Galois Representations

Galois representations

London Taught Course Centre

Lecture 3, 30 January 2017 5. Local Galois representations 5.1. Restricting global Galois representations For L/Q ﬁnite Galois, P|p, we deﬁned

IP ⊂ GP ⊂ Gal(L/Q), depending, up to conjugacy, only on p , where:

I IP = inertia group, trivial if p - Disc(L), ∼ I DP /IP = Gal(Fpf /Fp) = hFrobP i, where Fpf = OL/P. An Artin representation:

Gal(Q/Q) → Gal(L/Q) → GLm(C), is determined (up to equivalence) by the “local data”

p 7→ ρ(Frobp) := [ρ(FrobP )], for p - Disc(L). An `-adic representation (e.g., `-adic cyclotomic character):

ρ : Gal(Q/Q) → GLm(Q`), might not factor through Gal(L/Q) for a ﬁnite L/Q. Q ⊂ Q The inclusions ∩ ∩ induce a homomorphism: Qp ⊂ Qp

Gal(Qp/Qp) → Gal(Q/Q),

I well-deﬁned up to conjugacy in Gal(Q/Q), I injective by Krasner’s Lemma (which ⇒ Qp = QQp). As “local data” consider restrictions ρ| . Gal(Qp/Qp) For ﬁnite F over Qp, the p-adic metric extends uniquely to F (so also uniquely to Qp).

Let OF = { α ∈ F | |α| ≤ 1 }, complete DVR, with maximal ideal PF = { α ∈ F | |α| < 1 }. e ∼ Let pOF = PF and OF /PF = Fpf , so [F : Qp] = ef .

F is unramified if e = 1, define F0 ⊂ F, maximal unramified.

For L finite over Q, L ⊂ Q ⊂ Qp determines P|p. = = O / n Let LP closure of L in Qp, field of fractions of←− lim L P , Taking F = LP (finite over Qp, as above), we have

e = eP , OL/P = OF /PF , f = fP . Every ﬁnite (Galois) F/Qp is LP for some ﬁnite (Galois) L/Q. If L is Galois over Q, then so is F, and we have

Gal(Qp/Qp) Gal(F/Qp) ∩ ∩ Gal(Q/Q) Gal(L/Q), identifying Gal(F/Qp) = DP , Gal(F/F0) = IP , and

Gal(F0/Qp) = DP /IP = Gal(Fpf /Fp).

So for an Artin representation

Gal(Q/Q) → Gal(L/Q) → GLm(C), the restrictions ρ| for p Disc(L) Gal(Qp/Qp) - carry the same information as [ρ(Frobp)]. 5.2. Structure of local Galois groups Recall that ( / ) = ( / ) Gal Qp Qp ←−lim Gal F Qp where the limit is over ﬁnite Galois F/Qp. ∼ The surjections Gal(F/Qp) → Gal(F0/Qp) = Gal(Fpf /Fp) induce ( / ) ( / ) Gal Qp Qp ←−lim Gal F0 Qp

∼ ( f / ) = ( / ). = ←−lim Gal Fp Fp Gal Fp Fp ( / ) ∼ ( / ) = Recall that Gal Fp Fp = ←−lim Z f Z Zb, topologically generated by Frobp. Deﬁne the (absolute) inertia group Ip as the kernel of

Gal(Qp/Qp) Gal(Fp/Fp) = Zb. Denote its ﬁxed ﬁeld:

I ur = p = , Qp Qp −→lim F0 ur so Qp is the maximal unramiﬁed extension of Qp, and

ur Gal(Qp /Qp) = Gal(Fp/Fp) = Zb.

This gives an exact sequence:

1 → Ip → Gal(Qp/Qp) → Zb → 0.

ur We can describe Qp explicitly as [ [ Qp(ζn) = Qp(ζpf −1). p-n f We say F/Qp is tamely ramiﬁed if p - e. Kummer theory describes the maximal tamely ramiﬁed extension of Qp as

tr [ ur 1/e Qp = Qp (p ). p-e

ur ur 1/e ur Since ζe ∈ Qp , we see Gal(Qp (p )/Qp ) is generated by 1/e 1/e p 7→ ζep , giving

ur 1/e ur ∼ Gal(Qp (p )/Qp ) = µe(Qp)

(non-canonically isomorphic to Z/eZ). Taking limits gives: Y ( tr/ ur) ∼ µ ( ) ∼ ( ), Gal Qp Qp = ←−lim e Qp = Z` 1 `6=p

ur compatible with the actions of Gal(Qp /Qp). tr Let Pp = Gal(Qp/Qp), the wild inertia group, we have:

1 ⊂ Pp ⊂ Ip ⊂ Gal(Qp/Qp), so that Pp is a pro-p group, and we have an exact sequence:

tr ur tr ur 1 → Gal(Qp/Qp ) → Gal(Qp/Qp) → Gal(Qp /Qp) → 1. o k o k Y Z`(1) Zb `6=p

tr If φ is any lift of Frobp to Gal(Qp/Qp), then

φσφ−1 = σp

tr ur for all σ ∈ Gal(Qp/Qp ). Consider an `-adic local Galois representation:

ρ : Gal(Qp/Qp) → GLm(E), with E ﬁnite over Q`.

We say ρ is unramiﬁed if ρ|Ip is trivial, i.e., ρ factors through ur Gal(Qp /Qp), or equivalently a (possibly inﬁnite)

Gal(Qp/Qp) → Gal(F/Qp) → GLm(E)

ur with F ⊂ Qp .

In that case ρ corresponds to a representation of Gal(Fp/Fp), and is determined by ρ(Frobp).

We say ρ is tamely ramiﬁed if ρ|Pp is trivial, i.e. it factors through tr tr Gal(Qp/Qp) (or Gal(F/Qp) for some F ⊂ Qp).

Then ρ is determined by ρ(φ) and ρ(σ), where φ 7→ Frobp, and p σ is a topological generator of Ip/Pp, so ρ(φ)ρ(σ) = ρ(σ) ρ(φ). 5.3. Examples 5.3.1. `-adic cyclotomic character

( ) = µ n ( ) ( / ) Recall that Z` 1 ←−lim ` Qp has an action of Gal Qp Qp , deﬁning the `-adic cyclotomic character:

× × χ` : Gal(Qp/Qp) → Z` ⊂ Q` .

If ` 6= p, then χ` is unramiﬁed (since Qp(ζ`n ) is for all n), and

χ`(Frobp) = p.

× The image of χ` is the (open) subgroup of Z` topologically generated by p.

If ` = p, then χ`(= χp) is ramiﬁed (to be revisited. . . ) 5.3.2. `-adic Tate modules

Let A be an elliptic curve over Qp. Recall the `-adic Tate module of A is

( ) = ( )[`n], T` A ←−lim A Qp rank 2 over Z`, with an action of Gal(Qp/Qp), deﬁning:

ρA,` : Gal(Qp/Qp) → GL2(Z`) ⊂ GL2(Q`).

2 n ∼ The Weil pairing induces isomorphisms ∧ A[` ] = µ`n , giving

2 ∼ ∧ T`(A) = Z`(1), compatibly with Gal(Qp/Qp), so det(ρA,`) = χ`. If A has good reduction at p, then it’s deﬁned by an equation over Zp whose reduction is an elliptic curve A over Fp. We can then consider the `-adic Tate module of A:

( ) = ( )[`n] T` A ←−lim A Fp with its action of Gal(Fp/Fp). If ` 6= p (postponing ` = p. . . ), then reducing mod p induces an isomorphism

( ) = [`n] ∼ [`n] = ( ), T` A ←−lim A = ←−lim A T` A compatible with the action of Gal(Qp/Qp) (acting via Gal(Qp/Qp) → Gal(Fp/Fp) on T`(A)).

In particular ρA,` is unramiﬁed. Moreover in this case tr(ρA,`(Frobp)) = p + 1 − #A(Fp).

Idea of proof: #A(Fp) = deg(Frobp − 1) (viewed as an endomorphism of A), which = det(Frobp − 1) (now as an endomorphism of T`(A)). So

#A(Fp) = det(ρA,`(Frobp) − I2).

Combine this with det(ρA,`(Frobp)) = p to get formula.

So ρA,`(Frobp) has characteristic polynomial:

2 X − apX + p, where ap = p + 1 − #A(Fp). 5.3.3. `-adic cohomology Above generalizes to `-adic cohomology of any smooth projective X/Qp (or F ﬁnite over Qp): If X has good reduction and p 6= `, then

i H (X , `)(i) Qp Q

i is unramiﬁed, and isomorphic to H (X , `)(i) as a Fp Q representation of Gal(Qp/Qp).

Moreover, the characteristic polynomial of Frobp is in Z[x], is related to #X(Fpr ), and its roots are Weil numbers, i.e., satisfy |α| = pi/2. 5.3.4. The Tate curve

Back to an elliptic curve A over Qp.

It turns out that, for ` 6= p, ρA,` is unramiﬁed if and only if A has good reduction at p. Now suppose A has (potentially) multiplicative reduction. The theory of the Tate curve shows that

0 ∼ × A (Qp) = Qp /hqi

0 for some q ∈ pZp (where A = A or a quadratic twist). It follows that:

0 n ∼ 1/`n A (Qp)[` ] = hζ`n , q i/hqi. Same analysis as earlier non-semisimple example shows χ` c ψ ⊗ ρ ∼ ρ 0 ∼ A,` A ,` 0 1 for some ψ of order ≤ 2.

If ` 6= p, then χ` is unramiﬁed, and c restricts to a non-trivial homomorphism on Ip such that

I c is trivial on Pp, −1 I c(φσφ ) = pc(σ) for all σ ∈ Ip; ur i.e., c : Ip/Pp → Z`(1) as modules for Gal(Qp /Qp).

In particular, ρA0,` is tamely ramified, and ρA0,`(Ip) has infinite order. (For completeness: If A has potentially good reduction, then ρA,`(Ip) is finite.) 5.4. (Weakly) compatible systems 5.4.1. Independence of ` A general principle in the preceding examples was that the description of ρ was (in some sense) “independent of `” for ` 6= p. To make this more precise, recall the definition:

−s −1 Lp(ρ, s) = det(Im − ρIp (Frobp)p ) for an Artin representation ρ. This also now makes sense for an `-adic representation of Gal(Qp/Qp), provided the characteristic polynomial of

ρIp (Frobp) is in Q[x] (or F[x] for some number ﬁeld F ⊂ E, viewed also as ⊂ C). For example, χ`(Frobp) = p for all ` 6= p, so

1−s −1 Lp(χ`, s) = (1 − p ) .

If ρ : Gal(Qp/Qp) → GLm(E) for some finite E/Q, then each `-adic completion of E gives an `-adic ρ with the same Lp(ρ, s) (with coefficients in E). n Twisting by χ` gives Lp(ρ(n), s) = Lp(ρ, s − n). −1 Warning: In the definitions, should really have used Frobp Ip (on ρ ), so for example: Lp(ρ(n), s) = Lp(ρ, s + n). If A has good reduction, then for ` 6= p, 2 ρA,`(Frobp) has characteristic polynomial x − apx + p where ap = p + 1 − #A(Fp) is independent of `, as is

−s 1−2s −1 Lp(ρA,`, s) = (1 − app + p ) .

If A has multiplicative reduction, then ρIp is one-dimensional with Gal(Qp/Qp) acting via ψ, so

−s −1 Lp(ρA,`, s) = (1 − app ) with ap = ψ(Frobp) = ±1, again independent of `.

If A has additive reduction, then ρIp = 0, so Lp(ρA,`, s) = 1. 5.4.2. Global Galois representations Return now to global `-adic representations, but suppose we have one for each `, say:

ρ` : Gal(Q/Q) → GLm(Q`).

We say {ρ`} is a weakly compatible system if the following holds for all but finitely many p: There is a polynomial f (x) ∈ Q[x] such that for all ` 6= p: I the restriction ρ`| is unramified, Gal(Qp/Qp) I and the characteristic polynomial of ρ`(Frobp) is f (x). So {χ`} is an example of a weakly compatible system, as is {ρA,`} for an elliptic curve A over Q. So is the system of representations of Gal(Q/Q) on {Hi (X , )} for a smooth projective X over . Q Q` Q So is the system arising from a single Artin representation, (or associated to a modular form), but we need to allow larger coeffiicient fields: For a number field E, consider systems of representations

ρλ : Gal(Q/Q) → GLm(Eλ), for primes λ of E, and allow f (x) ∈ E[x]. The examples suggest we can ask for more: that Lp(ρ`, s) be independent of ` for all ` 6= p. Or even more: the `-adic representations associated to the Tate curve seem to have more in common than Lp(ρ`, s), even though the images of inertia were very different (each isomorphic to Z`, but for different `). We’ll make this precise next time in terms of Weil-Deligne representations.