Automorphic Representations and Deformation Theory in Arithmetic Geometry

AUTOMORPHIC REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC GEOMETRY

DANIEL MILLER

Abstract. This is a brief expository note, motivated by [Mor12], on the analogy between the character variety of the fundamental group of a hyperbolic knot, and the p-ordinary deformation space of a two-dimensional modular Galois representation. Hopefully this analogy should generalize to arbitrary reductive groups, or at least GL(n). In light of this, much of this note is an introduction to the relevant objects from a general perspective. So we introduce notion of a p-adic family of automorphic representations, and the (conjecturally) corresponding family of p-adic Galois representations in full generality.

Contents 1. The analogy between knots and primes1 2. Automorphic representations2 3. Shimura varieties5 4. Modular representations6 5. Deformation theory9 6. Deformations of hyperbolic structures and Hida theory 12 References 13

1. The analogy between knots and primes Recall that a morphism f : X → Y of schemes is étale if it is flat and unramified. Equivalently, f is flat and for each y ∈ Y , the fiber Xy is the spectrum of a finite product of finite separable extensions of k(y). If X is a connected noetherian scheme with chosen base point x ∈ X, the category of finite étalecovers of X is canonically equivalent to the category of finite sets with continuous action of π1(X, x). Here π1(X, x) is the étalefundamental group of X at x, which we will denote π1(X) when x is clear. To save space, we will write π1(A) instead of π1(Spec A). A good reference for all of this is [Mil13]. We start by recalling the analogy described in [Mor12, ch.3-4]. We should think 1 of the circle S as a K(Z, 1)-space. The arithmetic analogue is Spec(Fq) for any prime power q. Indeed, as is the case for any field, π1(Fq) = Gal(Fq/Fq) ' Zb, q generated by the Frobenius frq(x) = x . The arithmetic analogue of a 3-manifold

Date: July 2014. 2000 Mathematics Subject Classification. 11F41,11F70,11F80,11G18,13D10,14G35. Key words and phrases. Automorphic representation, Shimura variety, deformation theory, modular form, Hecke algebra, Hida theory. 1 REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 2 is X = Spec(OF ) r S for any number field F and finite set S of primes in OF . Indeed, for constructible sheaves F on X, there is a “3-dimensional duality” • 3−• ∨ Ext (F , Gm) = Hc (X, F ) [Maz73, 2.4] similar to the classical Artin-Verdier • 3−• ∨ duality Ext (L , Q) = Hc (M, L ) for local systems on a 3-manifold M. In topology, a knot is an embedding K : S1 ,→ S3. More generally, we could consider any K(Z, 1) ,→ M, where M is a 3-manifold. The arithmetic analogue is a map Spec(Fq) → Spec(OF ) coming from a prime ideal p ⊂ OF with residue 1 3 field Fq. In topology, the standard way of studying a knot K : S ,→ S is to 1 3 consider a small tubular neighborhood VK of S in S . For simplicity, we assume F = Z; then the arithmetic analogue is an “infinitesimal étaleneighborhood” of Spec(Fp) ,→ Spec(Z), namely Spec(Zp) → Spec(Z). The peripheral group of K is 2 π1(VK rK) ' Z , corresponding to π1(Spec(Zp)rSpec(Fp)) = π1(Qp). Finally, the 3 1 knot group is ΓK = π1(S r K), corresponding to ΓQ,p = π1(Z[ p ]). The peripheral map π1(VK r K) → ΓK corresponds to the map π1(Qp) → ΓQ,p. Unfortunately, this analogy is not perfect. While the topological periphery group π1(VK r K) is abelian, free on generators l, m (l for latitude, m for meridian) and π1(VK r K) → π1(VK ) has kernel hmi, the arithmetic case is a more complicated. The group Dp = π1(Qp) is a very complicated pro-solvable group. It can be written down in terms of generators and relations as in [NSW08, VII §5], but this is not very helpful. The group Dp does have a canonical quotient classifying tame covers of Spec(Qp) which has a presentation tame −1 p π1 (Qp) = hσ, τ : στσ = τ i. We can think of the degenerate case “p = 1” as being in exact analogy with topology. The kernel of Dp → π1(Zp) is written Ip, and (in analogy with topology) called the inertia group at p.

2. Automorphic representations

2.1. Adeles. Let F be a number field. If v is a place of F , write Fv for the completion of F at v, and Ov for the ring of integers of Fv. Write A = AF for the ring of adeles of F ; the most important fact about A is that it is a locally compact F -algebra. It can be defined in many ways: • The topological direct limit lim A(S), where S ranges over all finite sets −→S of valuations of F , and Y Y A(S) = Fv × Ov. v∈S v∈ /S • The topological tensor product (R × Zb) ⊗ F . Q0 • A restricted direct product (Fv,Ov), consisting of those tuples (av) ∈ Q v v Fv for which av ∈ Ov for almost all v. • Via [GS66], an initial object in the category of locally compact F -algebras with no proper open ideals, and with the intersection of all closed maximal ideals being 0.

Write Af = Zb ⊗ F for the ring of finite adeles. It is totally disconnected. In [Con12], it is shown that there is a unique fiber-product-preserving functor (−)(A) from affine schemes of finite type over F to topological spaces, compatible with closed embeddings, for which (Spec F [t])(A) = A with its usual topology. In particular, if G is an algebraic group over F , the abstract group G(A) carries the REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 3 structure of a locally compact topological group. As such, it has a unique (up to scalar) Haar measure dg. 2.2. Hecke algebras. A good reference for this section is [Fla79]. Let G be a connected reductive group over F , and let g = Lie(G). There exists an open subset of U ⊂ Spec(OF ) and a “spreading out” of G to a reductive group scheme on U. Up to finitely many places, this spreading out is well-defined. In particular, for almost all finite places v, the group G(Ov) is well-defined for almost all v. It is a maximal (open) compact subgroup of G(Fv). We normalize Haar measures on G(Fv) so that G(Ov) has volume 1, and choose the Haar measure on G(A) to be the product of measure on each G(Fv). Let v be a finite place. Write Hv for the Hecke algebra consisting of continuous, locally constant, compactly supported functions G(Fv) → Q. Multiplication is convolution: Z (f ? g)(x) = f(g)g(y−1x) dy. G(Fv ) Even though this is written as an integral, it is a finite sum over double cosets of open compact subgroups of G(Fv), so no analysis is involved. For almost all v, the algebra Hv comes with a canonical idempotent, ev = χG(Ov ). Write Hf for the restricted tensor product (in the sense of [Fla79, §2]) of the Hv with respect to the e . It is the direct limit lim H(S), where H(S) = N H , and for T ⊃ S, the v −→S v∈S v injection H(S) → H(T ) is induced by the ev for v ∈ T r S. We will also think of Hf as the algebra of locally constant, compactly supported functions f on G(Af ). The ring F∞ = F ⊗ R is a finite product of copies of R and C, so G(F∞) is naturally a Lie group. Fix a maximal compact subgroup K∞ ⊂ G(F∞). The Hecke algebra H∞ = H∞(G) is the convolution algebra of K∞-finite distributions on G(F∞) with support in K∞. There is an isomorphism ∼ U(gC) ⊗U(kC) M(K∞), −→H∞ D ⊗ µ 7→ D ? µ, where k = Lie(K∞) and M(K∞) is the algebra of measures on K∞. The global Hecke algebra of G is H = H∞ ⊗ Hf . We will be interested in special classes of representations of H. For a sufficiently large set S of finite places, it makes sense to define eS ∈ Hf to Q be the characteristic function of v∈ /S G(Ov). An admissible representation of Hf is an Hf -module V such that for each v ∈ V , there is a finite set S of places for which eS · v = v. 2.3. Automorphic representations. A good reference for this section is [BJ79]. Let F , G, . . . be as above. Let Z be the center of G, and choose a character ω : Z(F )\Z(A) → C×. Write L2(G, ω) for the space of measurable functions f : G(F )\G(A) → C such that f(zx) = ω(z)f(x) z ∈ Z(A) Z kfk2 = |f(x)|2 dx < ∞. G(F )Z(A)\G(A) 2 2 The space L (G, ω) is a representation of G(A) in the obvious way. Write Ldisc(G, ω) for the closed subspace generated by all irreducible closed subrepresentations. Let 2 A(G, ω) ⊂ Ldisc(G, ω) be the space of K-finite vectors which are also Z(g∞)-finite. (It is a non-trivial fact that K-finite vectors are smooth.) REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 4

Then A(G, ω) is naturally a H-module, and as such, decomposes as a countable direct sum of irreducible representations with ﬁnite multiplicities: M (1) A(G, ω) = m(π)π.

π∈Gbc.a(ω) We call the irreducible admissible representations of H appearing in (1) automorphic representations of G. By [Fla79, th.4], each automorphic representation π N decomposes as a restricted tensor product πv of irreducible admissible representations of the Hv. In the remainder, we will often pass without comment between admissible representations of G(Af ) and admissible representations of Hf . This is not hard. Suppose π : G(Af ) → GL(V ) is an admissible representation. For f ∈ Hf and v ∈ V , put Z f ? v = f(x)π(x) · v dx. G(Af ) This integral is actually a finite sum. Indeed, we can write f as a finite sum of scalars multiples of characteristic functions χgK , where K ⊂ G(Af ) is open, compact, and fixes v. For such a function, we see that Z Z χgK ? v = χgK (x)π(x)v dx = gv dx = vol(K)gv. G(Af ) K

So the action of Hf on V makes sense. Going the other way is also easy. If V is an admissible Hf -module and g ∈ G(Af ), choose open compact normal K such that −1 χK ? v = v. Inspired by the above, put gv = vol(K) χgK ? v.

2.4. Hecke eigensystems and L-functions. Let π be an automorphic representation of G and choose a nonzero vector u in π. For almost all places v, the idempotent ev = χG(Ov ) in Hv fixes u (in this case, we say that π is unramified at v). In particular, the action of Hv on π factors through that of ∞ Hv(Ov) = evHvev = Cc (G(Ov)\G(Fv)/G(Ov)). N Let S be a set of places outside which ev fixes u. Let H(S) = Hv(Ov). N v∈ /S Then π is an irreducible admissible module over H(S) ⊗ v∈S Hv. Since H(S) is central in this algebra, it must act via a character χ : H(S) → C. The system of homomorphisms {χv : Hv(Ov) → C : v∈ / S} is called a Hecke eigensystem. In the case G = GL(n), Hecke eigensystems have a particularly easy description. A character χ : Hv(Ov) → C is uniquely determined by a semisimple conjugacy N class σv(χ) ∈ GL(n, C). If π = v πv is an automorphic representation of GL(n), put σv(π) = σ(χπv ) and (for finite v): −s−1 Lv(s, π) = det 1 − N(v) · σv(π) . For S sufficiently large, we can define the partial L-function of π as Y LS(s, π) = Lv(s, π). v∈ /S This has the expected properties including analytic continuation, a functional equa- tion. . . . In the case G = GL(n), an automorphic representation π is determined by L(s, π). REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 5

3. Shimura varieties For the rest of this note, the reader should keep in mind the example F = Q, G = GL(2). Many of the deﬁnitions work in greater generality, but technicalities (which we wish to avoid) multiply endlessly.

3.1. Locally symmetric spaces and their cohomology. Classically, one stud- ies representations of a real semisimple group G by fixing a maximal compact K, setting X = G/K, and studying the regular representation of G on C∞(Γ\X) for Γ ⊂ G a discrete group. Big examples are the (affine) modular curves Y0(n), coming from Γ0(n) ⊂ SL(2, R). We will carry out this construction adelically. Let G be a connected reductive group over Q. Put X = Z∞\G(F∞)/K∞. Let K ⊂ G(Af ) be an open compact subgroup. We define

ShK (G) = G(Q)\(X × G(Af ))/K.

A priori, this is only a topological space, but the quotient map X × G(Af )/K → ShK (G) gives ShK (G) the structure of a Riemannian manifold. Let (V, ρ) be a representation of G. There is an induced local system Vρ of F -vector spaces on ShK (G), whose (global) sections are locally constant sections s : X×G(Af )/K → V such that s(γx) = ρ(γ)s(x) for γ ∈ G(Q). • The cohomology H (ShK (G), Vρ) is naturally an admissible Hf -module. For open compact C ⊂ K, the function χgC acts via the correspondence ·g ShK (G) ShK∩C (G) −→ ShK∩g−1Cg(G) ShK (G).

There is a standard compactification of ShK (G), namely its Borel-Serre compactification ShK (G). Define the cuspidal cohomology to be • • • Hcusp(ShK (G), Vρ) = ker H (ShK (G), Vρ) → H (∂ ShK (G), Vρ) .

The cuspidal cohomology is also an admissible Hf -module. In fact, we have a generalized Eichler-Shimura isomorphism [Sch09, 4.1].

• M • K Hcusp(ShK (G), Vρ) = H (H∞, π∞ ⊗ ρ) ⊗ πf .

π∈Acusp(G,χρ) K-spherical • The notation H (H∞, −) needs explanation. There is a good category of admissible H∞-modules, and hom(C, −) is left-exact. Its derived functor is the (g∞,K∞)- • cohomology H (H∞−). Better: • M • Hcusp(Sh(G), Vρ) = H (H∞, π∞ ⊗ ρ) ⊗ πf .

π∈Gbc(ω) 3.2. Canonical models. A good reference for this is [Moo98]. In subsection 3.1 we constructed ShK (G) as a Riemannian manifold. It turns out that there is a good definition of “canonical model” for Shimura varieties over number fields, and in that sense, all Shimura varieties ShK (G) have a canonical model over a number field called the reflex field. (We have been intentionally avoiding use of the Shimura datum necessary to define ShK (G) in full generality – the reflex field depends on this.) Let E be the reflex field. Not only does the projective system Sh(G) = lim Sh (G) ←− K descend to the reflex field, but the action of G(Af ) via correspondences descends in REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 6 a canonical way. Moreover, in [Har85] it is shown that our local systems Vρ descend in a functorial way to G(Af )-equivariant local systems on Sh(G). The main reason we care about this is that if ShK (G) is smooth and E = Q, then general theorems about étalecohomology tell us that • • Hsing,c(ShK (G), Vρ) = Hét,c(ShK (G)Q, Vρ(Ql)) after choice of an isomorphism C ' Ql. The choice of an arithmetic compactification of ShK (G) lets us extend the action of Hf to the étale cohomology of ShK (G).

4. Modular representations 4.1. Modular curves. In this section, algebraic groups, adeles, etc. will be taken over Q. Let n > 1 be an integer. We deﬁne the following congruence subgroup of SL2(Z): a b Γ (n) = ∈ SL(2, Z): c ≡ 0 (mod n) . 0 c d

Let K0(n) be the induced subgroup of GL2(Af ). Write Y0(n) for the induced locally symmetric space:

Y0(n) = ShK0(n)(GL2) = GL2(Q)\ GL2(A)/Z∞K∞K0(n). + Here Z∞ = Z(GLd(R)) and K∞ = SO(2) ⊂ GL2(R). Put X = GL2(R) /Z∞K∞. + Note that GL2(R) /Z = SL2(R), so + ∼ GL2(R) /Z∞K∞ −→ H = {z ∈ C : =z > 0} via γ 7→ γ · i. The strong approximation theorem tells us that

GL2(A) = GL2(Q) GL2(R)K0(n), so the quotient ShK0(n)(GL2) is just

(GL2(Q) ∩ K0(n))\H = Γ0(n)\H = Y0(n). This will be a singular complex-analytic orbifold. There are two ways of realizing Y0(n) and its compactication X0(n) as curves over Q:

(1) Interpret Y0(n) as a moduli space for elliptic curves with level structure. This moduli problem makes sense over Q, so Y0(n) descends in a canonical way to Q. (2) Use the general theory of canonical models of Shimura varieties. The former approach generalizes to a special class of Shimura varieties consisting of those of PEL type (standing for Polarization, Endomorphism, and Level structure). The theory of PEL-type Shimura varieties is interesting and useful, but we won’t go into it here. Instead, note that the space ± H = Z∞\ GL2(R)/ SO2(R) can be interpreted as the set of GL2(R)-conjugacy classes of homomorphisms S = RC/R Gm → GL(2)R containing x y h :(x, y) 7→ . −y x REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 7

This is all defined over Q, so the theory of canonical models discussed in subsection 3.2 tells us that if K ⊂ GL2(Af ) is any open compact subgroup, the quotient ± ShK (GL2) = GL2(Q)\(H × GL2(Af ))/K descends to a uniquely determined curve over Q. Moreover, this curve has a well- defined smooth compactification also defined over Q, so we don’t need to worry about the difference between minimal and toroidal compactifications.

4.2. The Eichler-Shimura construction. Let n > 3. As above, write Y0(n) for the Shimura variety ShK0(n))(GL2), and write X0(n) for its arithmetic compactiﬁca- 1 tion. We are interested in the cohomology Hcusp(X0(n), Vsymk−2 ). At the moment, this is just a C-vector space with an action of Hf . However, in subsection 3.1, there is an automorphic decomposition

1 M 1 k−2 Γ0(n) Hcusp(X0(n), Vsymk−2 ) = H (gl2, SO(2), π∞ ⊗ sym ) ⊗ πf .

π∈A(GL2) Γ0(n)-spherical

1 k−2 The computation in [Har87, §3.4-3.6] tells us that H (gl2, SO(2), π∞ ⊗sym ) = 0 unless π is the automorphic representation coming from a weight-k cuspidal eigen- 1 k−2 form of level n, in which case H (gl2, SO(2), π∞ ⊗ sym ) = C ⊕ C. In particular, 1 M Hcusp(X0(n), Vsymk−2 ) = C ⊕ C = Sk(Γ0(n)) ⊕ Sk(Γ0(n)). f eigen-cusp Recall that our modular curves are defined over Q. So we can consider the cohomology spaces 1 Hét,cusp(X0(n)Q, Ql) ' S2(Γ0(n), C). 1 These have commuting actions of ΓQ,ln = π1(Z[ ln ]) and Hf . So ΓQ,ln acts on each Hf -irreducible piece. These pieces are 2-dimensional, so we get, for each cuspidal eigenform f, a Galois representation ρf,l :ΓQ,nl → GL2(Ql). The Eichler-Shimura relation basically tells us that the Hecke and Frobenius parameters for πf and ρf,l match up. That is, for all p - ln, we have ρf,l(frp) = σp(πf ), or equivalently Lp(ρf,l, s) = Lp(πf , s). It is known that ρf,l :ΓQ,ln → GL2(Ql) factors through GL2(Kf,λ), where Kf = Q(ap(f): p prime) is a number field and λ is a place of Kf dividing l. An elementary argument shows that we can conjugate the image of ρf,l to lie in

GL2(Of,λ), where Of = OKf . In particular, we can reduce ρf,l modulo λ to get a continuous representation

ρ¯f,l :ΓQ,ln → GL2(Of,λ/λ) = GL2(Fλ).

We say that a mod-l representation of ΓQ is modular (better, automorphic) if it is of the formρ ¯f,l for some f. Similarly, we say that an l-adic representation of ΓQ is modular (better, automorphic) if it is of the form ρf,l for some l. In what follows, we will ignore the modular form f and just write π for the corresponding cuspidal automorphic representation of GL(2), keeping in mind that for some automorphic representations (those corresponding to Maass forms) we still don’t know how to construct the associated Galois representations. For an automorphic representation π, write ρπ,l for the corresponding l-adic representation (assuming it exists). REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 8

4.3. Interpolating modular representations. The Hida families we will see later on are essentially p-adic families of cuspidal automorphic representations of GL(2). Even better, Hida constructed the corresponding p-adic family of Galois representations. Here we will give the (conjectural) bigger picture. Let G be a connected reductive group over Q. Recall we have a Hecke algebra H = Hf ⊗ H∞, and automorphic representations of G are, by definition, a special class of irreducible representations of H. Remember that Hf is the convolution algebra of locally constant, compactly supported functions on G(Af ). If 1 K ⊂ G(Af ) is open compact, then eK = µ(K) χK is an idempotent in Hf , and we write H(K) = eK Hf eK for algebra of locally constant, compactly supported, K-bi- invariant functions on G(Af ). If eK acts trivially on an automorphic representation π, we say π is K-spherical. The basic idea of a p-adic family of automorphic representations is that we p p should fix K ⊂ G(Af ) (the tame level) and consider families of automorphic p representations that are K Kp-spherical, where Kp ranges over a special class of subgroups of G(Qp). One makes this rigorous via a modified Hecke algebra. Here we mainly follow [Urb11, 4.1]. Fix a prime p and assume G is split at p. Let (T,B) be a Borel pair, and let N be the unipotent radical of B. Define

r Ir = {g ∈ G(Zp) :g ¯ ∈ B(Z/p )} − −1 T = {t ∈ T (Qp): tN(Zp)t ⊂ N(Zp)} − − ∆r = IrT Ir.

− ∞ − There is an isomorphism u : Zp[T /T (Zp)] → Cc (Ir\∆r /Ir) by t 7→ ut = χIr tIr . Here it is crucial that we normalize the Haar measure so that Ir has volume 1. − So we call Up = Zp[T /T (Zp)] and think of Up as a single avatar for all of the ∞ − Cc (Ir\∆r /Ir, Zp). Our big Hecke algebra is ∞ p p p h = Cc (K \G(Af )/K , Qp) ⊗ Up.

Note that h ,→ Hf ⊗ Qp. So, if σ is an irreducible representation of h, we will call p K Ir σ automorphic if σ = πf |h for some honest automorphic representation π. Note that if G 6= GL(n), the representation π may not be unique. p A p-adic family of automorphic representations of G will contain K Ir-spherical representations for varying r. We will also require the weight to vary p-adically. So, let W be the p-adic weight space determined by

× W(A) = homcts(T (Zp),A ). Let Wcl = X∗(T ) be the set of classical weights. We say an automorphic represen- • p tation π is cohomological of weight λ if π appears in some Hcusp(ShK Ir (G), Vλ(C)). Given such a representation, we have the trace map trπ : h → Qp, which charac- terizes π as an h-module. Definition. A p-adic family σ of automorphic representations of G of tame level Kp consists of: • An rigid subset U ⊂ W. • A finite flat surjection w : T → U. • A Qp-linear map J : h → O(T). • A dense set σcl ⊂ Ucl. REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 9

We require that for each σ ∈ σcl, the weight λ = w(σ) is dominant and the composite evσ Jσ : h → O(T) −−→ Qp is m(π, λ) trπ for a cohomological representation π of weight λ. Here m(π, λ) is the Euler-Poincar´echaracteristic

p X i K Ir i p m(π, λ) = (−1) dim homh(π , Hcusp(ShK Ir (G), Vλ(C)). In [Urb11], Urban constructed p-adic families containing the “finite slope” representations, for groups satisfying the Harish-Chandra condition (having discrete series at infinity). Recall that for “nice” automorphic representations π, there should be a Ga- L lois representation ρπ,p :ΓQ → G(Qp) unramified almost everywhere, such that for unramified v, ρπ,p(frv) is conjugate to the Satake parameter of π at v. For G = GL(n), this will just be a representation ΓQ → GLn(Qp). Note that even if multiplicity-one theorems fail for G, the trace trπ determines ρπ,p. So we will speak of “the Galois representation associated to trπ,” bearing in mind that we don’t currently know how to construct ρπ,p for general G. Conjecture. Let σ be a p-adic family of automorphic representations of G with L-algebraic classical points. Then there is a continuous representation ρσ :ΓQ → LG(O(T)) such for all σ ∈ σcl, the composite

L evσ L ρσ :ΓQ → G(O(T)) −−→ G(Qp) is the Galois representation associated to Jσ. This conjecture is wide open – we don’t even know how to construct individ- ual ρπ,p for most G. In [SU14], Skinner and Urban construct p-adic families of pseudorepresentations for “unitary similitude groups” associated to an imaginary quadratic ﬁeld. For G = GL(2) and π corresponding to a p-ordinary form, Hida has constructed a big p-adic family containing π, and the associated p-adic family of Galois representations.

5. Deformation theory 5.1. Motivation. First let’s consider the motivation for studying deformations of Galois representations. If X is a nice (that is smooth, projective and geometrically • integral) variety over Q, its étalecohomology Hét(XQ, Ql) carries a continuous action of ΓQ. The “right” way to see this is as follows. Spread out X to a smooth proper scheme X over an open U = Spec(Z)rS. Write π : X → U for the structure • map. Then R π∗Ql is a local system on Uét. The étaleversion of covering space theory tells us that local systems correspond to representations of π1(U) = ΓQ,S. The motivating example was the representation ρE,l coming from an elliptic curve π 1 E −→ U via R π∗Ql. Langlands’ conjectural framework tells us that there should exist an automorphic cuspidal representation π of GL(2) for which ρE,l ' ρπ,l. We don’t know how to construct π this directly. However, we do know (via Serre’s conjecture) thatρ ¯E,l is automorphic. One of the main applications of deformation theory is to prove that the automorphy ofρ ¯E,l implies that of ρE,l. More generally, given an l-adic Galois representation ρ :ΓQ,S → GLn(Ql) that is suitably nice (geometric, in the sense of Fontaine-Mazur [FM95]), Langlands’ REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 10 program tells us that we should expect there to be a cuspidal automorphic representation π of GL(n) such that ρ ' ρπ,l (assuming we knew how to construct ρπ in general). Proving that ρ is automorphic is very hard! However, we have a much better chance (in theory and in practice) of showing thatρ ¯ :ΓQ,S → GLn(Fl) is automorphic. A theorem to the effect that “¯ρ automorphic ⇒ ρ automorphic” is known as a automorphy lifting theorem. In practice, one has to impose many technical conditions onρ ¯ and the automorphic representation withρ ¯π ' ρ¯, and one uses groups like GSp(n) instead of GL(n).

5.2. Representations of knot groups. Our exposition here follows that of [Mor12, ch.13-14]. Let K ⊂ S3 be a hyperbolic knot, M = S3 r K the knot complement, 3 π = π1(M) the knot group. The uniformization H → M induces a represen- 3 tation π → Aut(H ) = PSL2(C) which lifts to ρ : π → SL2(C). Introduce the representation variety Rep(π, SL2) of homomorphisms π → SL2(C). There are two ways of describing Rep(π, SL2). One elementary but useful approach is to write m π = hg1, . . . , gm|r1, . . . , rni. The variety Rep(π, SL2) is just the subset of SL2(C) cut out by the relations r1, . . . , rn. A more functorial deﬁnition is to require that for all C-algebras A, a natural isomorphism

homgrp(π, SL2(A)) ' homsch/C(Spec A, Rep(π, SL2)). The character variety of K is the geometric quotient SL2(C) XK = Rep(π, SL2)// SL2 = Spec C[Rep(π, SL2)] , via the obvious action of SL(2) on Rep(π, SL2) via conjugation. The representation ρ is a point in XK , and one is interested in the connected component XK (ρ).

5.3. Deformation functors. The analogous situation in number theory is much more complicated, partly because ΓS = π1(Spec(Z) r S) is not a finitely presented group – it’s a compact topological group which is (conjecturally) topologi- cally finitely presented. So instead of looking for representations ΓS → GL2(C), we should look for continuous representations ΓS → GL2(A), where A is a topological Zp-algebra. Briefly, a scheme X over k can be thought of in terms of its functor of points X(−): k-Alg → Set. In the arithmetic context, our deformation spaces will be formal schemes over Zp. For us, this just means that the test category consists of complete local pro-artinian Zp-algebras with residue field Fp. If R is such an ring, we write Spf(R) to denote the functor A 7→ hom(R,A). There is a way of making Spf(R) into a topological space with structure sheaf, but we will not need this. The functorial approach to defining representation schemes works well. If π is an arbitrary profinite group, there is a formal scheme Rep(d π, GLn), satisfying

Rep(d π, GLn)(A) = homcts(π, GLn A), for any local pro-artinian Zp-algebra A with residue field Fp. The problem is, Rep(d π, GLn) is really horrible as a space – it generally has infinitely many different connected components. So before we do anything else, let’s restrict to the connected component ofρ ¯, whereρ ¯ : π → GLn(Fp) has been fixed beforehand. The component X(¯ρ) represents continuous homomorphisms π → GLn(A) that reduce toρ ¯ modulo p. REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 11

As before, we will quotient out by the natural action of GL(n), but here we should be careful because GL(n) does not preserve the component X(¯ρ). The correct thing to do is to ﬁrst deﬁne

GLc n(A) = {g ∈ GLn(A): g ≡ 1 (mod p))} = ker (GLn(A) → GLn(Fp)) .

The action of GL(c n) on Rep(d π, GLn) preserves X(¯ρ). Now a miracle happens. Define X(¯ρ)(A) = X(A)/GLc n(A). Then in [Maz99, pr.1], it is proved that ifρ ¯ is absolutely irreducible and π satisfies a certain technical hypothesis (which will hold for all our examples), the functor X(¯ρ) is represented by a complete local noetherian Zp-algebra Rρ¯ with residue field Fp. That is, there is a representation ρ : π → GLn(Rρ¯) liftingρ ¯ such that any GLc n(A)-equivalence class of lifts π → GLn(A) is induced by a unique continuous homomorphism Rρ¯ → A. Suppose we have fixed a subgroup I ⊂ π. We call a representation ρ : π → I GL2(A) I-ordinary if ρ is a free, rank-one, direct summand of ρ. Supposeρ ¯ is absolutely irreducible and I-ordinary. Then we can define a subfunctor X◦(¯ρ) of X(¯ρ) by X◦(¯ρ)(A) = {ρ ∈ X(¯ρ)(A): ρ is I-ordinary}. ◦ By [Maz99, pr.3], X (¯ρ) is represented by a complete local noetherian Zp-algebra ◦ Rρ¯ with residue field Fp. 5.4. The case n = 1. The easiest example is when our representations take values 1 in GL(1). Let π = π1(Z[ p ]) andρ ¯ =κ ¯ : π → GL1(Fp) be the mod-p cyclotomic character. This is defined, for σ ∈ ΓQ, by κ¯(σ) σ(ζp) = ζp . × Let’s start by computing Rep(d π, GL1). Deformations ρ : π → A factor through ab ab × π . Class field theory tells us that π ' Zp . So × a Rep(d π, GL1) = Spf Zp qZp y ' Spf (Zp Zp ) , × J K ε:π→Fp where each Spf(Zp Zp ) is the connected component of some ε. We see that Rκ¯ ' Zp Zp ' Zp X , viaJ [Kp] ↔ X + 1. J K J K 5.5. The main example. Let U ⊂ Spec(Z) be open, and put π = π1(U). Let E −→e U be an elliptic curve. Choose a prime p∈ / U. The p-torsion subscheme E[p] is an étalecover of U, so we get an action of π on the underlying set E[p] ' (Z/p)2. This action preserves the group structure, so we have a representation

ρ¯ =ρ ¯E,p : π1(U) → GL2(Fp).

Another way of constructingρ ¯ is to use the equivalence between ´etale-local Fp- systems on U and Fp-representations of π to realize Re∗Fp as a mod-p representation of π. We could consider the universal deformation ring Rρ¯ and its associated formal spectrum X(¯ρ). Recall that if K,→ S3 is a hyperbolic knot, the “knot decomposition group” DK = π1(torus) is abelian, so ε ∗ ρ | ∼ , K DK ε−1 REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 12

× for some character ε :ΓK → C . In particular, ε(IK ) = 1. So to make the analogy between knots and primes more precise, we should require that our residual Galois representationρ ¯ : π → GL2(Fp) be p-ordinary in the sense that ϕ ∗ ρ¯| ∼ , Dp ψ where ψ(Ip) = 1. This is exactly “Ip-ordinary” as deﬁned above. So there is ◦ ◦ a p-ordinary representation ρ : π → GL2(Rρ¯) that is universal for p-ordinary ◦ deformations, i.e. Spf(Rρ¯) represents the functor X◦(¯ρ)(A) = {ρ ∈ X(¯ρ)(A): ρ is p-ordinary}.

6. Deformations of hyperbolic structures and Hida theory We put together all the machinery we’ve developed to see an analogy between the space of hyperbolic structures on a 3-manifold and the formal spectrum of Hida’s big Hecke algebras.

6.1. Deformation of hyperbolic structures. Let K be a hyperbolic knot, M = 3 S r K the knot complement. Put Γ = π1(M). Let TK = Teich(Γ) be the space of injective homomorphisms Γ → Iso(H3) with discrete image, taken up to conjugacy; 3 this is the space of hyperbolic structures on M. Since Iso(H ) = PSL2(C), we get a map

φ : TK → XK . It’s not quite straightforward – you have to fudge for this to be well deﬁned. On a ◦ ◦ small neighborhood, TK → XK is an isomorphism. ◦ ◦ ◦ Also, XK → C, ρ 7→ tr ρ(m) is locally biholomorphic. Similarly, T (¯ρ) → X (¯ρ), 1 choose a generator τ of the Zp-quotient of π1(Z[ p ]). Then ρ 7→ tr ρ is p-adically bianalytic nearρ ¯.

6.2. Hida theory. We showed explicitly how to construct the Galois representations associated to cuspidal eigenforms of weight 2. In fact, in [Del73], Deligne showed how to construct ρf,l for any cusp-eigenform f of weight k > 2. Recall that such forms have a Fourier expansion

X 2πnz f(z) = an(f)e . n>1

Say that f is p-ordinary if ap(f) is a p-adic unit. This is equivalent toρ ¯f,p being an extension of an unramiﬁed character by a character. In [Hid86a, Hid86b], Hida p-adically interpolated the Galois representations ρf,p coming from p-ordinary f of varying weight and level. Fix a prime p > 5 and an integer n prime to p. Let f be a p-ordinary modular form of level npr. Then there is a p-adic family f of automorphic representations containing f. In fact, Hida explicitly constructs a completion h◦ of h such that for T◦ = Spf(h◦), all p-ordinary forms of tame level n which are congruent to f lie in ◦ ◦ T . Moreover, there is the associated Galois representation ρf :ΓQ → GL2(h ), such that all p-ordinary modular Galois representations congruent to ρf,p come from h◦. REPRESENTATIONS AND DEFORMATION THEORY IN ARITHMETIC 13

6.3. The analogy. Let f be a p-ordinary cuspidal eigenform. Put ρ = ρf,p. Let ◦ ◦ T = Spf(h ) be the corresponding p-adic family of modular forms, and ρf :ΓQ → ◦ GL2(h ) the Galois representation. Since ρf ≡ ρ¯f,p (mod p) and ρf is p-ordinary, we get a map R◦ → h◦, equivalently ρ¯f φ : T◦ → X◦(¯ρ). The very hard theorem of Wiles, etc. is that φ is an isomorphism if f satisﬁes ◦ ◦ some technical hypotheses. This is the analogy of the map TK → XK being an isomorphism in a small neighborhood.

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Department of Mathematics, Cornell University, Malott Hall, Ithaca, NY 14853, USA E-mail address: [email protected] URL: http://www.math.cornell.edu/~dkmiller