h−1 h which is to say that (1 − aϕ) is an isomorphism on (1 − e)M with inverse (1 − aϕ) ψ1 . Also if ψ is such that

(1 − aϕ|(1−e)M )(1 − ψ) = id(1−e)M , 62 COURSE BY RICHARD TAYLOR

then we can solve

−aϕ|(1−e)M − ψ + aϕ|(1−e)M ψ = 0

ψ = ϕ(1−e)M (aψ − a) ∈ CCA((1 − e)M, (1 − e)M), from which we conclude that det(1 − ψ) is a well-defined element of A such that det(1 − × aϕ|(1−e)M ) det(1 − ψ) = 1, and hence det(1 − aϕ|(1−e)M ) ∈ A . It remains to check that eM is finite projective of rank h and that a ∈ A× if h > 0. From h (1−aϕ) e = 0, we see that e = haϕe+··· is ϕ composed with something, so e ∈ CCA(M,M). Let fi be a sequence of finite rank maps such that fi → e. Then

efie ∈ CCA(eM, eM),

and efie is finite rank and goes to ideM . Therefore we can find g = 1 − efie ∈ LA(eM, eM) arbitrarily small with 1−g of finite rank. But 1−g is invertible with inverse 1+g +g2 +··· for g sufficiently small, so eM must be finitely generated over A. From M = eM ⊕ (1 − e)M, we see that eM is Banach-projective; by Lemma 12.2.3 we conclude that it is actually projective over A. If eM = 0, we have det(1 − aϕ) ∈ A×, so h = 0. If not, we have

h 0 = tr(1 − aϕ|eM ) = tr(1 − a(hϕ|eM + ··· )) = rank(eM) + ab for some b ∈ A; since rank(eM) 6= 0, we conclude that a ∈ A×. Now we can compute rank(eM). Let f(T ) = det(1 − T ϕ|eM ) ∈ A[T ] and g(T ) = det(1 − × × T ϕ|(1−e)M ). We have f(T )g(T ) = det(1 − T ϕ) and g(a) ∈ A . Since a ∈ A , by polynomial −1 h1 division, we can write f(T ) = (1 − a T ) f1(T ) with f1(a) 6= 0. So

−1 h1 det(1 − T ϕ) = (1 − a T ) f1(T )g(T ). Therefore d k det(1 − T ϕ)| = 0 dT T =a

for k = 0, . . . , h1 − 1, and d h1 det(1 − T ϕ)| = h !(−a−1)h1 f (a)g(a), dT T =a 1 1

h1 where h1!(−a−1) is a unit, f1(a) 6= 0, and g(a) is a unit. We conclude that h1 = h and × f1(a) ∈ A . If A were a ﬁeld, we would have charpoly (T ) = (T − a)rank(eM). Instead, we can write ϕ|eM charpoly (T ) ≡ (T − a)rank(eM) (mod m) ϕ|eM f(T ) ≡ (1 − a−1T )rank(eM) (mod m)

−1 h1 for all maximal ideals m of A. But we also have f(T ) = (1 − a T ) f1(T ), so

−1 rank(eM)−h1 f1(T ) ≡ (1 − a T ) (mod m)

× for all maximal ideals m of A. But f1(a) ∈ A , so this is only possible if rank(eM) = h1 = h, −1 h as desired. Note that this means we also have det(1 − T ϕ|eM ) = (1 − T a ) . p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 63

15.2. Polynomial factors of characteristic power series. Now we would like to generalize the previous discussion beyond powers of linear polynomials. Lemma 15.2.1. Let P (T ) ∈ A[T ] with P (0) = 1 and leading coefficient in A×. (Remember that the leading coefficient restriction we keep putting in is automatic if A is a field.) Let 1 Q(T ) ∈ O(AA) such that P (T )Q(T ) = det(1 − T ϕ). 1 Suppose that (P,Q) = O(AA). Then there is a unique e ∈ A[ϕ] ⊂ LA(M,M) such that • e2 = e, • P ∗(ϕ)eM = (0), and • P ∗(ϕ) is an isomorphism on (1 − e)M. Furthermore, • rank(eM) = deg P , • det(1 − T ϕeM ) = P (T ), and • det(1 − T ϕ|(1−e)M ) = Q(T ).

Proof. Uniqueness: suppose e1 and e2 both satisfy the given conditions. Since both lie in A[ϕ], which is commutative, they must commute. Then we can expand

M = e1e2M ⊕ e1(1 − e2)M ⊕ (1 − e1)e2M ⊕ (1 − e1)(1 − e2)M. ∗ Since P (ϕ) is both 0 and invertible on e1(1 − e2)M, we ﬁnd that e1(1 − e2)M = 0, so e1 = e1e2. Similarly using (1 − e1)e2M, we ﬁnd that e2 = e1e2 as well, so e1 = e2. P ∗(ϕ) Existence: again let ψ = 1 − P ∗(0) = CCA(M,M). Then P ∗(T ) det(1 − ψT ) = D 1 − , det(1 − ϕT ) P ∗(0) P ∗(T ) P ∗(T ) = D 1 − ,P D 1 − ,Q P ∗(0) P ∗(0) P ∗(T ) = (1 − T )deg P D 1 − ,Q P ∗(0)

P ∗(T ) where we know that D 1 − P ∗(0) ,Q evaluated at 1 is a unit in A. Then by Lemma 15.1.3 2 deg P for a = 1, we can ﬁnd e ∈ A[ψ] ⊂ A[ϕ] ⊂ LA(M,M) such that e = e, (1 − ψ) eM = 0 (that is, P ∗(ϕ)deg P eM = 0), (1 − ψ) is an isomorphism on (1 − e)M, P ∗(ϕ) is an isomorphism × on (1 − eM), det(1 − ψ|(1−e)M ) ∈ A , and rank(eM) = deg P . ∗ We still need to check that P (ϕ)eM = 0, det(1−T ϕ|eM ) = P (T ), and det(1−T ϕ|(1−e)M ) = Q(T ). We have P ∗(T ) det(1 − ψ| ) = D 1 − , det(1 − T ϕ| ) (1). (1−e)M P ∗(0) (1−e)M 1 Since the LHS is a unit, we conclude that (P (T ), det(1 − T ϕ|(1−e)M )) = O(AA). So we can 1 ﬁnd λ, µ ∈ O(AA) with λP + µ det(1 − T ϕ|(1−e)M ) = 1.

Multiplying by det(1 − T ϕ|eM ), we get

P (λ det(1 − T ϕ|eM ) + µQ) = det(1 − T ϕ|eM ) 64 COURSE BY RICHARD TAYLOR

1 so P divides det(1 − T ϕ|eM ) in O(AA), hence in A[T ]. But det(1 − T ϕ|eM ) has degree at most rank(eM) = deg P and is nonzero, and both constant terms are 1, so in fact ∗ P (T ) = det(1 − T ϕ|eM ). Then Cayley-Hamilton gives P (ϕ)eM = 0. Finally, we have

P (T )Q(T ) = det(1 − T ϕ) = det(1 − T ϕ|eM ) det(1 − T ϕ|(1−e)M )

so P (T )(Q(T ) − det(1 − T ϕ|(1−e)M )) = 0, so Q(T ) = det(1 − T ϕ|(1−e)M ). Intuitively, if A = L is a ﬁeld, we can factor

Y ei det(1 − T ϕ) = Pi(T ) i where Pi(T ) is an irreducible polynomial in L[T ] with Pi(0) = 1, Pi 6= Pj if i 6= j, and if αi is a root of Pi and i < j then |αj| < |αi|. In this case we can easily ﬁnd Mi ⊂ M such that ∗ ei 0 Mi is preserved by ϕ and has rank (deg Pi)ei, Pi (ϕ) = 0 on Mi, and M = Mi ⊕ Mi . For general A, we are trying to interpolate what happens at the various maximal ideals of A, with varying degrees of success.

Lemma 15.2.2. Let M1,M2 be Banach projective Banach A-modules, ϕi ∈ CCA(Mi,Mi) for i = 1, 2, and ψ ∈ LA(M1,M2) such that ψ ◦ ϕ1 = ϕ2 ◦ ψ. Suppose that

det(1 − ϕiT ) = Pi(T )Qi(T ) × where Pi(T ) ∈ A[T ], Pi(0) = Qi(0) = 1, Pi(T ) has leading coeﬃcient in A , and (P1P2,Q1Q2) 1 = O(AA). Take ei as in Lemma 15.2.1. Then ψ ◦ e1 = e2 ◦ ψ.

Proof sketch. Let M = M1 ⊕ M2 and ϕ = ϕ1 ⊕ ϕ2 ∈ CCA(M,M). One may show that 0 0 ϕ1 0 e = e1 ⊕ e2 ∈ A[ϕ]. Then one concludes that commutes with = ϕ, hence ψ 0 0 ϕ2 e1 0 with . 0 e2

15.3. Locally analytic p-adic manifolds and groups. References: Bourbaki, General topology ([2]) for manifolds and Lie groups and Lie algebras ([1]) for groups. Let X be a second countable (meaning that it has a countable basis) Hausdorﬀ topological space. By a p-adic locally analytic chart for X, we mean a pair (U, ϕ) where U ⊂ X is open and ∼ n ϕ : U −→ ϕ(U) ⊂ Qp n is a homeomorphism from U onto an open subset ϕ(U) of Qp . We call two charts (U, ϕ) and (U 0, ϕ0) compatible if ϕ0 ◦ ϕ−1 : ϕ(U) → ϕ0(U) is locally analytic, meaning that in a neighborhood of any point it is given by a convergent power series. S An n-dimensional atlas A = {(Ui, ϕi)} is a set of compatible charts with Ui = X. We call two atlases A , A 0 equivalent if A ∪ A 0 is an atlas (i.e. every chart in one is compatible with every chart in the other). This is an equivalence relation.

Deﬁnition 15.3.1. A locally analytic manifold of dimension n over Qp is a pair (X, [A ]) where [A ] is an equivalence class of atlases. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 65

n n (Note that we can also do this for L0 in place of Qp for any finite extension L0/Qp. We will always work over Qp, but most things work fine for L0.) Such a space X is locally compact and paracompact, so we can find an atlas A = ` n {(Ui, ϕi)}i∈I such that I is countable, X = i Ui, and ϕiUi = Zp for all i. We will call such an atlas strict. 0 We say that a strict atlas A = {(Vj, ψj)} is a refinement of a strict atlas A = {(Ui, ϕi)} if for all j there is some i such that Vj ⊂ Ui, and −1 ϕi ◦ ψj : ψj(Vj) → ϕi(Ui) n m m n takes x ∈ Zp to a + p x for some a ∈ (Z/p Z) , where m is independent of j. Any strict A n has a proper refinement by cutting up Zp into residue discs. Next we will discuss locally analytic functions and maps on such spaces, and their duals.

16. 2/27/20: Functions and distributions on locally analytic manifolds and groups n Last time, we defined p-adic locally analytic manifolds X with charts to Qp . We also defined ` “strict atlases”, those of the form A = {(Ui, ϕi)}i∈I such that I is countable, X = Ui, and ∼ n ϕi : Ui −→ D(1) . 16.1. Locally analytic functions on manifolds. We wish to define what it means for a map X → V , where V is a LCTVS, to be locally analytic.

∗ Deﬁnition 16.1.1. An augmented atlas is a collection A = {(Ui, ϕi,Vi ,→ V )}i∈I where A = {(Ui, ϕi)}i∈I is a strict atlas for X, Vi is a Banach space, and each Vi ,→ V is continuous linear. Now let Y n LAA ∗ (X,V ) = O(D(1)) ⊗b πVi i where, explicitly,   n  X m  O(D(1)) ⊗b πVi = xmT : xm ∈ Vi | kxmkVi → 0 as m → ∞ m=(m1,...,mn) 

m m1 mn where T := T1 ··· Tn , and the norms are given by X m xmT = max kxmkV m i (because we can write ∼ ⊥J,0 (L, | · |)⊗b πV =⊥J,0 (V, k · kV ) if V is a Banach space). We have a map

LA ∗ (X,V ) → LA(X,V ) := lim LA ∗ (X,V ). A −→ A A ∗ Also, using the maps n n O(D(1)) ⊗b πVi → C(Zp ,Vi), we get the inclusion Y LAA ∗ (X,V ) → CA (X,V ) = C(Ui,V ). i 66 COURSE BY RICHARD TAYLOR ` Here if A = {(Ui, ϕi)}, so that X = Ui and Ui ⊂ X is open compact, the topology on Q CA (X,V ) = i C(Ui,V ) is given by the norms

kfkΩ,k·k = sup kf(ω)k ω∈Ω

for all Ω ⊂ X compact and seminorms k · k on V . Actually, CA (X,V ) is independent of A as an LCTVSs, which you can see by checking that the deﬁnition is preserved by reﬁnements. So we get a map

LA(X,V ) → C(X,V ) := CA (X,V ). If X is compact, C(X,V ) recovers our previous definition. Then the spaces LA(X,V ) are LCTVSs of compact type, since if r > 1 then O(D(r))n → O(D(1))n is compact. 16.2. Locally analytic distributions. The space of (locally analytic) distributions on X (of compact support, though the literature doesn’t mention that part) is la ∨ Dc (X,L) = LA(X,L)b . We have a continuous bijection M LA(X,L)∨ → lim (D(1)n)∨. b ←− O b A (Ui,ϕi)∈A la If X is compact, Dc (X,L) is then a nuclear Fréchet space. We can also define cts ∨ Dc (X,L)s/b = C(X,L)s/b. These are called “measures” (of compact support).

cts Example 16.2.1. For any x ∈ X, we have the evaluation measure δx ∈ Dc (X,L) given by δx(f) = f(x). ` Lemma 16.2.2. If X = i∈I Xi where Xi ⊂ X is compact open, we can write Y C(X,V ) = C(Xi,V ) I Y LA(X,V ) = LA(Xi,V ) I cts M cts Dc (X,L)b/s = Dc (Xi,L)b/s I la M la Dc (X,L) = Dc (Xi,L). I la cts Corollary 16.2.3. (1) Dc (X,L) and Dc (X,L)b are bornological and barrelled. la cts (2) hδx | x ∈ XiL is dense in Dc (X,L) and in Dc (X,L)s. Proof. (1) The properties of being bornological and barrelled are preserved under L. (2) Reduce to the case where X is compact. If the claim is false, then by Hahn-Banach, there is 0 6= ϕ : D → L continuous linear such that ϕ(δx) = 0 for all x ∈ X. But in both cases under consideration dualization is reﬂexive (dualizing again gives back what we started with), so there is f ∈ LA(X,L) or C(X,L)s such that ϕ(δ) = δ(f) for all δ. Then so f(x) = 0 for all x ∈ X, so f ≡ 0, so ϕ ≡ 0. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 67

We have maps la Dc (X,L) ⊗ι LA(X,V ) → V cts Dc (X,L) ⊗ι C(X,V ) → V for V complete. This is not obvious because the deﬁnition of LA is complicated, containing both a direct limit and a product, which don’t commute. To construct the map for LA (C is easier and left as an exercise), we start with the natural map n ∨ n O(D(1) )b ⊗π (O(D(1) )⊗b πVi) → Vi where we can put ⊗b instead of ⊗ because V is complete. This gives a map n ∨ Y n O(D(1) )b ⊗π O(D(1) )⊗b πVj → Vi ,→ V ∗ (Uj ,ϕj ,Vj ,→V )∈A n ∨ where O(D(1) )b acts by pairing with the ith factor. Using the general fact that (lim W ) ⊗ U ∼ lim(W ⊗ U), −→ i ι = −→ i ι we can combine these to get   ! M n ∨ Y n  O(D(1) )b  ⊗ι O(D(1) )⊗b πVi → V. (Ui,ϕi) i But we also have a map M M Dla(X,L) → lim (D(1)n)∨ → (D(1)n)∨ c ←− O b O b A (Ui,ϕi)∈A (Ui,ϕi) la so putting in Dc (X,L) and taking limits, we get !! Y lim Dla(X,L) ⊗ (D(1)n)⊗ V → V −→ c ι O b π i A ∗ i la Dc (X,L) ⊗ι LA(X,V ) → V which is our desired map.

16.3. Maps between manifolds, products, and tangent spaces. Deﬁnition 16.3.1. Let X,Y be locally analytic manifolds. We say that f : X → Y is locally analytic if for one (therefore all) atlases B = {(Vj, ψj)} for Y , the maps −1 n ψj ◦ f : f Vj → Qp are locally analytic. Deﬁnition 16.3.2. If f : X → Y is locally analytic, we have a pullback map f ∗ : LA(Y,V ) → LA(X,V ) g 7→ g ◦ f.

∗ We can similarly deﬁne f : C(Y,V ) → C(X,V ) (even if f is just continuous), f∗ : la la cts cts Dc (X,L) → Dc (Y,L), and f∗ : Dc (X,L)b/s → Dc (Y,L)b/s. 68 COURSE BY RICHARD TAYLOR

If j : U,→ X is an open set, we also have the extension-by-0 maps

j! : LA(U, V ) → LA(X,V ), j! : C(U, V ) → C(X,V ) where a function on U is extended to be identically 0 outside U, and similarly the restriction maps la la cts cts Dc (X,L) → Dc (U, L),Dc (X,L)s/b → Dc (U, L)s/b given by δ 7→ δ|U . Lemma 16.3.3. (1) We have an isomorphism n m ∼ n+m O(D(1) )⊗b πO(D(1) ) −→ O(D(1) ). (2) We have maps

C(X,L) ⊗π C(Y,L) → C(X × Y,L) (which is a isomorphism if X,Y compact) and

LA(X,L) ⊗ι LA(Y,L) → LA(X × Y,L) such that (f ⊗ g)(x, y) = f(x)g(y). (3) We have a map n ∨ m ∨ n+m ∨ O(D(1) )b ⊗b πO(D(1) )b → O(D(1) )b and if X,Y are compact then we have a map ∨ ∨ ∨ C(X,L)b ⊗b πC(Y,L)b → C(X × Y,L)b . (4) We have maps la la la ⊗b : Dc (X,L) ⊗ι Dc (Y,L) → Dc (X × Y,L) cts cts cts ⊗b : Dc (X,L)b/s ⊗ι Dc (Y,L)b/s → Dc (X × Y,L)b/s such that (δ⊗b)((x, y) 7→ f(x)g(y)) = δ(f)(g).

Proof. (1) This follows from ⊥I,0 (L, | · |)⊗b πV =⊥I,0 (V, k · kV ). (2) Exercise. ∨ ∨ ∨ (3) This follows from the general fact that Vb ⊗π Wb → (V ⊗b πW )b . (4) For the LA case (as usual, C is easier) we have maps ! ! M M Dla(X,L) ⊗ Dla(Y,L) → lim (D(1)n)∨ ⊗ lim (D(1)m)∨ c ι c ←− O b ι ←− O b A i B j ! !! M M → lim (D(1)n)∨ ⊗ (D(1)m)∨ ←− O b ι O b A ,B i j M = lim ( (D(1)n)∨ ⊗ (D(1)n)∨) ←− O b ι O b A ,B i,j !∨ M Y → lim (D(1)n+m)∨ = lim (D(1)n+m) . ←− O b ←− O , , A B i,j A B i,j b la The last space takes a map from Dc (X,Y ) which is a continuous bijection, but not necessarily a homeomorphism. That gives us a linear map of the desired form; now p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 69

we have to back up and check that it is separately continuous. So we need to check la that if δ ∈ Dc (X,L), then la la (−⊗bδ): Dc (Y,L) → Dc (X × Y,L) is continuous. Dualizing, this becomes a map LA(X × Y,L) → LA(Y,L). The source Q n+m is a limit of terms of the form i,j O(D(1) ), whose map to LA(Y,L) factors Q m through j O(D(1) ), so we need the jth factor of this map to be continuous for all B and j. But it turns out (check yourself) that the jth factor can be further rewritten as M Y n m Y n+m m δA ,i ⊗ id : δA ,i : O(D(1) )⊗b πO(D(1) ) = O(D(1) ) → O(D(1) ) i i L which is continuous since δA = δA ,i is continuous. n Deﬁnition 16.3.4. If X has an atlas A with a chart (Ui, ϕi), so that ϕi : Ui ,→ Qp , and x ∈ Ui, we deﬁne n Tx,(Ui,ϕi)X = Qp .

If (Uj, ϕj) is a compatible chart also containing x, we get a map −1 ∼ d(ϕi ◦ (ϕj) )x : Tx,(Uj ,ϕj )X −→ Tx,(Ui,ϕi)X by diﬀerentiating −1 ϕi ◦ (ϕj) : ϕj(Ui ∩ Uj) → ϕi(Ui ∩ Uj).

Identifying all the Tx,(Ui,ϕi)Xs via these maps gives us TxX, the tangent space to X at x. If we have a map f : X → V where V is a LCTVS, or a locally analytic map f : X → Y , we get corresponding diﬀerential maps dfx : TxX → V and dfx : TxX → Tf(x)Y . 16.4. Locally analytic groups. A locally analytic group G is a locally analytic manifold plus an identity element e ∈ G and locally analytic inverse and multiplication maps −1 : G → G, µ : G × G → G, satisfying the group axioms.

Example 16.4.1. GLn(Qp), or G(L) for any reductive group G over L.

We define Lie(G) := TeG. There is a bracket [−, −] : Lie(G) × Lie(G) → Lie(G) making Lie(G) a Lie algebra, defined as follows. If (U, ϕ) is a chart with e ∈ U and ϕ(e) = 0, and V ⊂ U is an open set such that V 2 ⊂ U, then n m : ϕ(V ) × ϕ(V ) → Qp (x, y) 7→ ϕ(ϕ−1(x)ϕ−1(y)) P is locally analytic, so given by a power series near 0. If m = mi,j, where mi,j is the bidegree i, j term in the power series, then we define

[x, y] := m1,1(x, y) − m1,1(y, x) ∈ TeG. This is independent of choices. If U ⊂ Lie(G) is a suﬃciently small neighborhood of 0, then there is an exponential map exp : U → G with exp(0) = e, d exp = id : Lie(G) → Lie(G), and exp((a + b)x) = 70 COURSE BY RICHARD TAYLOR

exp(ax) exp(bx) for all x ∈ U and a, b ∈ L suﬃciently small. Any two such maps agree near 0. Near e, exp has an inverse called log. If x, y ∈ Lie(G), we have log(exp(ax) exp(ay)) x + y = lim a→0 a log(exp(ax) exp(ay) exp(−ax) exp(−ay)) [x, y] = lim . a→0 a2 The following lemma can be found in Bourbaki.

Lemma 16.4.2. There are charts (Hm, ϕm) such that

• Hm is an open compact subgroup of G, •{ Hm} is a basis of neighborhoods of e ∈ G, • the group structure on Hm is analytic, that is, there are continuous algebra homomorphisms n n n µm : O(D(1) ) → O(D(1) )⊗b πO(D(1) ) n n im : O(D(1) ) → O(D(1) ) m such that for any g1, g2 ∈ Hm and f ∈ O(D(1) ),

f(ϕm(g1g2)) = µm(f)(ϕm(g1), ϕm(g2)) −1 f(ϕm(g )) = im(f)(ϕm(g)). Deﬁnition 16.4.3. The convolution product la la la ∗ : Dc (G, L) ⊗ι Dc (G, L) → Dc (G, L) is given by the composition

la la la m∗ la Dc (G, L) ⊗ι Dc (G, L) → Dc (G × G, L) −→ Dc (G, L).

17. 3/3/20: Distributions and representations for locally analytic groups Last time, we started discussing locally analytic p-adic groups. If G is such a group, we saw that we had maps exp : Lie(G) → G and log : G → Lie(G), and that we can ﬁnd p-adic ∼ ⊕n analytic subgroups Hm ⊂ G, with Hm = Zp , on which the group structure is given by convergent power series. N ∼ n2 Example 17.0.1. In G = GLn(Qp), we have neighborhoods 1 + p Mn×n(Zp) = Zp of the N identity, with the following group structure: matrices in 1 + p Mn×n(Zp) multiply as (1 + pN A)(1 + pN B) = 1 + pN (A + B + pN AB)

n2 so the group structure on Zp is given by (A, B) 7→ A + B + pN AB. Fact: if G is compact, then G ∼= lim G/H, so ←−H

For example, G = GLn(Zp) is compact and contains the open pro-p subgroup 1 + pMn×n(Zp). p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 71

la 17.1. Distributions on locally analytic groups. Last time, we deﬁned algebras Dc (G, L) cts and Dc (G, L)b/s, with multiplication given by convolution, that is,

⊗b m∗ ∗ : D(G, L) ⊗ι D(G, L) −→ D(G × G, L) −→ D(G, L). They also have an anti-involution

(−1) † : D(G, L) −−−→∗ D(G, L) −1 † (where (−1) : G → G is g 7→ g ). We have δgδh = δgh and δg = δg−1 . Also, ∗ is associative † † † and distributive over addition, with identity δ1, and we have the identity (δ ∗ ) = ∗ δ . la cts If G is compact, Dc (G, L) is a Fr´echet algebra, and Dc (G, L)b is a Banach algebra. We have a map la Lie(G) → Dc (G, L) δ − 1 x 7→ lim exp(ax) a→0 a which takes [x, y] to x ∗ y − y ∗ x, hence extends to a map from the universal enveloping algebra U(Lie(G)). If H < G is an open subgroup then

la/cts M la/cts Dc (G, L) = δg ∗ Dc (H,L). g∈G/H

la/cts la/cts (Note that δg ∗ Dc (H,L) = Dc (gH, L).) This is useful because it allows one to reduce questions to the case where G is compact. If G is compact, hence proﬁnite, we can deﬁne the completed group algebra G = lim [G/H], O ←− O J K HCG open normal with the inverse limit topology, and

[ −n L G = O G ⊗O L = π O G J K J K n J K (note that this means we are requiring bounded denominators), with the direct limit topology. Usually, O G is not open in L G (certainly not if G is inﬁnite). J K J K Lemma 17.1.1. If G is compact, Dcts(G, L) ∼= L G . The strong topology on Dcts(G, L) corresponds to the normable topology on L G with OJ GK as a bounded open lattice ( not with its usual topology, but with the p-adic topology).J K The weakJ K topology on Dcts(G, L) corresponds to the usual topology on L G . J K Proof. Let µ ∈ O G , say J K X µ = lim µ gH ←− gH HCG g∈G/H where µ ∈ is such that if H ⊂ H then µ = P µ . Given f ∈ C(G, L), we gH O 1 2 gH2 h∈H2/H1 ghH1 would like to “integrate f against µ”. P Certainly for each H the sum g∈G/H µgH f(g) makes sense. We claim that this is a Cauchy sequence as H and the choice of coset representatives vary. Proof: given > 0, there is 72 COURSE BY RICHARD TAYLOR

an open compact subgroup H C G such that f varies by less than on each coset of H. If H > H1 > H2, then X X X X X 0 0 µgH1 f(g) − µgH2 f(g) = µghH2 f(g) − µg H2 f(g ) 0 g∈H1\G g∈H2\G g∈H1\G h∈H2\H1 g ∈H2\G X X = µghH2 (f(g) − f(gh)).

g∈H1\G h∈H2\H1 0 Here |µghH | ≤ 1 and |f(g) − f(g h)| ≤ , so the whole thing has norm ≤ . 2 P So the sums µgH f(g) approach a limit µ(f). Furthermore, if kfk ≤ , then |µ(f)| ≤ , so µ ∈ Dcts(G, L). This gives us a map cts O G → Dc (G, L) J K cts which extends naturally to L G . If δ ∈ Dc (G, L), then it has an inverse image J K X µ = lim δ(1 )gH ∈ L G ←− gH g∈H\G J K

(because 1gH has bounded denominator as g, H vary). The strong topology on Dcts(G, L) has open unit ball consisting of the distributions sending O-valued functions into O. This just means that the distribution sends all functions of the form 1gH to O, which is to say that it is in O[[G]]. For the weak topology, take a typical open set Λ({f}, O). If N is such that kfk < |π|−N and H C G is an open normal subgroup such that f varies by less than 1 on the cosets of H (so if g1, g2 are in the same coset then |f(g1) − f(g2)| ≤ 1), then the unit ball Λ({f}, O) contains the preimage of πN O[G/H] ⊂ O[G/H]. On the other hand, that preimage contains 1 N Λ({ gH }g∈G/H , π O). So the two topologies are the same. Theorem 17.1.2. Let G be compact and locally analytic. (1) (Lazard) O G and L G are noetherian. J K J K la (2) (Schneider-Teitelbaum [9]) Dc (G, L) is a Fréchet-Steinalgebra and cts la (Dc (G, L) = L G ) → Dc (G, L) J K is faithfully flat. 17.2. Representations of locally analytic groups. G is still locally analytic. Definition 17.2.1. By a Banach representation of G we mean a Banach space V/L with an action of G (in particular a map G × V → V ) which is separately continuous. That is, for all g ∈ G, the map g : V → V is in L (V,V ), and for all x ∈ V , the map G → V , g 7→ gx, is continuous. One can check that this is equivalent to saying that the map π : G → Ls(V,V ) is continuous (and, of course, satisfies π(gh) = π(g)π(h)). Lemma 17.2.2. If V is a Banach representation of G, then (1) G × V → V is actually jointly continuous. That is, V → C(G, V ) is continuous, or equivalently, G → Lb(V,V ) is continuous. cts (2) There is a unique jointly continuous action of Dc (G, L)b on V such that δgx = gx. Proof. (1) Let H < G be any compact open subgroup. Then the image of the continuous map H → Ls(V,V ) is compact, hence bounded. By Banach-Steinhaus, the image of H is equicontinuous, that is, for any open lattice Λ ⊂ V , there is a second open p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 73

lattice M ⊂ V such that hM ⊂ Λ for all h ∈ H. That is, the image of H × M is in Λ. This gives us joint continuity at (1, 0), which is all we need. (2) The natural map V → C(G, V ) takes x to (g 7→ gx). By tensoring, this gives us a map cts cts Dc (G, L)b ⊗π V → Dc (G, L)b ⊗π C(G, V ) → V.

It is easy to check that δgx = gx and that (δ ∗ )(x) = δ((x)) (check the latter on cts δ-functions, which are dense in Dc (G, L)b). ∨ −1 Remark 17.2.3. G also acts on Vb , which is also a Banach space, by (gf) = f ◦ g , and cts ∨ this action is separately, therefore jointly, continuous. Similarly, Dc (G, L) acts on Vb by δf = f ◦ δ†. Definition 17.2.4. We call a Banach space representation V of G admissible if there is a G-invariant bounded open lattice Λ ⊂ V such that if H < G is an open subgroup, then H HomO ((V/Λ) , L/O) is a finitely generated O-module. That is, (V/Λ)H ∼= (L/O)h ⊕ (a finite cardinality O-module), where h < ∞. This property of (V/Λ)H is sometimes called being “cofinite” over O.

Let MO be the category of compact Hausdorﬀ linearly topologized torsion-free O-modules (“linearly topologized ” means the topology has a basis consisting of translates of O- submodules). The reference for the following results is Schneider-Teitelbaum ([8] and [7]). Q Lemma 17.2.5. The objects of MO are all of the form I O for some index set I (with the product topology).

Lemma 17.2.6. Let BanL be the category of Banach spaces over L. We have a contravariant functor

MO → BanL cts M 7→ HomO (M,L) Y O 7→⊥I,0 (L, | · |). I cts Q Proof. To be precise about the last line, the isomorphism between HomO ( I O,L) and ⊥I,0(L, | · |) is given by f 7→ (f(ei)), where ei is the ith standard basis element. This map −1 N Q makes sense because if f is continuous and O-linear then f (π O) contains i/∈J O for N some J ⊂ I with #J < ∞, so for i∈ / J, f(ei) ∈ π O. This functor induces an anti-equivalence of categories

MO ⊗ L → BanL

(where MO ⊗ L has the same objects as MO but the homomorphisms are tensored up to ∨ L), whose inverse functor takes V ∈ BanL to the unit ball in V considered with the weak topology. Similarly we can deﬁne MO G , the category of topological O G -modules (where O G has the weak topology) which areJ compact,K Hausdorﬀ, O-torsion-free,J K and linearly topologizedJ K 74 COURSE BY RICHARD TAYLOR

over O. Then we get a functor

MO G ⊗ L → {Banach space representations of G} J K M 7→ HomO,cts(M,L). Definition 17.2.7. An O G - or L G - module M (no topology needed) is called coadmissible if for one (hence every) compactJ K openJ K subgroup H < G, M is finitely generated over O H or L H . J K J K coad coad Lemma 17.2.8. (1) ModO G ⊗ L = ModL G . (2) Any coadmissible O G -moduleJ K has a canonicalJ K topology, that is, a unique Hausdorff topology such that OJHK × M → M is continuous for one (hence every) open compact J K subgroup H. If M is O-torsion-free, this makes M an object in MO G . (3) There is an anti-equivalence of categories J K {coadmissible L G -modules} → {admissible Banach space representations of G} J K M 7→ HomL,cts(M,L) ∨ Vs ← V. [ Next time we’ll talk about locally analytic representations, and then the eigencurve!

18. 3/5/20: Locally analytic representations and more on compact operators Last time, we defined admissible Banach representations V of G. We saw that we could ∨ equivalently look at V in terms of the coadmissible L G -module Vs . J K 18.1. Locally analytic representations. Definition 18.1.1. A locally analytic representation V of G is a Hausdorff barrelled LCTVS V (e.g. Fréchet or compact type) together with a separately continuous action G × V → V such that for all x ∈ V the map G → V given by g 7→ gx is locally analytic. Lemma 18.1.2. (1) G × V → V is jointly continuous. la (2) The map Dc (G, L) × V → V given by (δ, x) 7→ δ(g 7→ gx) is a separately continuous action on V . la (3) If X ∈ Dc (G, L) is in the image of Lie(G), it acts on x ∈ V via exp(aX)x − x d Xx = lim = exp(tX)x. a→0 a dt la (4) Dc (G, L) → Lb(V,V ) is continuous. Proof. (1) Same proof as before—the conditions suffice to make Banach-Steinhaus apply. (2) Since δgx = gx, we can check that (δ ∗ )(x) = δ((x)) by continuity, so this map is la la an action. Continuity in Dc (G, L) follows from Dc (G, L) × LA(G, V ) → V being la separately continuous by a similar argument as before. Let θ : Dc (G, V ) → Ls(V,V ) be the resulting continuous map. Continuity in V is slightly more complicated because it’s not clear that V → LA(G, V ) is continuous. To check it, reduce to the case where G is compact. Fix la δ ∈ Dc (G, V ). We need to check that the map V → V given by x 7→ δx is continuous. la Because Dc (G, V ) is metrizable (indeed Fréchet), we can find a sequence δn → δ such that δn is a finite sum of δ-functions; for these we know that the map V → V p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 75

given by x 7→ δnx is continuous for all n. Since δn → δ, {δn} is bounded, and since la θ : Dc (G, V ) → Ls(V,V ) is continuous, {θ(δn)} is also bounded. Since V is barrelled, the {θ(δn)} are uniformly continuous. So for any open lattice Λ ⊂ V , there is an open lattice M ⊂ V with δnM ⊂ Λ for all n. So δM ⊂ Λ (since Λ is closed), and therefore δ is continuous. P∞ Xnx (3) This is because we can write exp(X)x = n=0 n! for X ∈ Lie(G) sufficiently small. la θ (4) We know that the composition Dc (G, L) −→ Lb(V,V ) → Ls(V,V ) is continuous. la la Since Dc (G, L) is bornological, it suffices to check that for any bounded B ⊂ Dc (G, L), θ(B) is bounded. That is, for all bounded C ⊂ V and open lattices V ⊂ Λ, we want × to find a ∈ L such that θ(B) ⊂ aΛ(C, Λ). But we know that θ(B) ⊂ Ls(V,V ) is bounded, so there is a lattice M ⊂ V such that (θ(B))(M) ⊂ Λ (by equicontinuity, since V is barrelled). Then we can find b ∈ L× with C ⊂ bM. Then (θ(B))(C) ⊂ bΛ, so θ(B) ⊂ Λ(C, bΛ) = bΛ(C, Λ), so we can take a = b. Lemma 18.1.3. There is an anti-equivalence of categories between {locally analytic represen- la tations of G on V of compact type} and {separately continuous actions of Dc (G, L) on nuclear ∨ la ∼ ∨ ∨ Fréchetspaces}, sending V to Vb , where the action of Dc (G, L) → Lb(V,V ) = Lb(Vb ,Vb ) on the latter is δ(f) = f ◦ δ†. la ∨ Remark 18.1.4. If G is compact, the action of Dc (G, L) on Vb is necessarily jointly continuous. la Definition 18.1.5. (1) A coadmissible Dc (G, L)-module is a module M such that M is la coadmissible over Dc (H,L) for one (hence every) compact open subgroup H ⊂ G. la (Recall that Dc (H,L) is a Fréchet-Stein algebra, but not usually noetherian, so coadmissible means the following: we can write Dla(H,L) as the lim of a sequence c ←− · · · → A2 → A1 where the maps are flat and the Ai are noetherian Banach algebras, and a module M is coadmissible if it is of the form M = lim M where M is a finitely ←− i i generated module over Ai and Ai−1 ⊗Ai Mi = Mi−1. Note that we are not sure whether la if M/D (H,L) is such that A ⊗ la M is finitely generated for all i, this should c i Dc (H,L) imply that M is coadmissible.) Also recall that any such M has a canonical topology. (2) A locally analytic representation V of G is called admissible if V is of compact type ∨ la and Vb is coadmissible over Dc (G, L). Corollary 18.1.6. There is an anti-equivalence of categories between {admissible locally la ∨ analytic representations of G} and {coadmissible Dc (G, L)-modules} taking V to V . Lemma 18.1.7. The admissible locally analytic representations of G form an abelian category in which • images and kernels take the subspace topologies, • any morphism is strict with closed image (meaning that given ψ : V → W , the bijection V/ ker ψ → im ψ is a homeomorphism between the quotient and subspace topologies), and • a closed invariant subspace of an admissible locally analytic representation is again admissible. Suppose V is a Banach representation of G. Let V la := {x ∈ V | G → V : g 7→ gx is locally analytic} ⊂ V. 76 COURSE BY RICHARD TAYLOR

There are diﬀerent ways of topologizing this in the literature (the subspace topology is a bad idea). We will follow Emerton (see [5]). Let H ⊂ G be a compact open subgroup with analytic multiplication (i.e. H ∼= D(1)n and multiplication is given by convergent power series). Let

V H−an := {x ∈ V | h 7→ hx is analytic on H} = {x ∈ V | h 7→ hx is analytic on gH for all g ∈ G} ∼ M n = O(D(1) )⊗b πV. g∈G/H H−an la S H−an 0 Then V is naturally a LCTVS and we have V = H V , and if H ⊃ H we have a natural map V H−an → V H0−an. So we can give V la = lim V H−an the direct limit topology. −→ (This is the same as the Schneider-Teitelbaum deﬁnition if V is admissible.) Lemma 18.1.8. Suppose V is an admissible Banach representation of G. (1) V la ⊂ V is dense. la la ∨ (2) V is an admissible locally analytic representation of G, and (V )b is ﬁnitely generated la over Dc (H,L) for all compact open subgroups H ⊂ G (this is sometimes called being “strictly admissible”). la ∨ la ∨ (3) We have (V )b = Dc (G, L) ⊗L G Vb . (4) The functor from admissible BanachJ K representations to admissible locally analytic representations taking V to V la is exact.

18.2. More on compact operators. We have seen, if u ∈ CCA(M,M) where A is a Banach algebra and M is a Banach projective Banach A-module, how to define det(1 − uT ). If 1 det(1 − uT ) = Q(T )P (T ) where Q ∈ A[T ], P ∈ O(AA), Q(0) = P (0) = 1, Q has unit leading 1 coefficient, and (P,Q) = O(AA), we have seen that we can decompose M = M1 ⊕ M2 so that ∗ ∗ Q (u) = 0 on M1 and Q (u) is an isomorphism on M2. Now let A be a noetherian Banach algebra and k · k a defining norm with kabk ≤ kakkbk. 2 If f(T ) = f0 + f1T + f2T + · · · ∈ A T , recall that we obtain the Newton polygon of f as J K the lower convex hull of the points (n, − logp kfnk) in the xy-plane. We refer to the slopes of 1 the sides of the Newton polygon as “the slopes of f”. Then f(T ) ∈ O(AA) if and only if the slopes of f go to ∞. Furthermore, we have seen that if A = L and f ∈ O(A1), h is a slope of f if and only if there is α ∈ L such that f(α) = 0 and |α| = ph. We say that f “has slopes ≤ h” or “> h” if all slopes of f are ≤ h or > h. We say that an A-module M has a slope ≤ h decomposition if we can decompose

M = M≤h ⊕ M>h so that

(1) if x ∈ M≤h, there is Q ∈ A[T ] with unit leading coefficient (we call such Q “multiplicative”) and all slopes ≤ h such Q∗(u)x = 0 (recall by definition Q∗(T ) = T deg QQ(1/T )), (2) M≤h is finitely generated over A, and ∗ (3) for any multiplicative Q ∈ A[T ] with slopes ≤ h, Q (u)|M>h is an isomorphism. If M is a Banach space, A = L, and u is completely continuous, then there is a slope ≤ h decomposition of M for all h. Remark 18.2.1. (1) If M has a slope ≤ h decomposition, then it is unique. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 77

(2) Let D≤h = {Q ∈ A[T ] | Q multiplicative and has slopes ≤ h} ⊂ A[T ]. Note that D≤h contains 1 and is closed under multiplication. Then an A[T ]-module M has a −1 slope ≤ h decomposition if and only if there is a surjection M D≤hM with finitely generated kernel (over A or equivalently A[T ]). (3) If ψ : M → N is a map of A[T ]-modules and M,N have slope ≤ h decompositions, then ψ takes M≤h to N≤h and M>h to N>h, and im ψ and ker ψ have slope ≤ h decompositions. (4) Taking the ≤ h and > h parts is exact when defined. Lemma 18.2.2. Suppose A = O(D(1)n). Let M be a Banach projective Banach A-module. n Let u ∈ CCA(M,M). Let x ∈ Max(A) (by which we approximately mean x ∈ D(1) , although really x is a Galois orbit of such). Let h ∈ Q≥0. For an integer N, d ∈ {0,...,N}, and b0, . . . , bN ∈ A, we may define the Banach algebra 0 N+2 A = O(D(1)A )/(Ti − bi, bdT∗ − 1) N+2 (where T0,...,TN ,T∗ are the variables on D(1)A ); this defines an affinoid subdomain (in fact a Laurent subdomain) of A. Then we can choose such an A0 such that the resulting map 0 0 Max(A ) ,→ Max(A) has x in its image and A ⊗b A,πM has a slope ≤ h decomposition for u.

(The reason for the extra condition on the value of bd is that bd will be chosen to enforce 0 |ad(y)| being constant on Max(A ).)

19. 3/10/20: Automorphic forms on definite quaternion algebras I want to illustrate the theory of p-adic automorphic forms in the simplest possible case. I don’t have time to go beyond that. To that end let D denote a definite quaternion algebra with centre Q. Thus D is characterized by the finite set of places S at which D is not split. The set S will contain ∞ (as D is assumed definite), and any finite set S of places of Q which contains ∞ and has even cardinality can arise. We will denote by ∗ the anti-involution on D characterized by ∗ d + d equals the reduced trace tr d of d for all d ∈ D. Choose a maximal order OD of D. It will be stable under ∗ (as tr d ∈ Z for d ∈ OD). For v 6∈ S pick an isomorphism ∼ OD ⊗Z Zv = M2×2(Zv). Under this isomorphism ∗ is carried to the map that takes a matrix to its adjugate. We define an algebraic group G over Z by setting × G(R) = (R ⊗Z OD) . ∼ It becomes reductive over Z[1/S]. If v 6∈ S then we get an isomorphism G × Zv = GL2/Zv. The topological group G(R) is compact mod centre. 19.1. Classical automorphic forms. Before discussing p-adic automorphic forms we will quickly review the classical theory of automorphic forms which is particularly simple on G. If A∞ denotes the ring of finite adeles then the quotient G(Q)\G(A∞) is compact. If R is a ring (commutative with 1) and M is an R-module we will write AM for the set of locally constant functions ∞ G(Q)\G(A ) −→ M. These are the ‘weight 2’ automorphic forms on G. We could discuss other weights as well, but for lack of time I won’t for now. This has a natural smooth action of G(A∞) via (gϕ)(x) = ϕ(xg). 78 COURSE BY RICHARD TAYLOR

(‘Smooth’ means that the stabilizer of any function in AM is open.) If M is finitely generated over R, then this action is also ‘admissible’ over R in the sense that for any open compact ∞ U subgroup U ⊂ G(A ) the U-fixed points AM is a finitely generated R-module. In fact U M AM = M, G(Q)\G(A∞)/U and G(Q)\G(A∞)/U is a finite set. We see that AU AU M = Z ⊗Z M and that [ U AM = AM . U The most classical case is the case M = R = C. In this case ∼ M AC = π where π runs over irreducible admissible representations π of G(A∞) each of which has a factorization as a restricted tensor product ⊗vπv with πv an irreducible admissible represen- G(Zv) tation of G(Qv) with dim πv = 1 for all but finitely many v (and = 0 for the other v). 0 G(Zv) (The tensor product ⊗vπv is restricted with respect to the lines πv .) We say that π is G(Zv) unramified at v if v 6∈ S and πv 6= (0). It turns out that either πv is infinite dimensional (indeed ‘generic’) for all v 6∈ S, or dim πv = 1 for all v and π = χ ◦ det for some continuous ∞ × × × character χ :(A ) /Q → C . (Here det denotes the reduced norm G → Gm.) Moreover the strong multiplicity one theorem tells us that if π and π0 are two irreducible submodules ∼ 0 0 of AC with πv = πv for all but finitely many v, then π = π . If U, V are open compact subgroups of G(A∞) and g ∈ G(A∞) then we can write a UgV = giV i a finite disjoint union. There is a well defined R-linear map [UgV ]: AV −→ AU M PM ϕ 7−→ i giϕ, called a Hecke operator. In the particular case V ⊂ U and g = 1 then we will write

X V U trU/V = [UV ] = g : AM −→ AM . g∈U/V ∞ We call an open compact subgroup U of G(A ) unramiﬁed at v if v 6∈ S and U ⊃ G(Zv). If U is unramiﬁed at v we will write v 0 v 0 T = U U and S = U U . v 0 1 v 0 v

These operators commute as v varies over all unramified primes for U. We will write TR(U) U for the R-subalgebra of EndR(AR) generated by all the Hecke operators Tv and Sv as v runs over unramified primes for U. It is a finite commutative R-algebra, called the unramified U Hecke algebra. It acts on AM for any R-module M. If R → S is a ring morphism then

S ⊗R TR(U) TS(U). p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 79

If S is ﬂat over R this map is an isomorphism. If R is a reduced Q-algebra then TR(U) is also reduced. In the case R = C and T ∈ TC(U) and π is an irreducible constituent of AC then T acts on πU by a scalar, which I will denote π(T ). There is an isomorpohism ∼ Q TC(U) −→ πU 6=(0) C T 7−→ (π(T ))π,

U where the product is over all irreducible constituents π of AC with π 6= (0). ∼ If π is an irreducible constituent of AC and ı : C = Qp then there is a continuous semi-simple representation

rπ,ı : GQ −→ GL2(Qp) with the following properties:

(1) rπ,ı is unramified outside p and the primes where π is ramified, and for such a place v the conjugacy class rπ,ı(Frobv) has characteristic polynomial 2 X − π(Tv)X + vπ(Sv). (2) If v 6= l then r | is determined by π (and ı). π,ı GQv v 1 0 (3) If c ∈ G is a complex conjugation then r (c) is conjugate to . Q π,ı 0 −1 (4) The restriction r | is de Rham with Hodge-Tate numbers {0, 1}. Moreover a π,ı GQp certain invariant WD(r | ) of r | is determined by π (and ı), however the π,ı GQp π,ı GQp p restriction r | is not itself in general determined by π . π,ı GQp p (5) rπ,ı is irreducible if and only if π is infinite dimensional. U ∗ Finally we need to recall the Petersson pairing on AR. Suppose that U = U. Then we define a perfect R-bilinear pairing U U h , iU : AR × AR −→ R (ϕ , ϕ ) 7−→ P #(xUx−1 ∩ G( ))−1ϕ (g)ϕ (g−∗). 1 2 x∈G(Q)\G(A∞)/U Q 1 2 Note that some mild assumption is needed for this definition to make sense: the term #(xUx−1 ∩ G(Q))−1 needs to make sense in R. This will be OK if for instance 6 is invertible in R, or if U is ‘sufficiently small’. We will not go into details here. If V ∗ = V as well then we have ∗ h[UgV ]ϕ1, ϕ2iU = hϕ1, [V g U]ϕ2iV . If V ⊂ U and g = 1 we get

htrU/V ϕ1, ϕ2iU = hϕ1, ϕ2iV .

Moreover if v is unramiﬁed for U, then Tv and Sv are self-adjoint with respect to h , iU .

19.2. p-adic modular forms. As usual L will denote a ﬁnite extension of Qp, O will denote its ring of integers, λ will denote the maximal ideal of O and F = O/λ the residue ﬁeld of O. We will normalize the absolute value | | on L by |p| = p−1. We will assume that p 6∈ S. Suppose that U is an open compact subgroup of G(A∞,l). U m • AO/λm is an admissible smooth representation of GL2(Qp) over O/λ . cts,U ∞ • We will set AO = C(G(Q)\G(A )/U, O) with the p-adic toplogy. We see that it is a separately continuous representation of GL ( ) isomorphic to lim AU . 2 Qp ←−m O/λm 80 COURSE BY RICHARD TAYLOR

cts,U ∞ • AL = C(G(Q)\G(A )/U, L) is a Banach representation of GL2(Qp), which can be cts,U identified with AO ⊗O L. If one chooses the defining norm to be the sup norm k k∞ cts,U cts,U then AL has unit ball AO . These spaces are sometimes referred to as ‘completed cohomology’. • Note that U cts,U cts,U AL/O = AL /AO , and so if V is an open compact subgroup of GL2(Qp) we have ∞ cts,U cts,U V U×V ∼ #G(Q)\G(A )/UV (AL /AO ) = AL/O = (L/O) . cts,U We deduce that AL is an admissible Banach representation of GL2(Qp). We may sometimes set cts [ cts,U AL = AL . U ∞ cts It has a separately continuous action of G(A ). The U fixed points of AL are identified cts,U with AL . n n We will write V (p ) for the kernel GL2(Zp) → GL2(Z/p Z), and define n AU = lim AU×V (p ) bO ←− O tr a torsion-free, compact Hausdorff, linearly topologized O-module with a compatible continuous action of O GL2(Qp) . We will also write J K U U AbL = L ⊗O AbO . At least for n sufficiently large we have

∞ n U ∼ n G(Q)\G(A )/UV (p ) AbO = O V (p ) , J K U U so that AbO and AbL are coadmissible. We have the important observation that cts U ∼ cts U HomO (AbO ,L) = HomO (AbO , O) ⊗O L ∼ cts U m −→ (lim HomO (AbO , O/λ )) ⊗O L ←−m n ←−∼ (lim lim Hom (AUV (p ), O/λm)) ⊗ L ←−m −→n O O O ∼ UV (pn) = (lim lim A m ) ⊗ L ←−m −→n O/λ O = (lim AU ) ⊗ L ←−m O/λm O cts,U = AL . U [The ﬁrst isomorphism follows from the fact that AbO is compact. The main point to check about the second is surjectivity, which follows from the completeness of O. The main point to check about the third is again surjectivity, which follows because the kernel of any element cts U m UV (pn) of HomO (AbO , O/λ ) is open and hence is the preimage of an open submodule of AO for some n. The fourth isomorphism follows from the duality described at the end of the last section.] Thus we also have that cts,U U ∨ AL = (AbL )b and U cts,U ∨ AbL = (AL )s . p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 81

For n ≥ 1 there is a natural map n+1 n TO(U × V (p )) TO(U × V (p )) compatible with the maps U×V (pn) U×V (pn+1) AR ,→ AR and U×V (pn+1) U×V (pn) tr : AR AR . It sends Tv (resp. Sv) to Tv (resp. Sv) for all unramified v. Thus if we define n O(U) = lim O(U × V (p )) T ←n T then there is an action of TO(U) on each of the spaces: U U U cts,U AO , AbO , AO/λm , AO , which commutes with the action of GL2(Qp). n Being finite and free over O we see that TO(U × V (p )) has finitely many maximal ideals and n ∼ Y n TO(U × V (p )) = TO(U × V (p ))m m n where the product runs over all maximal ideals m of TO(U × V (p )). Moreover each TO(U × n V (p ))m is a complete local noetherian ring finite and free as a O-module. We have a diagram n TO(U) TO(U × V (p )) TO(U × V (p)) ↓ ↓ n TF(U × V (p )) TF(U × V (p)) in which all the maps are surjective. It is not hard to see that pull back along the vertical arrows indices a bijection between maximal ideals. The same is true for the horizontal arrows. [To check this it suffices to check it for the lower horizontal arrow. Let us sketch the proof of this. We have an isomorphism n AUV (p ) ∼ {ϕ : G( )\G( ∞)/U → Ind V (p) : ϕ(xu) = u−1ϕ(x) ∀u ∈ V (p)}. F = Q A V (pn)F V (p) The V (p)-module Ind V (pn)F has a filtration by V (p) invariant submodules such that each graded piece is isomorphic to F as a V (p)-module. Thus the kernel I of n TF(U × V (p )) TF(U × V (p)) n preserves a filtration of AUV (p ) and acts trivially on each graded piece, from which we F conclude that I is nilpotent, and so contained in every maximal ideal.] We deduce that TO(U) has only finitely many maximal ideals and that ∼ Y TO(U) = TO(U)m m where the product runs over all maximal ideals m of TO(U). Similar considerations show that each TO(U)m is noetherian, and hence is a complete noetherian local ring. If m is a maximal ideal of TO(U) then there is a continuous semi-simple representation

rm : GQ −→ GL2(k(m)) such that: 82 COURSE BY RICHARD TAYLOR

(1) rm is unramiﬁed outside p and the primes where U is ramiﬁed, and for such a place v the conjugacy class rm(Frobv) has characteristic polynomial

2 X − TvX + vSv. 1 0 (2) If c ∈ G is a complex conjugation then r(c) is conjugate to . Q 0 −1

We call m Eisenstein if rm is reducible. We think of m being non-Eisenstein as the ‘generic’ situation. If m is non-Eisenstein then there is a continuous representation

rm : GQ −→ GL2(TO(U)m) such that:

(1) rm is unramiﬁed outside l and the primes where U is ramiﬁed, and for such a place v the conjugacy class rm(Frobv) has characteristic polynomial

2 X − TvX + vSv. 1 0 (2) If c ∈ G is a complex conjugation then r(c) is conjugate to . Q 0 −1 The following theorem is a culmination of the work of many people. Theorem 19.2.1. Suppose that

r : GQ −→ GL2(O) ⊂ GL2(L) is unramified outside finitely many primes and that for each finite prime v ∈ S the semi- ss ss ±1 simplified restriction r| is unramified and the eigenvalues of r| (Frobv) have ratio v . GQv GQv (This latter condition can be relaxed, but some modified condition is necessary to reflect the

fact that we are working with G and not GL2.) Suppose moreover that for c ∈ GQ a complex conjugation 1 0 r(c) ∼ . 0 −1 (We say r is odd.) Also suppose that (1) r mod λ is absolutely irreducible. (2) p > 2. (3) (r mod λ)| is not the extension of a character χ by a character χ with χ /χ GQp 2 1 2 1 either trivial or the cyclotomic character. (These latter conditions are presumably not necessary, and some progress has been made p,∞ on relaxing them.) Then for some open compact subgroup U ⊂ GL2(A ) ramiﬁed only at primes in S and primes where r ramiﬁes, and some maximal ideal m of TO(U)m there is a continuous homomorphism

θ : T(U)m −→ O such that θ(r ) = r. Moreover θ factors through some (UV (pn)) if and only if r| is de m TO GQp Rham with Hodge-Tate numbers {0, 1}. We also have the following conjecture: p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 83

Conjecture 19.2.2. Keep the notation and assumptions of the theorem. Then there is a Banach representation V = V (r| ) of GL ( ) which is determined by, and determines, GQp 2 Qp r| such that Acts,U [ker θ] ∼= V ⊕m for some m. Equivalently GQp L U ∼ ∨ ⊕m AbO ⊗TO(U),θ L = (Vs ) . Off the top of my head I am unsure of the status of this conjecture, but Emerton proved the corresponding conjecture when G is replaced by GL2. Notice that whereas the GL2(Qp) action on AU did not in general determine r| , the action on Acts,U or AU does. Also note L GQp L bL that the TO(U) see all odd, irreducible, continuous, two dimensional representations r of GQ n satisfying some conditions at the primes in S, while the TO(U × V (p )) only see the small subset for which r| is de Rham with Hodge-Tate numbers {0, 1}. GQp Finally let us mention the locally analytic versions of these spaces of automorphic forms. ∞,l la,U For U ⊂ G(A ) we will write AL for the space of locally analytic functions ∞ ϕ : G(Q)\G(A )/U −→ L. (Note that G(Q)\G(A∞)/U is a locally analytic manifold.) Thus la,U cts,U la AL = (AL ) is an admissible locally analytic representation of GL2(Qp) with a commuting faithful action of TO(U). We have la,U ∨ ∼ U la (AL )b = AbL ⊗L GL2(Qp) Dc (GL2(Qp),L) la J K and this is a coadmissible Dc (GL2(Qp),L)-module. 20. 3/12/20: The eigencurve 20.1. Weight space. Weight space parametrizes continuous characters of a compact abelian locally analytic group. We will consider noetherian Banach algebras A defined by a norm k k satisfying •k abk ≤ kakkbk, • and kank = kakn. (We say that k k is power multiplicative.) An important example is provided by reduced affinoid algebras. By an affinoid algebra we mean one of the form O(D(1)n)/I for some (necessarily closed) ideal I. If A is an affinoid algebra and m is a maximal ideal then A/m is a finite extension of L. If A is reduced its topology is defined by the (‘spectral’) norm kak = sup |a mod m|. m We will need a small amount of rigid geometry, which I unfortunately don’t have time to develop. Hopefully you have had some exposure to these ideas in some other context. I’ll briefly summarise some of the concepts we will need. If A is an affinoid algebra we will write Sp A for its set of maximal ideals. For instance Sp O(D(1)n) = Gal (L/L)\D(1)n. It has various useful topologies. The ‘canonical topology’ has a basis consisting of the sets of the form {m : |ai mod m| ≤ 1 for i = 1, ..., n}

for a1, ..., an ∈ A. If f : A → B is a L-algebra homomorphism between aﬃnoid algebras we get a continuous map f ∗ : Sp B → Sp A. There is a notion of B being an aﬃnoid subdomain of A. In this case Sp B is an open subset of Sp A and B is completely determined (as an 84 COURSE BY RICHARD TAYLOR

A-algebra) by the open set Sp B ⊂ Sp A. One example of aﬃnoid subdomains is provided by the Laurent domains n+m 0 0 A −→ (A⊗b πO(D(1) ))/(Ti − ai, ajTj − 1 : i = 1, ..., n; j = 1, ..., m). 0 0 0 Here T1, ..., Tn,T1, ..., Tm are the additional n + m variables and the ai and aj lie in A. One can ‘glue’ aﬃnoid spaces Sp A to form rigid analytic varieties. We will not explain S this in general but consider only the special case of X = Ai where

Sp A1 ⊂ Sp A2 ⊂ Sp A3... are aﬃnoid subdomains one of another. (The property of being an aﬃnoid subdomain is transitive.) For example we have [ (D(1)0)n = Sp O(D(r)n)

as r runs over pQ<0 and [ Affn = Sp O(D(r)n) as r runs over pQ. On the level of points one has (D(1)0)n = Gal (L/L)\(D(1)0)n and n Affn = Gal (L/L)\L . There is a general notion of an affinoid subdomain of a rigid analytic variety. In the special S case X = Ai as above it coincides with being an affinoid subdomain of some Sp Ai. There is also the notion of an admissible covering by affinoid subdomains. In the above special case a covering by affinoid subdomains Uj is one in which each Sp Ai is contained in a union of a finite number of Uj. A rigid analytic variety X has a structure sheaf OX , and OX (Sp A) = A for any affinoid subdomain Sp A ⊂ X. This structure sheaf has, in particular, the usual sheaf propery for admissible covers by affinoid subdomains. Lemma 20.1.1. For A as above there is a bijection cts × ∼ Hom (Zp,A ) −→ {a ∈ A : kak < 1} χ 7−→ χ(1) − 1. Proof: We sketch the proof. Firstly the map is well defined for if kak ≥ 1 then k(1 + a)pn − 1k = kakpn which does not tend to 0 as n → ∞. Secondly the map is injective because if χ and χ0 have the same image then χ and χ0 agree on Z and hence by continuity are equal. Finally we need to check surjectivity. Given a ∈ A with kak < 1 we get a map Z → A× which sends n to (1 + a)n. As A is complete, it suffices to show that it is continuous with respect to the p-adic n topology on Z. Concretely we need to show that (1 + a)p → 1 as n → ∞. For any b ∈ A with kbk < 1 we have k(1 + b)p − 1k ≤ max{|p|kbk, kbkp}. Inductively we conclude that k(1 + a)pn − 1k ≤ max{|p|nkak, |p|n−1kakp, ..., kakpn }, and the desired result follows. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 85

For a ∈ A with kak < 1 we have

kan/nk ≤ kakn|p|− logp n = |p|−n logp kak−logp n → 0 as n → ∞. Thus ∞ X log(1 + a) = (−1)n−1an/n n=1 converges to an element of A. If I didn’t make a mistake, we get kak if kak ≤ 1/e k log(1 + a)k ≤ [L : Qp]/(−e log kak) if kak ≥ 1/e. Thus ∞ X exp(pmt log(1 + a)) = tn(pm log(1 + a))n/n! n=0 converges for t ∈ Zp if log kak < − min{1, p(1/(p−1)−m)/e}. (Again assuming I didn’t make a computational error.) We conclude that:

× Lemma 20.1.2. Let A be as above and let χ : Zp → A be a continuous character. Then χ is locally analytic. More speciﬁcally there exists fχ ∈ Z≥0 such that ∞ fχ X n χ(1 + p t) = cχ,nt n=0

for all t ∈ Zp, where cχ,n ∈ A and kcχ,nk ≤ 1 for all n and cn,χ → 0 as n → ∞. d tor More generally consider an abelian compact locally analytic group Γ of the form Zp × Γ , with Γtor ﬁnite. The lemma immediately implies that any continuous character χ :Γ → A× (for A as above) is locally analytic, and on a suitable neighbourhood of the identity is given d by a power series in O(D(1)A) all of whose coeﬃcients have norm ≤ 1. Now moreover assume that L contains all #Γtor roots of unity. Then we set 0 d tor ∨ WΓ = (D(1) ) × (Γ ) . We have O(W ) = (lim O(D(r)d)) ⊗ L(Γtor)∨ . Γ −→ L r<1 There is a continuous character univ × χ :Γ −→ O(WΓ) , d tor which sends (γ1, ..., γd, γtor) ∈ Γ = Zp × Γ to

γ1 γd (1 + T1) ...(1 + Td) ⊗ (χ(γtor))χ. This has the universal property that if A is any reduced aﬃnoid algebra and χ :Γ → A× is a continuous character, then there is a unique map

Sp A → WΓ under which χuniv pulls back to χ. Note that χuniv is not itself locally analytic, but its pull back to any aﬃnoid algebra is locally analytic. Loosely speaking the closer you get to the ‘boundary’ of WΓ the smaller the radius of analyticity. 86 COURSE BY RICHARD TAYLOR

20.2. Finite slope p-adic modular forms. Let us write B for the subgroup of GL2 consisting of upper triangular matrices, and T for the subgroup of B consisting of diagonal matrices. Suppose that × χ : T (Zp) −→ L . We will sometimes think of χ as a character of B(Zp) and write a b χ = χ (a)χ (b). 0 d 1 2

fχ 2 There is an fχ ∈ Z>0 such that χ is analytic on (1 + p Zp) and that, more precisely, fχ χi(1 + p t) is given by a power series in O(D(1)) all whose coeﬃcients are in O. We deﬁne la,U la,U −1 AL (χ) = {ϕ ∈ AL : ϕ(xu) = χ(u) ϕ(x) ∀u ∈ B(Zp)}.

This space is preserved by TO(U). Let Iw denote the inverse image of B(Fp) in GL2(Zp). If a b ∈ Iw then c d a b 1 0 a b = . c d c/a 1 0 (ad − bc)/a ∞ Let XU be a (ﬁnite) set of representatives for G(Q)\G(A )/UIw. Then we can write ∞ a 1 0 G(A ) = G(Q)xU B(Zp). pZp 1 x∈XU la,U n We see that a function ϕ ∈ AL (χ) is analytic on left cosets of V (p ) if and only if, for each n x ∈ XU and each γ ∈ pZ/p Z the function

fx,γ : Zp −→ L 1 0 e 7−→ ϕ x γ + pnc 1

la,U V (pn)−an is analytic. Moreover a norm on the Banach space (AL ) (χ) is given by

kϕk = sup kfx,γk. n x∈XU ,γ∈Z/p Z la,U V (pn)−an Moreover (AL ) (χ) has a Banach basis consisting of the functions ϕx,γ,m for x ∈ XU , n−1 γ ∈ Z/p Z and m ∈ Z≥0 which is supported on 1 0 G(Q)xU n B(Zp) pγ + p Zp 1 and sends 1 0 x 7−→ tm. pγ + pnt 1

Note that kϕx,γ,mk = 1. la,U If ϕ ∈ AL (χ) we deﬁne a magical transformation X p α (U ϕ)(x) = ϕ x , p 0 1 α∈Z/pZ p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 87 where we think of p and α as elements of Qp. The deﬁnition does not depend on the choice la,U of representatives α for classes in Z/pZ. The image Upϕ again lies in AL (χ) because a b pa aα + b (U ϕ) x = P ϕ x p 0 d α 0 d a 0 p (aα + b)/d = P χ ϕ x α 0 d 0 1 a b = χ (U ϕ)(x). 0 d p

The operator Up commutes with the action of TO(U). For n ≥ fχ we have

la,U V (pn+1)−an la,U V (pn)−an Up :(AL ) (χ) −→ (AL ) (χ),

la,U V (pn+1)−an and so Up deﬁnes a completely continuous operator on (AL ) (χ). To see this note that (2) 1 0 X p α (U ϕ) x = ϕ x p e 1 ep eα + 1 α X p α 1 0 1 − αe −eα2/p = ϕ x 0 1 ep/(1 − αe) 1 0 1/(1 − αe) α X p α 1 0 = (χ /χ )(1 − αe)ϕ x . 1 2 0 1 ep/(1 − αe) 1 α

la,U V (pn)−an We can go further and look at the ‘matrix’ of Up on (AL ) (χ). Write p α aα bα x = gxαu 0 1 pcα dα with g ∈ G(Q), u ∈ U and a b α α ∈ Iw. pcα dα Then 1 0 (U ϕ) x p e 1 X aα bα 1 0 = (χ1/χ2)(1 − αe)ϕ xα pcα dα ep/(1 − αe) 1 α aα+(bαp−aαα)e !! X 1 0 bα = (χ /χ )(1 − αe)ϕ x 1−αe 1 2 α p(cα+(dα−cαα)e) 1 0 (aαdα−pbαcα)(1−αe) α aα+(bαp−aαα)e aα+(bαp−aαα)e X 1 0 = χ (a d − pb c )(χ /χ )(a + (b p − a α)e)ϕ x , 2 α α α α 1 2 α α α α p(cα+(dα−cαα)e) 1 α aα+(bαp−aαα)e 88 COURSE BY RICHARD TAYLOR

and so 1 0 (U ϕ 0 0 0 ) x p x ,γ ,m pγ + pne 1 (3) P n = α χ2(aαdα − pbαcα)(χ1/χ2)(aα + (bαp − aαα)pγ + p e(bαp − aαα)) 0 1−n 0 0 m p (cα+(dα−cαα)pγ−γ (aα+(bαp−aαα)pγ))+pe(dα−cαα−γ (bαp−aαα)) n aα+(bαp−aαα)pγ+p e(bαp−aαα)

0 where the sum is over those α such that xα = x and 0 n−1 cα + (dα − cαα)pγ ≡ γ (aα + (bαp − aαα)pγ) mod p .

We deduce that kUp| la,U V (pn)−an k ≤ 1. (AL ) (χ) la,U V (pn+1)−an la,U V (pn)−an Because Up(AL ) (χ) ⊂ (AL ) (χ) we see that det(1−TUp| la,U V (pn)−an ) (AL ) (χ) 1 Q is independent of n for n large. We will denote it det(1 − TUp| la,U ) ∈ O(AﬀL). If r ∈ p AL (χ) then there is a factorization

det(1 − TUp| la,U ) = Qr(T )P (T ) AL (χ) where

• Qr(T ) ∈ L[T ] with QA,r(0) = 1 and all slopes ≤ logp r; 1 • P (T ) ∈ O(AﬀL) with P (0) = 1 and all slopes > logp r.

For n suﬃciently large, there is a slope ≤ logp r slope decomposition

n n Ala,U (χ)V (pn)−an = Ala,U (χ)V (p )−an ⊕ Ala,U (χ)V (p )−an. L L ≤logp r L >logp r 0 This decomposition is preserved by the action of TO(U). If n > n and n is sufficiently large, then n n0 Ala,U (χ)V (p )−an = Ala,U (χ)V (p )−an, L ≤logp r L ≤logp r la,U and we will simply denote this AL (χ)≤logp r. [The inclusion ⊂ is clear. For the re- 0 la,U V (pn )−an verse inclusion use the fact that Up is an isomorphism on A (χ) such that L ≤logp r 0 n0−n la,U V (pn )−an la,U V (pn)−an la,U U A (χ) ⊂ A (χ) .] Then A (χ)≤log r is a O(D(r)L)/(Qr(T ))- p L ≤logp r L ≤logp r L p module (where T acts by Up) which is finite over L and has an action of TO(U). We write la,U fs [ la,U AL (χ) = AL (χ)≤logp r, r the space of finite slope locally analytic p-adic modular forms. (Non finite slope locally analytic p-adic modular forms still seem rather mysterious.) la,U Because kUpk ≤ 1 we see that AL (χ)≤h = (0) if h < 0. Suppose now that χ is locally algebraic in the sense that there exists n ∈ Z>0 and k1 ≥ k2 in Z such that a 0 χ = ak1 dk2 0 d n for all a, d ∈ 1 + p Zp. Then we set la,U V (pn)−alg AL (χ) p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 89

la,U to be the set of ϕ ∈ AL (χ) such that 1 0 ϕ x pnc 1 ∞ is a polynomial of degree ≤ k1 − k2 in c ∈ Zp for all x ∈ G(Q)\G(A )/U. This space is preserved by TO(U) and by Up. [The ﬁrst of these assertions is clear. For the latter we use equation 2.] n We will write V1(p ) for the subgroup of GL2(Zp) consisting of elements whose reduction a b modulo pn is upper triangular with diagonal entries 1. If v = ∈ V (pn) and e ∈ pn c d 1 Zp then a b 1 0 1 0 a + be b = . c d e 1 (de + c)/(be + a) 1 0 (ad − bc)/(a + be)

la,U V (pn)−alg Thus for ϕ ∈ AL (χ) we have 1 0 1 0 ϕ xv = det(v)k2 (a + be)k1−k2 ϕ x . e 1 (de + c)/(be + a) 1

If Wk1,k2 denotes the space of polynomials of degree ≤ k1 − k2 over L we can make it a representation of GL2(Qp) via the formula a b −1 ( P )(T ) = (ad − bc)k2 (a + bT )k1−k2 P ((dT + c)/(bT + a)). c d This is easily checked to be a representation isomorphic to det−k1 ⊗Symm k1−k2 L2. We conclude that n n la,U V (p )−alg ∼ U×V1(p ) AL (χ) = AL(Wk1,k2 ) , n U×V1(p ) where AL(Wk1,k2 ) denotes the space of ‘classical’ automorphic forms consisting of all functions ∞ ψ : G(Q)\G(A )/U −→ Wk1,k2 such that ψ(xv) = v−1ψ(x) n n la,U V1(p )−alg for all v ∈ V1(p ). The map sends ϕ ∈ AL (χ) to the function that sends 1 0 x 7−→ ϕ x t 1 n considered as a polynomial in t for t ∈ p Zp. This identification is equivariant for the action of the unramified Hecke operators Tv and Sv for v =6 p. Moreover it is equivariant for the n U×V1(p ) action of Up if we make Up act on AL(Wk1,k2 ) via the formula X p α p α (U ψ)(x) = pk1 ψ x . p 0 1 0 1 α∈Z/pZ la,U V (pn)−alg We deduce in particular that AL (χ) is finite dimensional. ∼ U×V (pn) 1 −1 If we fix an isomorphism ı : C → L then AL(Wk1,k2 ) ⊗L,ı C is isomorphic to the space of functions

n −k1 k1−k2 2 Ψ: G(Q)\G(A)/(U × V1(p )) −→ det ⊗ Symm C 90 COURSE BY RICHARD TAYLOR

such that Ψ(xh) = h−1Ψ(x)

for all h ∈ G(R) ,→ GL2(C). The map sends ψ to −1 ∞ x 7−→ x∞ (xpψ(x ) ⊗ 1). This may help explain the adjective ‘classical’. It is again equivariant for the action of the unramiﬁed Hecke operators Tv and Sv for v 6= p; and for the action of Up if we make Up act on Ψ as above by the formula X p α (U Ψ)(x) = pk1 Ψ x . p 0 1 α∈Z/pZ From the real analytic theory and some p-adic algebraic geometry, in particular Mazur’s n la,U V1(p )−alg (‘Newton over Hodge’) theorem, one can deduce that the slopes of Up on AL (χ) are all ≤ k1 − k2 + 1. la,U V (pn)−alg la,U V (pn)−an As AL (χ) and (AL ) (χ) have weight decompositions, so does la,U V (pn)−an la,U V (pn)−alg (AL ) (χ)/AL (χ) . It follows from equation 3 that

k1−k2+1 kUp| la,U V (pn)−an la,U V (pn)−alg k ≤ |p| . (AL ) (χ)/AL (χ) la,U V (pn)−an la,U V (pn)−alg Thus the slopes of Up on (AL ) (χ)/AL (χ) are all ≥ k1 − k2 + 1. Thus la,U la,U V (pn)−alg AL (χ)<(k1−k2+1) ⊂ AL (χ) . This is a version of Coleman’s famous ‘control theorem’: locally analytic p-adic forms of small slope are classical.

20.3. The eigencurve. A fundamental observation of Coleman and Mazur is that finite slope locally analytic p-adic forms move in p-adic families - these families are called eigenvarieties. If some property of finite slope locally analytic forms can be shown to vary continuously on an eigenvariety, one can hope to use this to prove the property for all such forms from knowing it for some especially easy dense set of points. I’m afraid I don’t have time to give examples of this, but will have to content myself with sketching the construction of the eigenvariety for automorphic forms on G. The main point is to copy what we have done over L over an arbitrary affinoid subdomain of weight space. la,U la,U V (pn)−an la,U If A is an affinoid algebra we can define AA and (AA ) just as we defined AL la,U V (pn)−an and (AL ) . They still have commuting actions of TO(U) and GL2(Qp) (because the cts,U cts,U × ambient space AA = A⊗b πAL does). If χ : T (Zp) → A is a continuous character we la,U la,U V (pn)−an la,U la,U V (pn)−an can also define AA (χ) and (AA ) (χ) just as we did AL (χ) and (AA ) (χ). These spaces are preserved by TO(U). We can also define la,U la,U Up : AL (χ) −→ AL (χ) by X p α (U ϕ)(x) = ϕ x , p 0 1 α∈Z/pZ p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 91 where we think of p and α as elements of Qp. For n-sufficiently large, this sends la,U V (pn+1)−an la,U V (pn)−an (AA ) (χ) to (AA ) (χ) and hence defines an A-linear completely continuous la,U V (pn+1)−an endomorphism of (AA ) (χ). We obtain a characteristic power series 1 det(1 − TUp| la,U ) ∈ O(AffA). AA (χ) If ψ : A → B then

ψ det(1 − TUp| la,U ) = det(1 − TUp| la,U ). AA (χ) AB (ψχ)

Now write W for WT (Zp). We get a universal 1 1 det(1 − TU | la,U univ ) ∈ O(W × Aff ) = lim O(Aff 2 ), p A (χ ) −→ O(D(r) ) 1 the limit taken as r → ∞. We can define a rigid analytic subvariety SU ⊂ W × Aff , the vanishing locus of det(1 − TUp|Ala,U (χuniv)). We call SU the spectral curve. Put another way for Q each affinoid subdomain Sp A ⊂ W and each r ∈ p the power series det(1 − TUp| la,U univ ) AA (χ ) defines a closed affinoid rigid analytic subvariety of Sp A × D(r), namely

Sp O(D(r)A)/(det(1 − TUp| la,U )). AA (χ) These aﬃnoid varieties agree on intersections and so provide an admissible covering of the spectral curve SU . We will call a pair (Sp A ⊂ W, r) where Sp A ⊂ W is a (necessarily reduced) aﬃnoid domain and r ∈ pQ suitable if there is a factorization

det(1 − TUp| la,U univ ) = QA,r(T )P (T ) AA (χ ) where

• QA,r(T ) ∈ A[T ] is multiplicative (i.e. its leading coefficient is a unit), has QA,r(0) = 1 and has all slopes ≤ logp r; 1 • P (T ) ∈ O(AffA) with P (0) = 1 and all slopes > logp r. (The terminology ‘suitable’ is non-standard - I made up the word.) In this case one can check that × • P (T ) ∈ O(D(r)A) ; • for any multiplicative R(T ) ∈ A[T ] with all slopes ≤ logp r (including QA,r(T )) we have 1 (P (T ),R(T )) = O(AffA). [The first assertion is proved by using the usual power series expansion of (1 + T f(T ))−1. To prove the second assertion one can argue by contradiction, and so assume that (P,R) 1 is contained in a maximal ideal m of O(AffA). One shows that the residue field of m is a finite extension of L and deduces that the preimage mc of m in A is also maximal. Working modulo mc one reduces to the special case that A is a finite field extension of L. In this case 1 it follows from our previous analysis of O(AffA) for A a finite extension of Qp.] Note that if (Sp A ⊂ W, r) is suitable and if Sp A0 ⊂ Sp A is an affinoid subdomain, then (Sp A0 ⊂ W, r) is also suitable. 1 The spaces D(r)A for which (Sp A ⊂ W, r) is suitable form an admissible cover of Aff × W by affinoid subdomains, which we will also refer to as ‘suitable’. [I alluded to this last week in lectures. Given x ∈ W and r ∈ pQ one defines a suitable affinoid neighbourhood Sp A of

x. Then one constructs a factorisation det(1 − TUp| la,U univ ) = Q(T )P (T ) by a successive AA (χ ) 92 COURSE BY RICHARD TAYLOR

1 approximation. Initially one just constructs P (T ) ∈ A T , and then proves it lies in O(AﬀA) by looking at its Newton polygon.] The aﬃnoid ringsJ K

O(D(r)A)/(QA,r(T ))

for (Sp A ⊂ W, r) suitable give an admissible covering of SU by aﬃnoid domains, which we will call ‘suitable aﬃnoid subdomains’.

If (Sp A ⊂ W, r) is suitable then, for n suﬃciently large, there is a slope ≤ logp r slope decomposition

n n Ala,U (χuniv)V (pn)−an = Ala,U (χuniv)V (p )−an ⊕ Ala,U (χuniv)V (p )−an. A A ≤logp r A >logp r (This is not true for general (non-suitable) (Sp A ⊂ W, r).) This decomposition is preserved 0 by the action of TO(U). If n > n and n is suﬃciently large, then, as in the case A = L,

n n0 Ala,U (χuniv)V (p )−an = Ala,U (χuniv)V (p )−an, A ≤logp r A ≤logp r

la,U univ la,U univ and we will simply denote this AA (χ )≤logp r. Then AA (χ )≤logp r is a module over −1 O(D(r)A)/(QA,r(T )) (where T acts by Up ) which is ﬁnite projective over A. These modules la,U univ fs patch to give a coherent sheaf A (χ ) of OSU -modules over SU , which also carries an 0 0 00 0 action of TO(U). [If (Sp A ⊂ W, r) and (Sp A ⊂ W, r ) are suitable and if Sp A ⊂ Sp A∩Sp A , then (Sp A00 ⊂ W, r) and (Sp A00 ⊂ W, r0) are again suitable. So we have two things to check: (1) If Sp A0 ⊂ Sp A ⊂ W and if both (Sp A ⊂ W, r) and (Sp A0 ⊂ W, r) are suitable, then

0 la,U univ la,U univ A ⊗A AA (χ )≤logp r = AA0 (χ )≤logp r. (2) If r0 ≤ r are in pQ and if (Sp A ⊂ W, r) and (Sp A ⊂ W, r0) are both suitable, then

0 la,U univ la,U univ 0 0 O(D(r )A)/(QA,r (T )) ⊗O(D(r)A)/(QA,r(T )) AA (χ )≤logp r = AA (χ )≤logp r . The ﬁrst property follows from the functoriality of slope decomposition with respect to the Banach algebra A. For the second one shows that there is a factorization

QA,r(T ) = QA,r0 (T )P (T ) where P (T ) ∈ A[T ] and (P (T ),QA,r0 (T )) = A[T ]. This can be done in the same way that we

factorize det(1 − TUp| la,U univ ) = QA,r(T )P (T ).] AA (χ ) la,U univ fs The sheaf A (χ ) is finite over SU . On any suitable affinoid subdomain it is finite la,U univ fs la,U univ fs projective over the image of that subdomain in W. The fibre A (χ )w of A (χ ) la,U univ fs la,U univ fs at a point w ∈ W is identified with AL (χw ) . Thus the sections of A (χ ) are p-adic families of finite slope locally analytic p-adic automorphic forms. As TO(U) acts on la,U univ fs A (χ ) it generates a sheaf of commutative OSU -algebras la,U univ fs T(U) ⊂ EndO (A (χ ) ). SU This is a version of the Hecke algebra on finite slope locally analytic p-adic automorphic

forms but spread out over weight space. It is ﬁnite over OSU . One can form the relative Sp of T(U) over SU . We get another rigid analytic variety EU → SU called the eigenvariety. Its points correspond to systems of Hecke eigenvalues on ﬁnite slope locally analytic p-adic automorphic forms, but they are organized into rigid analytic families. p-ADIC FAMILIES OF AUTOMORPHIC FORMS NOTES 93

References [1] Nicolas Bourbaki. Elements of mathematics: Lie groups and Lie algebras. Springer, 2005. [2] Nicolas Bourbaki. General Topology: Chapters 1–4, volume 18. Springer Science & Business Media, 2013. [3] Kevin Buzzard. Eigenvarieties. L-functions and Galois representations, London Math. Soc. Lecture Notes, 320:59–120, 2007. [4] P. Colmez. Fontaine’s rings and p-adic L-functions. Lecture notes, 32:33, 2004. [5] Matthew J Emerton. Locally analytic vectors in representations of locally p-adic analytic groups. American Mathematical Soc., 2017. [6] P. Schneider. Nonarchimedean Functional Analysis. Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2013. [7] Peter Schneider and Jeremy Teitelbaum. Banach space representations and Iwasawa theory. Israel journal of mathematics, 127(1):359–380, 2002. [8] Peter Schneider and Jeremy Teitelbaum. Locally analytic distributions and p-adic representation theory, with applications to GL2. Journal of the American Mathematical Society, 15(2):443–468, 2002. [9] Peter Schneider and Jeremy Teitelbaum. Algebras of p-adic distributions and admissible representations. Inventiones Mathematicae, 153:145–196, 2003.