Exercise 2 (due on October 26) Choose 4 out of 8 problems to submit. Problem 2.1. (Local Galois cohomology computation) Let K be a finite extension of Qp or Fp((t)) with residue field Fq. Let V be a representation of GK on an F`-vector space. 1 GK ∗ GK (1) Show that when ` 6= p, H (GK ;V ) = 0 unless V 6= 0 or V (1) 6= 0. When ` = p 1 and K = Qp, what is dim H (GQp ;V ) \usually"? (2) When ` 6= p, compute without using Euler characteristic formula, in an explicit way, i n dim H (GK ; F`(n)). Your answer will depend on congruences of q modulo `. Observe that the dimensions coincidence with the prediction of Tate local duality and Euler characteristic formula. i (3) When ` = p and K a finite extension of Qp, compute the dimension of dim H (GK ; Fp(n)). Problem 2.2. (Dimension of local Galois cohomology groups) Let K be a finite extension of Qp, and let V be a representation of GK over a finite dimensional F`-vector space. Suppose that V is irreducible as a representation of GK and dim V ≥ 2. 1 2 (1) When ` 6= p, show that H (GK ;V ) = 0. (Hint: compute H using local Tate duality and then use Euler characteristic.) 1 (2) When ` = p, what is dim H (GK ;V )? Problem 2.3. (An example of Poitou{Tate long exact sequence) Consider F = Q, and let S = fp; 1g for an odd prime p. Determine each term in the Poitou{Tate exact sequence for 2 the trivial representation M = Fp. (Hint: usually H (GF;S; Fp) is difficult to determine; but one can use Euler characteristic to help.) 1 × Problem 2.4. (Cohomology of OF S [ S ] ) Let F be a number field and S a finite set of places of F including all archimedean places and places above `. (1) Show that there is a natural exact sequence 1 ! O [ 1 ]× Q F × × Q O× ! F × × ! Cl(O [ 1 ]) ! 1; F S v2S v v2 =S Fv AF F S 1 1 where Cl(OF [ S ]) is the ideal class group of OF [ S ], namely the quotient of the ideal class group Cl(OF ) by the subgroup generated by ideals in S. (2) By studying the exact sequence 1 × Q S × 1 × Q S × 1 !OF S [ S ] ! v2S(Fv ⊗ F ) !OF S [ S ] v2S(Fv ⊗ F ) ! 1; show that 1 1 × ∼ 1 H GF;S; OF S [ S ] = Cl(OF [ S ]); and there is an exact sequence 8 >Q`=Z` v non-arch 2 1 × M < 1 0 ! H GF;S; OF S [ S ] ⊗ Z` ! 2 Z=Z v = R and ` = 2 ! Q`=Z` ! 0: v2S :>0 otherwise For i ≥ 3, we have ( 1 = ` = 2 and i even; i 1 × ∼ M i × ∼ M 2 Z Z H GF;S; OF S [ ] ⊗ Z` = H (R; C ) = S 0 otherwise. v real v real 1 2 1 × 1 × Remark: Using Kummer theory 1 ! µ` !OF S [ S ] !OF S [ S ] ! 1, we can then use this 1 to further compute H (GF;S; µ`). Problem 2.5. (A step in the proof of local Euler characteristic formula) Consider the fol- lowing situation. Let K be a finite extension of Qp such that K = K(µp). Let L=K be a finite cyclic extension with Galois group H of order relatively prime to p. Let N be a finite Fp[H]-module. Our goal is to compute H × × p dim OL =(OL ) ⊗ N(−1) (1) Consider the logarithmic map × logp : OL /µ(L) / L 1 pn a / pn logp(a ) pn 2 where n is taken sufficiently divisible so that a 2 1 + p OL so that logp(1 + x) = x2 x3 x − 2 + 3 − · · · makes sense. Show that logp is well-defined homomorphism (and independent of the choice of n), and that it induces an isomorphism × ∼ logp : OL /µ(L) ⊗Zp Qp = L: (2) Show that for any two OL-lattices Λ1; Λ2 2 L, we have H H dim(Λ1=pΛ1 ⊗ N) = dim(Λ2=pΛ2 ⊗ N) : H (In terms of writing, it might be better to compare (Λi=$LΛi ⊗ N) first.) (3) Recall that L ∼= K[H] as H-modules by Hilbert 90. From this and (2), deduce that H × × p H dim OL =(OL ) ⊗ N(−1) = dim N + dim N · [K : Qp]: Problem 2.6. (Comparing first Galois cohomology classes and extensions of Galois repre- sentations) If you are unfamiliar with the background of this problem, one can consult the short note on this topic, available on the webpage. Let k be a field with discrete topology, and let G be a finite group acting k-linearly on a finite dimensional k-vector space M. Let ρ : G ! GLk(M) be the representation. 1 (1) Given a cohomology class [c] 2 H (G; M), represented by cocycle g 7! cg 2 M, show that the following map defines a representation of G on Ec := M ⊕ k: ρ(g) c g 7! g : 0 1 0 (2) Show that if (cg)g2G and (cg)g2G define the same cohomology class, the representations Ec and Ec0 defined in (1) are isomorphic. (3) By definition, there exists an exact sequence 0 ! M ! Ec ! k ! 0. Taking the G-cohomology gives a connecting homomorphism kG = k −−−−!δ H1(G; M) Show that δ(1) = [c]. (4) (Optional) Given an exact sequence of k[G]-modules 0 ! M ! E1 ! E2 ! k ! 0; 3 we may write F as the image of E1 ! E2 and thus get two short exact sequences 0 ! M ! E1 ! F ! 0; and 0 ! F ! E2 ! k ! 0 This way, the boundary maps of the group cohomology defines two maps δ : k = H0(G; k) −! H1(G; F ) −! H2(G; M) and thus the image δ(1) defines a cohomology class [c] in H2(G; M). Now, suppose that we have a commutative diagram 0 / M / E1 / E2 / k / 0 0 0 0 / M / E1 / E2 / k / 0: Show that the second cohomology class defined by these two exact sequences are the same. Problem 2.7. (An explicit computation of local Galois cohomology when ` 6= p) Let K be a finite extension of either Qp or Fp((t)), with ring of integers OK and residue field kK . Let ` 1 be a prime different from p. Let M be a finite GK -module that is ` -torsion. Following the instruction below to give another proof of the Euler characteristic for local Galois cohomology when ` 6= p: 2 X (2.7.1) χ(G ;M) := (−1)i · length Hi(G ;M) = 0: K Z` K i=0 (1) Let IK and PK denote the inertia subgroup and the wild inertia subgroup of GK . >0 1 Show that H (PK ;M) = 0 for any GK -module M that is ` -torsion. Using the Hoshchild{Serre spectral sequence to deduce that, for every i ≥ 0, i ∼ i IK H (IK ;M) = H (IK =PK ;M ): tξ;` i ∼ (2) Let PK;` denote the kernel of IK ! IK =PK −−! Z`(1). Show that we have H (IK ;M) = i PK;` H (Z`(1);M ). PK;` (3) Put N := M , and write τ for a generator of IK =PK , then we have 0 ∼ τ=1 1 ∼ H (IK ;M) = N ;H (IK ;M) = N=(τ − 1)N: Note also that the second isomorphism is given by evaluating the cochain at τ; so the Frobenius action on N=(τ − 1)N is twisted by the inverse of cyclotomic character. Thus, we should have wrote N(−1)=(τ − 1)N(−1) instead. (4) Let φK denote a Frobenius element. Show that we have isomorphisms: N(−1) . H0(G ;M) ∼= (N τ=1)φK =1;H2(G ;M) ∼= (φ − 1): K K (τ − 1)N(−1) K 1 For H (GK ;M), we may describe the unramified part and singular part of it as follows: G 1 IK 1 1 kK 0 / H (GkK ;M ) / H (GK ;M) / H (IK ;M) / 0: =∼ =∼ φK =1 N(−1) τ=1 τ=1 (τ−1)N(−1) N =(φK − 1)N (5) From this, deduce Euler characteristic formula (2.7.1) directly. 4 (6) Using the discussion above to prove the following isomorphism of exact sequences: G 1 IK 1 1 kK 0 / H (GkK ;M ) / H (GK ;M) / H (IK ;M) / 0 =∼ =∼ =∼ G ∗ ∗ 1 ∗ kK 1 ∗ 1 ∗ IK 0 / H (IK ;M (1)) / H (GK ;M) / H (GkK ; (M (1)) ) / 0: 1 IK 1 ∗ IK (Hint: first show that the subgroup H (GkK ;M ) and H (GkK ; (M (1)) ) annihi- late each other. This is because such pairing factors through the cup product. 1 IK 1 ∗ IK 2 1 H (GkK ;M ) × H (GkK ; (M (1)) ) ! H (GkK ; µ` ) = 0: G 1 IK 1 ∗ kK After this, it is enough to show that #H (GkK ;M ) = #H (IK ;M (1)) , which makes use of the discussion above.) Problem 2.8. Fix a prime number `. Let F be a number field and S a finite set of places that includes all archimedean places and places above `. For an extension L of F , we write S SL for the set of places of L that lies over places in S. Let F denote the maximal Galois extension of F that is unramified outside S. We compare the cohomology groups (for i = 1; 2) i S;× × i 1 × Q S × H GF;S; F AF S ⊗ Z` with H GF;S; OF S [ S ] v2S(Fv ⊗ F ) ⊗ Z`: (1) For a finite extension L ⊂ F S, show that we have an exact sequence 1 ! O [ 1 ]× Q L× × Q O× ! L× × ! Cl(O [ 1 ]) ! 1; L S w2SL w w2 =SL Lw AL L S 1 1 where Cl(OL[ S ]) is the ideal class group of OL[ S ].