Galois Cohomology and Applications Shiva Chidambaram, discussed with Prof. Frank Calegari March 17, 2017 1 Notations and Preliminaries We start with some notation. For any field extension K F , GK F denotes the Galois group Gal(K F ). j j j For any field K , K denotes the separable closure of K , and G denotes Gal(K K ). If K is a local field, K j I denotes the inertia subgroup Gal(K K ur ), where K ur is the maximal unramified extension of K . K j If K is a number field and l is a prime in K , Gl denotes the absolute Galois group of the completion Kl , and if l is finite, Il denotes the inertia subgroup. Depending on our choice of embedding K , K , we can regard G as a subgroup of G in different ways. This choice doesn’t affect any ¡! l l K of our statements. Also, if S is a finite set of places of a number field K , KS denotes the maximal extension of K unramified outside S, and G denotes the corresponding Galois group Gal(K K ). K ,S Sj For any finite group G and a G-module M, unless we are dealing with Global Euler Characteristic formula or Wiles-Greenberg formula, H 0(G,M) will denote the modified Tate cohomology group G G 0 M /NG (M) rather than the usual M . In the two exceptional cases, we will use Hˆ (G,M) to denote the modified Tate cohomology groups, distinguishing it from the normal cohomology groups H 0(G,M). The dual of an abelian group A is A_ Hom(A,Q/Z). For any two G-modules M and N, the Æ ¡ 1 ¢ abelian group Hom(M,N) is a G-module with (g f )(m) g f (g ¡ m) . We will talk exclusively ¢ Æ ¢ ¢ about Galois modules. In that context, the notion of a dual module is as follows. If K F is a Galois j extension, and M is a GK F -module, then the dual module M ¤ is defined as M ¤ Hom(M,¹), where j Æ ¹ is the GK F -submodule of K £ consisting of the roots of unity. If M is finite, then M ¤ Hom(M,K £) j Æ Let G be a group, and M,N,P be G-modules such that there is a G-module pairing Á : M N P. ­ ¡! This induces a pairing between cohomology groups, called the cup product. i j i j H (G,M) H (G,N) H Å (G,P) £ ¡! We will only be interested in the case i j 2. If x H 0(G,M) and f H 2(G,N), then x f H 2(G,P) Å Æ 2 2 [ 2 is represented by the cocycle x f : G G P;(g ,g ) Á(x f (g ,g )). The cup product pairing [ £ ! 1 2 7! ­ 1 2 between H 2(G,M) and H 0(G,N) is defined similarly. If f H 1(G,M) and f H 1(G,N), then 1 2 2 2 f f H 2(G,P) is represented by the cocycle f f : G G P;(g ,g ) Á(f (g ) g f (g )). 1 [ 2 2 1 [ 2 £ ! 1 2 7! 1 1 ­ 1 2 2 2 Galois Cohomology This treatment of Galois Cohomology follows [1] and [5]. The cohomology H ¤(GF ,M) of a module M for GF , is sometimes denoted H ¤(F,M). This shouldn’t be confusing because we will only have Galois groups acting on modules. We will start by stating some fundamental theorems in the subject. Proofs can be found in [3]. 1 Theorem 1. Local Tate duality: Let K be a p-adic local field and M be a finite GK module. Then 1. the groups H i (G ,M) are finite for all i, and zero for all i 3. K ¸ 2. for i 0,1,2, the cup product pairing Æ i 2 i 2 H (G ,M) H ¡ (G ,M ¤) H (G ,¹) Q/Z K £ K ¡! K Æ induced by the natural pairing between M and M ¤ is non-degenerate. So, they are duals of each other. i IK 2 i IK 3. If p does not divide #M, then H (GK /IK ,M ) and H ¡ (GK /IK ,M ¤ ) are the exact annihila- tors of each other in the above pairing. If K is an Archimedean local field, 1 and 2 still hold. We note that since GK is finite in this case, these are actually the modified Tate cohomology groups. Theorem 2. Local Euler Characteristic formula: Let K be a finite extension of Qp , and M be a finite GK module. Then, #H 0(G ,M)#H 2(G ,M) K K [K :Qp ] ordp (#M) 1 #M K p¡ ¢ #H (GK ,M) Æj j Æ Theorem 3. Global Euler Characteristic formula: Let K be a number field and M be a finite GK module. Let S be a finite set of places consisting of all infinite places, all places at which M is ramified, and all places at which #M has positive order. Then GK ,S acts on M and we have Q 0 0 2 #H (Gv ,M) #H (GK ,S,M)#H (GK ,S,M) v j1 1 [K :Q] #H (GK ,S,M) Æ #M (We note that these are the unmodified cohomology groups.) Let § be a finite set of primes in Q and X be a G§ GQ,§-module. This is the same as saying, X Æ 1 is a GQ-module that is unramified outside §. It can be shown that the cohomology H (G§, X ) is 1 isomorphic to the subgroup of H (GQ, X ) consisting of classes that are unramified outside §, i.e., 1 ¡ 1 Y 1 ¢ H (G§, X ) ker H (GQ, X ) H (Il , X ) . (2.1) ' ¡! l § 62 1 If X is finite, then H (G§, X ) is finite. When X is a finite GQ-module, the kernel of the Galois action is a finite index subgroup and hence is equal to GL for some number field L. Hence, X is unramified at all primes unramified in L. So, if we let § to be the set consisting of all the infinite primes, primes diving #X , and primes at which X is ramified, then § is finite and X is a G§-module. In this case, there is a nine-term i i exact sequence involving H (G§, X ) and H (G§, X ¤)_ called the Poitou Tate sequence, that we will study next. Though we state the theorems in the situation with base field Q, all of it applies for any number field K . The general result will be used in the next section in an application to Iwasawa theory. Let X and § be as above. Then, for each i, we have a map i Y i ®i,X : H (G§, X ) H (Gv , X ) ¡! v § 2 2 We can consider the corresponding map ®i,X ¤ for the module X ¤. We remark here that X ¤ is also unramified at all v §. Dualizing this map, and using Local Tate duality for each v §, we get the 62 2 following map Y 2 i i ¯2 i : H ¡ (Gv , X ) H (G§, X ¤)_. ¡ v § ¡! 2 Proposition 1. There is a non-degenerate pairing ker® ker® Q/Z. 1,X ¤ £ 2,X ¡! Proposition 2. ® is injective, ¯ is surjective and ker¯ Im ® . 0,X 2 r Æ r,X The proofs of the above propositions can be found in [3]. Putting these together, we get the Poitou Tate exact sequence. Theorem 4. Poitou-Tate: The following nine term sequence is exact 0 ®0,X Q 0 ¯0 2 0 / H (G§, X ) / H (Gv , X ) / H (G§, X ¤)_ v § 2 γ1 1 ®1,X Q 1 ¯1 1 / H (G§, X ) / H (Gv , X ) / H (G§, X ¤)_ v § 2 γ2 2 ®2,X Q 2 ¯2 0 / H (G§, X ) / H (Gv , X ) / H (G§, X ¤)_ / 0 v § 2 where γ1 and γ2 are the following maps induced by the non-degenerate pairing in Proposition 1. 2 ¡ ¢ 1 γ : H (G , X ¤)_ ker® _ ' ker® , H (G , X ), 1 § 2,X ¤ ¡! 1,X ¡! § 1 ¡ ¢ 2 γ : H (G , X ¤)_ ker® _ ' ker® , H (G , X ). 2 § 1,X ¤ ¡! 2,X ¡! § Using the Poitou Tate exact sequence, one can deduce the Wiles-Greenberg formula for gen- eralised Selmer groups. Let X be a finite GQ-module. For each place v of Q, let Lv be a given 1 1 Iv subgroup of H (Gv , X ) such that for all but finitely many places, Lv is the subgroup H (Gv /Iv , X ) 1 of unramified cohomology classes. The generalized Selmer group HL (GQ, X ) for the given set L {L } of local conditions, is defined as Æ v µ ¶ 1 1 Y 1 ± HL (GQ, X ) ker H (GQ, X ) H (Gv , X ) Lv Æ ¡! v (2.2) © 1 ª x H (GQ, X ) res (x) L for all v Æ 2 j v 2 v 1 1 where res : H (GQ, X ) H (G , X ) is the restriction map for each place v. We will also sometimes v ! v 1 Q use HL ( , X ) to denote this Selmer group. For each v, let L?v be the exact annihilator of Lv under 1 Iv the local Tate pairing. By part 3 of Theorem 1, we have L? H (G /I , X ¤ ) for all but finitely many v Æ v v v. The collection {L?v } is called the dual set of local conditions, and denoted L ¤.