Galois representations 1 Introduction (Vladimir) 1.1 Galois representations Galois representations really mean representations of Galois groups. Denition 1.1. An Artin representation, ρ, over a eld K is a nite dimensional complex representation of Gal(K=K) which factors through a nite quotient (by an open subgroup). I.e., there exists nite Galois extension F=K, such that ρ comes from a representation of Gal(F=K) K G F G K Gal(K=K) ! Gal(F=K) ! GLn(C) Note. e.g., I the trivial representation is the same Artin representation for all F=K p Example. Let 3 , , 3 2 −1 . The character table F = Q(ζ3; 5) K = Q G = Gal(F=Q) = S3 = s; tjs = t = id; tst = s is id (12) (123) I 1 1 1 So: 1 −1 1 ρ 2 0 −1 • I(s) = I(t) = 1 • (s) = 1,(t) = −1 p ! − 1 − 3 1 0 • ρ(s) = p2 2 , ρ(t) = 3 1 0 −1 2 − 2 Example. Dirichlet characters: Z=NZ ! C multiplicative. Q(ζN ) ∗ G=(Z=NZ) Q Hence Dirichlet characters can be seen as representation 1 χ : G! C = GL1(C) 1 Denition 1.2. A mod Galois representation is the same thing with matrices in . l GLn(Fl) Example. Let E=Q be an elliptic curve. We know E(Q)[l] =∼ Z=lZ × Z=lZ. Set F = Q(E[l]), the smallest eld generated by the x-coordinates and y-coordinates of the points of order l. We end up with a Galois group F G Q acts on and preserves addition, i.e., . Therefore we get . G E[l] pg(P + Q)p = g(P ) +pg(Q) ρ :pG ! GL2(Fl) E.g.: Let y2 = x3−5, then E[2] = 0; ( 3 5; 0); (ζ 3 5; 0); (ζ2 3 5; 0) . So take F = (ζ ; 3 5), then G = Gal(F= ) 3 3 p Q 3 p Q 3 3 permutes E[2] (we see that G = S3). Let us write down the matrix, so let P = ( 5; 0) and Q = (ζ3 5; 0). 0 1 • g 2 S be a 3-cycle, ρ(g) = 2 GL ( ) 3 1 1 2 F2 1 1 • g 2 S , be a transposition, ρ(g) = 2 GL ( ) 3 0 1 2 F2 1.2 l-adic representations sep Denition 1.3. A continuous l-adic representation over K is a continuous homomorphism Gal(K =K) ! GLd(F) for some nite . F=Ql Remark. An l-adic representation is continuous if and only if for all n there exists a nite Galois extension Fn=K sep n n such that Gal(K =Fn) ! id mod l . I.e., ρ mod l factors through a nite extension Fn=K. sep 1 + lOF lOF So Gal(K =F1) map to . lOF 1 + lOF Example. Let E=Q be an elliptic curve: basis for . • P1;Q1 E(Q)[l] basis for 2 , with , • P2;Q2 E(Q)[l ] lP2 = P1 lQ2 = Q1 . • . basis for n , with , . • Pn;Qn E(Q)[l ] lPn = Pn−1 lQn = Qn−1 For dene by , , and g 2 Gal(Q=Q) 0 ≤ an; bn; cn; dn < l gP1 = a1P1 + bQ1 gQ1 = c1P1 + d1Q1 gPn = n−1 n−1 n−1 n−1 a1 + ··· + anl Pn + (b1 + ··· + bnl )Qn and gQn = (c1 + ··· + cnl )Pn + (d1 + ··· + dnl )Qn. Then n−1 n−1 a1 + ··· + l an + : : : c1 + ··· + l cn + ::: ρ(g) = n−1 n−1 2 GL2(Zl) b1 + ··· + l bn + : : : d1 + ··· + l dn + ::: Note that ρ(g) mod ln tells you what g does to E[ln]. This does give a 2d continuous l-adic representations. 2 Galois Representations: vocabulary (Matthew S) 2.1 Galois Theory of Innite Algebraic Extensions Notation. G(F=K) := Gal(F=K), GK = G(K=K) the absolute Galois group For this section we assume K is a perfect eld (so every extensions is separable) and F is a normal algebraic extension of K. 2 Example. Let be a prime, and , let be dened as p. is xed by . Naively p K = Fp F = Fp φp φp(x) = x Fp hφpi we would think ∼ , but this is not true at all. To see this, take such that an where GFp = hφpi = Z φ 2 GFp φjFpn = φp is a sequence such that where . This shows . fang an ≡ am mod m mjn GFp > hφpi Denition 2.1. Let F=K be a Galois extension. For each nite subextension K0 consider G(K0=K). When we have two of them, such that K0 ⊆ K00 consider G(K00=K) ! G(K0=K): This denes an inverse system of groups. G(F=K) = lim G(K0=K). −K0=K B = fleft=right cosets of finite index subgroupsg Fact. G(F=K) is Hausdor, compact and totally disconnected. Theorem 2.2. Let F=K be a Galois extension. The map K0 ! G(F=K0) is a bijective inclusion reversing corres- pondence between K0 and closed subgroups of G(F=K), H ! F H . Example. Back to the example, G( n = ) = =n , so G = lim =n = . Fp Fp Z Z Fp −n Z Z Zb 2.2 Galois groups of Q. Fix , : Q ! Qp Q ! Qp Qp Fp ur Qp Fp Qp Fp Note ur ∼ . G(Qp =Qp) = GFp ∼ µr = GQp / / G(Qp =Qp) / GFp C C The kernel of such a map is Ip. Ip admits a large normal p-subgroup, Wp;the wild inertia group. Ip=Wp tame inertia Let , for a Galois extension of if : Θ:GQp G(K=Qp) Qp • Θ(Ip) = 0 we say that K is unramied • Θ(Wp) = 0 then we say that K is tamely ramied • Θ(Wp) 6= 0 then we say that K is widely ramied Example. Cyclotomic extensions: ∗ ∗ ∗ ∼ , n we have an isomorphism . Let , G(Q(ζm)=Q) = (Z=mZ) Kl = [n2Z>0 Q(ζl ) G(Kl=Q) ! Zl l : GQ ! Zl dened as for: l(σ). is ramied at and at , For , recall , then , p. σ 2 GQ,σ(ζ) = ζ Kl 1 l p 6= l φp (φp) = p φp(ζ) = ζ Conjecture. Any nite group is a discrete quotient of GQ 3 2.3 Restricting the ramication Let be a set of primes including . Let be the maximal extension of unramied outside . Let S f1g QS Q S . GQ;S = G(QS=Q) Theorem 2.3 (Hermito-Minkowski). Let nite, a nite set of primes, . Then there exists nitely K=Q S d 2 Z>0 many degree d extensions F=K unramied outside F . In particular is nite. Homcont(GK;S; Z=pZ) Theorem 2.4 (p-niteness condition). Let p be a prime, K a number eld, S a nite set of primes (non- archimedean). Let G ⊂ GK;S which is open then there exists only nitely many continuous homomorphism from G to Z=pZ. Theorem 2.5. If is a nite extension then is topologically nite generated. K Qp GK Conjecture. If , the map is an inclusion • p 2 S GQp ! GQ;S If , the map has kernel exactly . So . • p2 = S GQp ! GQ;S Ip GQp =Ip ,! GQ;S Suppose now that we have not xed our embedding. Theorem 2.6 (Chebotarov). Let K=Q be a Galois extension unramied outside a nite set of primes S. Let T ⊇ S be a nite set of primes. For each there exists a well-dened , the union of these classes is p2 = T [φp] ⊂ G(K=Q) dense in G(K=Q) 2.4 Galois Representations Denition 2.7. A Galois representation over a topological ring A unramied outside S (a set of primes) is a continuous homomorphism, . ρ : GQ;S ! GLn(A) Let M be a free rank n A-module, we can equip it with a G action: g · a = ρ(g) · a. More formally: Suppose we have a free A-module M such that: • G (a pronite group) acts continuously • M = lim M H where H runs over open normal subgroups of G, −!H then we can make M into a A[[G]]-module: A[[G]] = lim A[G=H] where H is as before. We say , a representation of , is : −H ρ GQ • unramied at p if it is trivial on Ip. • tamely ramied at p if it is trivial on Wp • otherwise it is widely ramied. Proposition 2.8. Let S be any set of primes: 1. An Artin representation, , is determined by trace on such that is unramied ρ : GQ ! GLn(C) (ρ(φp)) p2 = S ρ at p. 2. A semisimple representation, , is determined by the values of trace i where mod l ρ : GQ ! GLn(k) (^ (ρ(φp))) i = 1; : : : ; n on primes p2 = S at which ρ is unramied. If l > n it is sucient to use trace(ρ(φp)) at the same primes. 3. A semisimple -adic representations, , is determined by trace on at which is l ρ : GQ ! GLn(A) (ρ(φp)) p2 = S ρ unramied. 4 2.5 Conductors of representation The inertia group is ltered by u , closed and for Ip Ip C GQ;p u 2 [−1; 1] If then u v • u ≤ v Ip ⊃ Ip If , then u and 1 • u ≤ 0 Ip = Ip Ip = f1g u • Wp = [u>0Ip u v • Ip = \v<uIp Denition 2.9. Conductor of ρ at p is the integer 1 u Ip I mp(ρ) = codim(ρ ) + codim(ρ p )du ˆ0 . The conductor of ρ is the integer Y N(ρ) = pmp(ρ) p where p runs over all p 6= l (unless its Artin) 3 Invariants of Artin and l-adic Representations (Céline) Notation.