Local Galois Cohomology Computation) Let K Be a ﬁnite Extension of Qp Or Fp((T)) with Residue ﬁeld Fq

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 = {p, ∞} 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 v∈S v v∈ /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 ] → v∈S(Fv ⊗ F ) → OF S [ S ] v∈S(Fv ⊗ F ) → 1, show that 1 1 × ∼ 1 H GF,S, OF S [ S ] = Cl(OF [ S ]), and there is an exact sequence  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. v∈S 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 following 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 ∈ 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 ∈ 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 representations) 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] ∈ H (G, M), represented by cocycle g 7→ cg ∈ 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)g∈G and (cg)g∈G 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 ` ∞ 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 ∞ 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 ∞ 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 ] v∈S(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 w∈SL w w∈ /SL Lw AL L S 1 1 where Cl(OL[ S ]) is the ideal class group of OL[ S ]. (This is Problem 2.4(1) earlier.) (2) Show that the limit lim Cl(O [ 1 ]) is trivial. (Hint: A property of Hibert class ←−L⊂F S L S field theory is that, if LHilb/L is the Hilbert class field of L, namely the maximal unramified abelian extension of L, every ideal of L becomes principal in LHilb. Using the commutative diagram for compatibility of Artin maps with ideal class groups

Art 0× × L0 ab L \AL0 / GL0 O O natural Ver

× × ArtL ab L \AL / GL , to show that this boils down to the following group theoretic statement: Let G be a pro-ﬁnite group and H its commutator group, then the transfer map Gab → Hab is the zero map. This is known as the Artin’s principal ideal theorem. The class ﬁeld theory by Artin–Tate has a proof of this, on their page ) (3) Deduce that

0 1 × Q S × × × Q × H G , O S [ ] (F ⊗ F ) ∼ F \ / O , F,S F S v∈S v = AF v∈ /S Fv

1 1 × Q S × H GF,S, OF S [ S ] v∈S(Fv ⊗ F ) = 0 and 2 1 × Q S × ∼ H GF,S, OF S [ S ] v∈S(Fv ⊗ F ) ⊗ Z` = Q`/Z`.