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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, Pjp, 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 ρj . Gal(Qp=Qp) For finite F over Qp, the p-adic metric extends uniquely to F (so also uniquely to Qp). Let OF = f α 2 F j jαj ≤ 1 g, complete DVR, with maximal ideal PF = f α 2 F j jαj < 1 g: 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 Pjp. = = 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 ρj 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 2 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 2 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 ρjIp 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 ρjPp 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 jαj = 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 2 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 ⊗ ρ ∼ ρ 0 ∼ A;` A ;` 0 1 for some 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 2 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 , so −s −1 Lp(ρA;`; s) = (1 − app ) with ap = (Frobp) = ±1, again independent of `.
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