Compatibility of arithmetic and algebraic local constants II. The tame abelian potentially Barsotti–Tate case.

Jan Nekovář

Dedicated to the memory of Ian Cassels

Abstract: We prove the compatibility of arithmetic local constants of Mazur and Rubin with the usual local constants for pairs of congruent self-dual Galois representations that become Barsotti–Tate over a tamely ramified abelian extension. This allows us to complete the proof of the p-parity conjecture (p > 2) for Selmer groups of Hilbert modular forms of parallel weight two and abelian varieties with real multiplication (in particular, elliptic curves) over totally real number fields.

Introduction

Let L be a finite extension of Q_{ℓ}, let K be a finite extension of Q_{p} (where p > 2) with ring of integers O, and let V be a finite-dimensional vector space over K equipped with a continuous K-linear action of Γ_{L} = Gal(\overline{L}/L). Assume that V is self-dual in the sense that there exists a Γ_{L}-equivariant nondegenerate skew-symmetric pairing ⟨ , ⟩ : V × V ⟶ K(1) = K ⊗_{Z_{p}} Z_{p}(1) and that V contains a Γ_{L}-stable O-lattice T ⊂ V for which ⟨ , ⟩ induces an isomorphism T ⟶^{∼} T^{∗}(1) := Hom_{O}(T, O(1)). The residual representation \overline{T} := T/πT (where π ∈ O is a uniformiser) has coefficients in the finite field F := O/πO and the skewsymmetric pairing \overline{⟨ , ⟩} : \overline{T} × \overline{T} ⟶ F(1) induced by ⟨ , ⟩ is again nondegenerate.

The Bloch–Kato subspace H_{f}^{1}(L, V) ⊂ H^{1}(L, V) = H^{1}(Γ_{L}, V) (equal to H_{ur}^{1}(L, V) if ℓ ≠ p) defines, by propagation, subspaces

H_{f}^{1}(L, T) := Ker (H^{1}(L, T) ⟶ H^{1}(L, V)/H_{f}^{1}(L, V)) ⊂ H^{1}(L, T)

and

ℱ := Im (H_{f}^{1}(L, T) ⟶ H^{1}(L, \overline{T})) ⊂ H^{1}(L, \overline{T}).

According to Tate local duality, the cup product attached to \overline{⟨ , ⟩} induces a nondegenerate symmetric pairing

∪ : H^{1}(L, \overline{T}) × H^{1}(L, \overline{T}) ⟶ H^{2}(L, F(1)) ≃ F

under which ℱ is a Lagrangian (i.e., maximal isotropic) subspace of H^{1}(L, \overline{T}) if ℓ ≠ p. This is still the case if ℓ = p, provided V is a de Rham representation of Γ_{L}, which we assume henceforth.

Assume that V′, T′, ⟨ , ⟩′ is another triple as above which has the property that the pair (\overline{T}′ = T′/πT′, \overline{⟨ , ⟩}′) is isomorphic to (\overline{T} , \overline{⟨ , ⟩}). Fix such an isomorphism and use it to identify the quadratic spaces (H^{1}(L, \overline{T}), ∪) and (H^{1}(L, \overline{T}′), ∪).

The arithmetic local constant of Mazur and Rubin [MR, Def. 4.5], [N5, (2.2.1)] attached to the above data is defined as

δ_{L}(T, T′) = δ_{L}(T′, T) := dim_{F}(ℱ /ℱ ∩ ℱ′) (mod 2) ∈ Z/2Z.

Denote by W D(V) the representation of the Weil-Deligne group of L attached to V (resp. to D_{pst}(V), [Fo2], [FoPR, I.1.3.2]) if ℓ ≠ p (resp. if ℓ = p). Let ψ be a nontrivial additive character of L and dx_{ψ} the Haar measure on L that is self-dual with respect to ψ. The local constant ε(V) := ε(W D(V), ψ, dx_{ψ}) does not depend on ψ and is equal to ±1 [N2, Prop. 2.2.1].

Mazur and Rubin expect their arithmetic local constant δ_{L}(T, T′) to be related to ε(V)/ε(V′). This was checked by explicit calculations in certain cases arising from abelian varieties [Ch]. It was subsequently proved [N6, Thm. 2.17] that

Keywords: local constants, Selmer groups, parity conjecture, Hilbert modular forms, elliptic curves. MSC: 11G05, 11G40, 11S40

δ_{L}(T, T′) = ε(V)/ε(V′)

holds in full generality if ℓ ≠ p. One expects this relation to be true even if ℓ = p, provided the Hodge–Tate weights of V and V′ are the same (including the multiplicities). This was verified in two very special cases in [N6, §3].

The main local result of the present article is the following.

Theorem A (= Theorem 5.5). If ℓ = p ≠ 2 and if there exist abelian tamely ramified finite extensions K/L and K′/L and p-divisible groups (equipped with an O-action) G over O_{K} (resp. G′ over O_{K′}) such that T |_{Γ_{K}} = T_{π}G and T′|_{Γ_{K′}} = T_{π}G′, then δ_{L}(T, T′) = ε(V)/ε(V′).

If K = K′ = L, then the representations V and V′ are crystalline and ε(V) = ε(V′) = 1. In this case dim_{F}(ℱ /ℱ ∩ ℱ′) ≡ 0 (mod (p − 1)[k : k ∩ F]), without assuming that T and T′ are self-dual, by Theorem 2.16.

In general, one can assume that K′ = K and that the abelian extension K/L is totally tamely ramified (by Proposition 5.3). Theorem A is then a consequence of the fact that the subspace

ℱ = H_{fl}^{1}(O_{K}, G[π])^{Gal(K/L)} ⊂ H^{1}(K, \overline{T})^{Gal(K/L)} = H^{1}(L, \overline{T})

can be expressed explicitly in terms of the Breuil–Kisin module with descent data with respect to K/L attached to the finite flat F-vector space scheme G[π] over O_{K}, combined with some very simple linear algebra (and a slightly more involved symplectic algebra if K ≠ L).

It is very likely that the method of proof will extend to the case when K/L is an arbitrary Galois extension of order prime to p, but our assumption that K/L is totally tamely ramified greatly simplifies the description of the descent data and is suficient for the global applications discussed below.

In the global case we need to slightly change the notation and assume that L is a number field, p ≠ 2 and T, T′ are representations (self-dual as before) of the group Γ_{L,S} = Gal(L_{S}/L), where L_{S} is the maximal extension of L unramified outside a fixed finite set S of primes of L which contains all primes dividing p∞. We assume that V and V′ are de Rham representations of Γ_{L_{v}}, for all primes v | p.

As above, we obtain, for all finite primes v of L, local epsilon constants ε_{v}(V), ε_{v}(V′) ∈ {±1}, Lagrangian subspaces ℱ_{v}, ℱ′_{v} ⊂ H^{1}(L_{v}, \overline{T}) = H^{1}(Γ_{L_{v}}, \overline{T}) and arithmetic local constants δ_{v}(T, T′) = δ_{L_{v}}(T, T′) ∈ Z/2Z.

The Selmer group attached to the Selmer structure ℱ := (ℱ_{v}) is defined as

H_{ℱ}^{1} (L, \overline{T}) := Ker(H^{1}(Γ_{L,S}, \overline{T}) ⟶ ⨁_{v∈S_{f}} H^{1}(L_{v}, \overline{T})/ℱ_{v}),

where S_{f} is the set of finite primes of L contained in S (and similarly for ℱ′ = (ℱ′_{v})). As explained in [N5, (2.2.1)], the formula of Mazur and Rubin [MR, Thm. 1.4]

dim_{F} H_{ℱ}^{1} (L, \overline{T}) − dim_{F} H_{ℱ′}^{1} (L, \overline{T}) ≡ ∑_{v∈S_{f}} δ_{v}(T, T′) (mod 2)

together with the existence of a cohomological version of the Cassels–Tate pairing constructed by Flach [Fl, Thm. 1] imply that

χ_{f}(L, V) − χ_{f}(L, V′) ≡ ∑_{v∈S_{f}} δ_{v}(T, T′) (mod 2),

where

χ_{f}(L, V) := dim_{K} H_{f}^{1}(L, V) − dim_{K} H^{0}(L, V) = h_{f}^{1}(L, V) − h^{0}(L, V)

(h^{0}(L, V) = 0 if the Galois representation V is pure – necessarily of weight −1 – at some finite prime of L).

The compatibilities (−1)^{δ_{v}(T,T′)} = ε_{v}(V)/ε_{v}(V′) proved for v ∤ p in [N6, Thm. 2.17] and for v | p in Theorem A imply the following result (the assumption of tame ramification is no longer necessary, since the statement is invariant under base change to an arbitrary abelian extension of odd degree).

Theorem B (= Theorem 6.7). Assume that, in the global situation considered above, for every prime v | p of L there exist finite abelian extensions K(v)/L_{v}, K(v)′/L_{v} and p-divisible groups G (resp. G′) (equipped with an O-action) over O_{K(v)} (resp. O_{K(v)′}) such that T |_{Γ_{K(v)}} = T_{π}G and T′|_{Γ_{K(v)′}} = T_{π}G′. Then

(−1)^{χ_{f} (L,V)}/ε(V) = (−1)^{χ_{f} (L,V′)}/ε(V′),

where ε(V) := ∏_{v} ε_{v}(V) and ε(V′) := ∏_{v} ε_{v}(V′) (the product is taken over all primes of L and ε_{v}(V) = ε_{v}(V′) := (−1)^{dim_{K}(V)/2} for all v | ∞).

Let F be a totally real number field and f a cuspidal Hilbert modular newform over F of level 𝔫, parallel weight 2 and trivial character. The field K_{f} generated over Q by the Hecke eigenvalues λ_{f}(v) = λ_{f}(T (v)) of f is a totally real finite extension of Q. For every prime number p and every prime 𝔭 | p in K_{f} one can attach to f an irreducible two-dimensional representation V_{𝔭}(f) of the Galois group Γ_{F} = Gal(\overline{F} /F) with coefficients in any fixed finite extension K of (K_{f})_{𝔭}, characterised by the fact that

∀v ∤ p𝔫 det(1 − X Fr_{geom}(v) | V_{𝔭}(f)) = 1 − λ_{f}(v)X + (Nv)X^{2}.

The Tate twist V := V_{𝔭}(f)(1) admits a nondegenerate Γ_{F} -equivariant skew-symmetric pairing V × V ⟶ K(1) and any Γ_{F} -stable O-lattice T ⊂ V (O = O_{K}) becomes self-dual T ⟶^{∼} T^{∗}(1) if we multiply the pairing by a suitable scalar in K^{∗}.

Let π(f) be the automorphic representation of P GL_{2}(A_{F}) attached to f. The conjectures of Bloch and Kato ([BK], [FoPR]) predict that h_{f}^{1}(F, V_{𝔭}(f)(1)) := dim_{K} H_{f}^{1}(F, V_{𝔭}(f)(1)) should be equal to

h_{f}^{1}(F, V_{𝔭}(f)(1)) =^{?} ord_{s=1}L(f, s) = ord_{s=1/2} L(π(f), s).

The following result was proved in [N5, Thm. 1.4] if 2 ∤ [F : Q] or if π(f)_{v} is not a principal series representation for some finite prime v of F (and then generalised to almost all non CM cases and many CM cases in [N5, Thm. 3.5]). Thanks to Theorem B (and a separate argument in the case when the residual representation \overline{T} is reducible) we obtain the following definitive result.

Theorem C (= Theorem 8.7). If p ≠ 2, then

dim_{K} H_{f}^{1}(F, V_{𝔭}(f)(1)) ≡ ord_{s=1}L(f, s) (mod 2)

holds, for every cuspidal Hilbert modular newform f over F of parallel weight 2 and trivial character.

As explained in [N1, 12.11.6] and [N5, 4.2], potential modularity results of [BLGGT] and [W] imply, respectively, the following consequences of Theorem C (which were proved in [N5, Thm. 4.3, Thm. A] and [N6, Thm. 5.10] under additional hypotheses).

Theorem D (= Theorem 8.9). Let A be an abelian variety with real multiplication by O_{M} (where M is a totally real number field of degree [M : Q] = dim(A)) defined over a totally real number field F. If 𝔭 | p in M and p ≠ 2, then

dim_{M_{𝔭}} H_{f}^{1}(F, V_{𝔭}(A)) = rk_{O_{M}} A(F) + cork_{O_{M𝔭}} �(A/F)[𝔭^{∞}] ≡ ord_{s=1}L(ιA/F, s) (mod 2)

holds, for every ι : M ↪ R.

Theorem E (= Theorem 8.10). Let E be an elliptic curve defined over a totally real number field F. If p ≠ 2, then

dim_{Q_{p}} H_{f}^{1}(F, V_{p}(E)) = rk_{Z}E(F) + cork_{Z_{p}}�(E/F)[p^{∞}] ≡ ord_{s=1}L(E/F, s) (mod 2)

(this is also known to be true for p = 2 in all non CM cases and in many CM cases, by [N5, Thm. A] and an earlier result of [DD2, Thm. 2.4]).

The proof of Theorem C goes as follows. Firstly, one can assume that 2 | [F : Q] and that π(f)_{v} is a principal series representation for all finite primes v of F. Secondly, after replacing F by a suitable abelian extension of p-power degree (which does not change the statement, since p is odd) one can assume that π(f)_{v} becomes an unramified principal series representation after a base change to an abelian extension of F_{v} of degree prime to p, for all finite primes v of F. In particular, π(f)_{v} is a tame principal series representation of P GL_{2}(F_{v}), for all v | p. A refinement of Taylor’s level raising [T1, Thm. 1] involving types [BDJ, Lemma 2.9 and its proof] produces another eigenform f′ of parallel weight 2 and trivial character with the following properties:

• π(f′)_{v_{0}} is an unramified twist of the Steinberg representation for some v_{0} ∤ 𝔫p;

• the level of f′ divides 𝔫v_{0};

• for all v | p, π(f′)_{v} is a tame principal series representation of P GL_{2}(F_{v});

• the residual representations \overline{T} and \overline{T}′ of V and V′, respectively, have isomorphic semisimplifications.

In particular, if \overline{T} is (absolutely) irreducible, then \overline{T} ≃ \overline{T}′. Theorem B then applies and yields

dim_{K} H_{f}^{1}(F, V_{𝔭}(f)(1)) − ord_{s=1}L(f, s) ≡ dim_{K} H_{f}^{1}(F, V_{𝔭}(f′)(1)) − ord_{s=1}L(f′, s) (mod 2),

but the right hand side is conguent to 0 (mod 2), by [N5, Thm. 1.4].

If \overline{T} is reducible, one applies a different argument based on a combination of an Euler characteristic formula for Breuil–Kisin modules (equipped with tame descent data) with a generalisation of a formula of Cassels [C1, Thm. 1.1]. This yields the following abstract result (see [DD1, Thm. 2], [DD2, Thm. 5.7], [Č, Thm. 1.4] and [CFKS, Thm. 2.1] for analogous results for Selmer groups of elliptic curves and abelian varieties, respectively).

Theorem F (= Theorem 7.10). If, in the global situation discussed before Theorem B (with p ≠ 2), the following conditions are satisfied:

(i) \overline{T} contains a Γ_{L}-stable Lagrangian subspace; (ii) for each finite prime v ∤ p of L there exist a finite abelian extension L(v)/L_{v} and a finite Galois extension K(v)/L(v) such that p ∤ [K(v) : L(v)] and T |_{Γ_{K(v)}} is unramified; (iii) for each finite prime v | p of L there exists a finite abelian extension K(v)/L_{v} such that T |_{Γ_{K(v)}} = T_{π}G for some p-divisible group G (equipped with an O-action) over O_{K(v)}; then

(−1)^{χ_{f} (L,V)} = ε(V).

If dim_{K}(V) = 2, then the statement still holds if there exist finite primes v of L not satisfying (ii) or (iii), but for which W D(V |_{Γ_{Lv}}) has nontrivial monodromy.

This article is dedicated to the memory of Ian Cassels, who passed away in July 2015. His celebrated survey [C2] and the series of articles “Arithmetic of curves of genus 1, I–VIII” formed a very important part of the author’s mathematical education. It is no coincidence that the Cassels–Tate pairing in its various incarnations features prominently in all proofs of parity results such as Theorems B, C, D, E and F above. Acknowledgements. The author is grateful to James Newton for patiently answering questions about Breuil–Kisin modules with tame descent data and to the referee for a very careful reading of the article and a number of helpful comments. Notation and conventions. The cardinality of a finite set X is denoted by |X|. We abbreviate ⊗_{Z} as ⊗.

1. Flat cohomology of finite flat group schemes

(1.1) Let p ≠ 2 be a prime number, let K (the base field) be a finite extension of Q_{p}, with ring of integers O_{K} and residue field k. Denote by Γ_{K} := Gal(\overline{K}/K) and I_{K} := Gal(\overline{K}/K^{ur}), respectively, its absolute Galois group and inertia group, and by ∥ · ∥_{K} : K^{×} ⟶ |k|^{Z} the normalised valuation (∥ϖ∥_{K} = |k|^{−1}, for any uniformiser ϖ ∈ O_{K}). Let K (the coefficient field) be another finite extension of Q_{p}, with ring of integers O ⊂ K and residue field F := O/πO, where π ∈ O is a fixed uniformiser.

For an arbitrary finite extension L of Q_{p} we denote by rec_{L} : L^{×} ⟶ Gal(L^{ab}/L) = Γ_{L}^{ab} the reciprocity map, normalised by letting uniformisers correspond to geometric Frobenius elements. If ℓ is any prime number, let

χ_{cycl,L,ℓ} : Γ_{L}^{ab} ⟶ Aut_{Z_{ℓ}}(μ_{ℓ^{∞}}(\overline{L})) = Z_{ℓ}^{×}

be the ℓ-adic cyclotomic character and \overline{χ}_{cycl,L,ℓ} : Γ_{L}^{ab} ⟶ Aut_{F_{ℓ}}(μ_{ℓ}(\overline{L})) = F_{ℓ}^{×} its reduction modulo ℓ. Recall that χ_{cycl,L,ℓ} = χ_{cycl,Q_{p},ℓ} ∘ N_{L/Q_{p}} and

∀u ∈ Z_{p}^{×} ∀n ∈ Z  χ_{cycl,Q_{p},ℓ}(rec_{Q_{p}}(p^{n}u)) = \{ \begin{matrix}u, & ℓ = p, \\ p^{−n}, & ℓ ≠ p \end{matrix}  (1.1.1)

([Se2, Thm. 2], with a sign change arising from our normalisation of the reciprocity map). (1.2) Denote by (p − Gr/O_{K}) the exact category of finite flat abelian group schemes of p-power rank over Spec(O_{K}) and by (F − vs/O_{K}) the F-linear subcategory of finite flat F-vector space schemes over Spec(O_{K}) (in particular, (F_{p} −vs/O_{K}) consists of all objects of (p−Gr/O_{K}) killed by p). Similarly, let (p−div/O_{K}) be the Z_{p}-linear exact category of p-divisible groups (Barsotti–Tate groups) over Spec(O_{K}) and (O − div/O_{K}) the O-linear subcategory of π-divisible O-modules (i.e., p-divisible groups equipped with an O-action). If G is an object of any of the above categories, we denote by G^{D} its Cartier dual. (1.3) The generic fibre G_{K} of any G ∈ (p − Gr/O_{K}) can be identified with the finite Γ_{K}-module G_{K}(\overline{K}). The flat cohomology groups H_{fl}^{i}(O_{K}, G) (for the fppf or the fpqf topologies) have the following properties ([M], [Mi, III.1]).

(1.3.1) The canonical map

H_{fl}^{i}(O_{K}, G) ⟶ H_{fl}^{i}(K, G_{K}) = H_{et}^{i}(K, G_{K}) = H^{i}(Γ_{K}, G_{K}(\overline{K})) = H^{i}(K, G_{K}(\overline{K})) = H^{i}(K, G_{K})

is bijective (resp. injective) for i = 0 (resp. for i = 1). (1.3.2) H_{fl}^{i}(O_{K}, G) = 0 for i > 1. (1.3.3) An exact sequence 0 ⟶ G_{1} ⟶ G_{2} ⟶ G_{3} ⟶ 0 in (p − Gr/O_{K}) gives rise to an exact sequence of

finite p-primary abelian groups

0 ⟶ H_{fl}^{0}(O_{K}, G_{1}) ⟶ H_{fl}^{0}(O_{K}, G_{2}) ⟶ H_{fl}^{0}(O_{K}, G_{3}) ⟶ H_{fl}^{1}(O_{K}, G_{1}) ⟶
⟶ H_{fl}^{1}(O_{K}, G_{2}) ⟶ H_{fl}^{1}(O_{K}, G_{3}) ⟶ 0.

(1.3.4) The orthogonal complement of H_{fl}^{1}(O_{K}, G) ⊂ H^{1}(K, G_{K}) under the pairing

∪ : H^{1}(K, G_{K}) × H^{1}(K, G_{K}^{D}) ⟶ H^{2}(Γ_{K}, G_{m}(\overline{K})) ⟶^{∼} Q/Z

is equal to H_{fl}^{1}(O_{K}, G^{D}). In particular, if G ⟶^{∼} G^{D} is self-dual, then H_{fl}^{1}(O_{K}, G) is a Lagrangian subspace of H^{1}(K, G_{K}).

(1.3.5) The Euler characteristic formula [MRo, Prop. 8.1]:

|H_{fl}^{0}(O_{K}, G)|/|H_{fl}^{1}(O_{K}, G)| = ∥disc_{G/O_{K}} ∥_{K}^{1/rk(G)},

where disc_{G/O_{K}} ∈ (O_{K} − {0})/O_{K}^{×2} is the discriminant of the finite free (generically étale) O_{K}-algebra Γ(G, O_{G}).

(1.4) If G = (G[π^{n}])_{n≥1} ∈ (O − div/O_{K}), then T := T_{π}G = \underleftarrow{lim}_{n} G[π^{n}](\overline{K}) = \underleftarrow{lim}_{n} T/π^{n}T is a Γ_{K}-stable

O-lattice in V := V_{π}G = T ⊗_{O} K, which is a crystalline representation of Γ_{K} with coefficients in K and whose Hodge–Tate weights are contained in {0, 1}. Conversely, any such V is of the form V = V_{π}G for some G ∈ (O − div/O_{K}), by [B3, Thm. 5.3.2]. Note that \overline{T} := T/πT = G[π](\overline{K}).

The maps H_{fl}^{1}(O_{K}, G[π^{n+1}]) ⟶ H_{fl}^{1}(O_{K}, G[π^{n}]) induced by multiplication by π are surjective (by (1.3.3)); their projective limit

\underleftarrow{lim}_{n} H_{fl}^{1}(O_{K}, G[π^{n}]) ⊂ \underleftarrow{lim}_{n} H^{1}(K, G[π^{n}](\overline{K})) = \underleftarrow{lim}_{n} H^{1}(K, T/π^{n}T) = H^{1}(K, T)

is equal to the Bloch–Kato subspace

\underleftarrow{lim}_{n} H_{fl}^{1}(O_{K}, G[π^{n}]) = H_{f}^{1}(K, T)  (1.4.1)

and H_{f}^{1}(K, T)/π^{n}H_{f}^{1}(K, T) = H_{fl}^{1}(O_{K}, G[π^{n}]) for all n ≥ 1 (see [N4, Prop. A.2.6]). In particular,

ℱ := Im(H_{f}^{1}(K, T) ⟶ H^{1}(K, \overline{T})) = H_{f}^{1}(K, T)/πH_{f}^{1}(K, T) = H_{fl}^{1}(O_{K}, G[π]) ⊂ H^{1}(K, \overline{T}).  (1.4.2)

(1.5) Let G′ = (G′[π^{n}])_{n≥1} ∈ (O − div/O_{K}). Assume that there exists an isomorphism of the generic fibres G[π]_{K} ⟶^{∼} G′[π]_{K} (equivalently, an isomorphism of F[Γ_{K}]-modules \overline{T} = T/πT ⟶^{∼} T′/πT′ =: \overline{T}′, where T′ := T_{π}G′), which we fix. As in Introduction, we are interested in the relative position of ℱ and

ℱ′ := Im(H_{f}^{1}(K, T′) ⟶ H^{1}(K, \overline{T}′)) = H_{fl}^{1}(O_{K}, G′[π]) ⊂ H^{1}(K, \overline{T}′) ⟶^{∼} H^{1}(K, \overline{T}).

According to [R, Prop. 2.2.2], the two flat models G[π] and G′[π] of \overline{T} admit an infimum, i.e., a universal object H ∈ (p − Gr/O_{K}) for which H_{K}(\overline{K}) = \overline{T} and for which there exist morphisms

G[π] ⟶ H ⟵ G′[π].

This is again an object of (F − vs/O_{K}). We show in Proposition 2.13(1) below that the sum of the two canonical maps

ℱ = H_{fl}^{1}(O_{K}, G[π]) ⟶ H_{fl}^{1}(O_{K}, H) ⟵ H_{fl}^{1}(O_{K}, G′[π]) = ℱ′  (1.5.1)

(which are injective, since all three cohomology groups inject into the same space H^{1}(K, \overline{T}), thanks to (1.3.1) for i = 1) is surjective. As a result,

ℱ /(ℱ ∩ ℱ′) ⟶^{∼} (ℱ + ℱ′)/ℱ′ = H_{fl}^{1}(O_{K}, H)/H_{fl}^{1}(O_{K}, G′[π]).  (1.5.2)

In Theorem 2.16 below we show that

dim_{F} ℱ /(ℱ ∩ ℱ′) ≡ 0 (mod (p − 1)[k : k ∩ F]),

which implies Theorem A in the case K = L.

2. Breuil–Kisin modules and flat cohomology

(2.1) Let K and K be as in 1.1 (in particular, p ≠ 2). We need some additional notation: W := W (k), K_{0} := W [1/p], e := e(K/Q_{p}) = [K : K_{0}]. Fix a uniformiser ϖ of O_{K} and denote by E(X) ∈ W [X] the minimal polynomial of ϖ over K_{0}. This is an Eisenstein polynomial: E(X) ≡ X^{e} (mod pW [X]), E(0)/p ∈ W^{×}.

Let φ : W ⟶ W be the lift of the absolute Frobenius φ : a ↦ a^{p} on k and define φ : W [[u]] ⟶ W [[u]] by φ(A(u)) = (^{φ}A)(u^{p}), where^{φ}(∑ a_{i}u^{i}) = ∑ φ(a_{i})u^{i}. The same formula defines φ on the p-adic completion O_{E} of W ((u)) = W [[u]][1/u]. (2.2) Fix a compatible system of p-power roots ϖ^{1/p^{n}} of ϖ and let K_{∞} = ⋃_{n≥1} K(ϖ^{1/p^{n}}). Fontaine’s construction [Fo1, Thm. 3.4.3] based on the field of norms k((u)) of K_{∞}/K rather than that of K(μ_{p^{∞}})/K ([B1], [K2, 1.1.12]) attaches to every continuous representation of Γ_{K_{∞}} = Gal(\overline{K}/K_{∞}) on a Z_{p}-module of finite type T_{0} (not necessarily free) an O_{E}-module of finite type D(T_{0}) (free if T_{0} is free over Z_{p}) equipped with a φ-semilinear map φ : D(T_{0}) ⟶ D(T_{0}) for which

φ ⊗ id : φ^{∗}D(T_{0}) = D(T_{0}) ⊗_{O_{E},φ} O_{E} ⟶ D(T_{0})

is an isomorphism (D(T_{0}) is an étale φ-module over O_{E}). The functor

T_{0} ↦ D(T_{0}) := (T_{0} ⊗_{Z_{p}} O_{E^{ur}})^{Γ_{K∞}}

defines an equivalence of abelian categories (compatible with tensor products and duals)

{continuous representations of Γ_{K_{∞}} on finite type Z_{p}−modules}

↓^{≈}

{étale φ−modules of finite type over O_{E}}

with quasi-inverse given by

M ↦ T_{0}(M) := (M ⊗_{O_{E}} O_{E^{ur}})^{φ=id}

Above, O_{E^{ur}} denotes a certain Cohen ring with uniformiser p and residue field k((u))^{sep}. (2.3) Kisin ([K1, Thm. 2.2.7], [K2, Cor. 2.2.22]), building upon an earlier work of Breuil [B3, Thm. 4.2.2.9], established an exact antiequivalence of exact categories between (p−div/O_{K}) and the category BT_{φ} of pairs (M, φ), where:

(2.3.1) M is a free W [[u]]-module of finite type; (2.3.2) φ = φ_{M} : M ⟶ M is a φ-semilinear map; (2.3.3) M^{∘} = Im(φ ⊗ id : φ^{∗}(M) ⟶ M) (i.e., the W [[u]]-submodule of M generated by φ(M)) contains

E(u)M; (2.3.4) morphisms are W [[u]]-linear maps commuting with φ.

(2.4) It will be more convenient for us to use covariant functors. The composition of Kisin’s functor with Cartier duality yields an exact equivalence of exact Z_{p}-linear categories

M : (p − div/O_{K}) ⟶^{≈} BT_{φ}

with the following properties [K3, Thm. 1.2.1, Thm. 1.4.2]. If G = (G[p^{n}])_{n≥1} ∈ (p − div/O_{K}), T_{p}G := \underleftarrow{lim} G[p^{n}](\overline{K}) and V_{p}G := T_{p}G ⊗_{Z_{p}} Q_{p}, then:

(2.4.1) there is a functorial isomorphism

M(G) ⊗_{W [[u]]} O_{E} ⟶^{∼} D((T_{p}G)(−1)|_{Γ_{K∞}});

(2.4.2) the filtered φ-module of the crystalline representation (V_{p}G)(−1) of Γ_{K} (whose Hodge–Tate weights are contained in {−1, 0}) is given by functorial isomorphisms

D_{cris}((V_{p}G)(−1)) ⟶^{∼} (M(G)/uM(G))[1/p],

D_{dR}((V_{p}G)(−1)) = F il^{0} ⟶^{∼} φ^{∗}(M(G))⊗_{W [[u]]}K ⊃ F il^{1} ⟶^{∼} (φ⊗id)^{−1}(E(u)M(G))⊗_{W} K_{0} ⊃ F il^{2} = 0,

where the morphism W [[u]] ⟶ K is given by sending u to ϖ; (2.4.3) if G = (μ_{p^{n}})_{n≥1}, then M(G) = W [[u]] and φ(1) = 1; (2.4.4) if G = \underline{Z}_{p} = (\underline{Z/p^{n}Z})_{n≥1}, then M(G) = W [[u]](−1), whose underlying W [[u]]-module is equal to W [[u]]

and where φ(1) = pE(u)/E(0).

(2.5) Proposition. Let G ∈ (p − div/O_{K}), M = M(G) ∈ BT_{φ}. The cohomology groups H^{i}(C^{•}(M)) of the complex

C^{•}(M) := [φ_{M} − pE(u)/E(0) : M ⟶ M^{∘}]

concentrated in degrees 0 and 1 are functorially isomorphic to

H^{i}(C^{•}(M)) ⟶^{∼} Ext_{BT_{φ}}^{i}(M(\underline{Z}_{p}), M) = Ext_{(p−div/O_{K})}^{i}(\underline{Z}_{p}, G) (i = 0, 1).

Proof. For i = 0 the isomorphism Hom_{BT_{φ}}(M(\underline{Z}_{p}), M) ⟶^{∼} Ker(φ_{M} − pE(u)/E(0) : M ⟶ M^{∘}) is obtained by sending α : W [[u]](−1) ⟶ M to α(1). For i = 1 we need to study exact sequences

0 ⟶ M ⟶ N ⟶ W [[u]](−1) ⟶ 0,

where N is a free W [[u]]-module equipped with a φ-semilinear map φ_{N} : N ⟶ N and where the two morphisms commute with φ.

A choice of a W [[u]]-linear splitting s : W [[u]](−1) ⟶ N identifies (N, φ_{N}) with

M ⊕ W [[u]](−1), φ_{N}(m, f) = (φ_{M}(m) + φ(f)m_{s}, φ(f)pE(u)/E(0)),

where

m_{s} = φ_{N}(s(1)) − pE(u)/E(0) · s(1) ∈ M.

If we choose another splitting s′, then m_{s′} − m_{s} ∈ (φ_{M} − pE(u)/E(0))M. We claim that

N^{∘} ⊃ E(u)N ⟺ (M ⊕ W [[u]](−1))^{∘} ⊃ E(u)(M ⊕ W [[u]](−1)) ⟺ m_{s} ∈ M^{∘}.

Indeed, if N^{∘} ⊃ E(u)N, then there exist f_{i}, g_{i} ∈ W [[u]] and m_{i} ∈ M such that

(0, pE(u)/E(0)) = ∑_{i=1}^{n} g_{i} φ_{N}(m_{i}, f_{i}) = (∑_{i} g_{i} φ_{M}(m_{i}) + (∑_{i} g_{i} φ(f_{i})) m_{s}, (∑_{i} g_{i} φ(f_{i})) pE(u)/E(0)) ,

hence ∑_{i} g_{i} φ(f_{i}) = 1 and m_{s} = − ∑_{i} g_{i} φ_{M}(m_{i}) ∈ M^{∘}. Conversely, if m_{s} ∈ M^{∘}, then m_{s} = ∑_{i} h_{i} φ_{M}(m_{i}) for some h_{i} ∈ W [[u]] and m_{i} ∈ M, which implies that

(m_{s}, 0) = ∑_{i} h_{i} φ_{N}(m_{i}, 0), (0, pE(u)/E(0)) = φ_{N}(0, 1) − ∑_{i} h_{i} φ_{N}(m_{i}, 0) ∈ N^{∘}.

As (E(u)M, 0) ⊂ M^{∘} ⊂ N^{∘}, it follows that E(u)N ⊂ N^{∘}, as claimed.

As a result, the map Ext_{BT_{φ}}^{1}(M(\underline{Z}_{p}), M) ⟶ H^{1}(C^{•}(M)) sending the class of N to the class of m_{s} is well defined and surjective. It is clearly additive and injective.

(2.6) Corollary. For G ∈ (p − div/O_{K}) and M = M(G) there is a functorial isomorphism

H_{f}^{1}(K, T_{p}G) ⟶^{∼} H^{1}(C^{•}(M)) = M^{∘}/(φ_{M} − pE(u)/E(0))M.

Proof. The subspace H_{f}^{1}(K, T_{p}G) ⊂ H^{1}(K, T_{p}G) classifies those extensions of continuous Z_{p}-representations of Γ_{K}

0 ⟶ T_{p}G ⟶ T ⟶ Z_{p} ⟶ 0

for which T = T_{p}G′, for some G′ ∈ (p − div/O_{K}) (by [BK, 3.7] and [B3, Thm. 5.3.2]). As the functor G ↦ T_{p}G is fully faithful [Ta, Thm. 4], the group of such extensions is isomorphic to the group of extensions

0 ⟶ G ⟶ G′ ⟶ \underline{Z}_{p} ⟶ 0

in (p − div/O_{K}), which is, in turn, isomorphic to H^{1}(C^{•}(M)), by Proposition 2.5.

As the referee pointed out, the above argument (which gives another proof of (1.4.1), when combined with Proposition 2.9 below) can be replaced by a direct appeal to the equality (1.4.1) (and Proposition 2.9). (2.7) The above discussion has a version with coefficients, namely, an exact equivalence of categories

M : (O − div/O_{K}) ⟶^{≈} BT_{O,φ},

where BT_{O,φ} is the O-linear category of (O ⊗_{Z_{p}} W)[[u]]-modules of finite type, free as W [[u]]-modules, equipped with an O-linear and (id ⊗ φ)-semilinear map φ_{M} : M ⟶ M for which M^{∘} ⊃ E(u)M.

It turns out that such a module is automatically free as an (O ⊗_{Z_{p}} W)[[u]]-module; this follows from the following two facts. Firstly, the O-algebra O ⊗_{Z_{p}} W is isomorphic to a product of r copies of the ring of integers O′ of a finite unramified extension K′ of K of degree [K′ : K] = [k : F_{p}]/r. Secondly, the semigroup of all nonnegative powers (id ⊗ φ)^{n} (n ≥ 0) of the Frobenius map id ⊗ φ : O ⊗_{Z_{p}} W ⟶ O ⊗_{Z_{p}} W acts transitively on the set of O-algebra morphisms O ⊗_{Z_{p}} W ⟶ O′, since φ generates Gal(k/F_{p}).

For G ∈ (O − div/O_{K}) and M = M(G), the complex C^{•}(M) from Proposition 2.5 satisfies

H^{i}(C^{•}(M)) ⟶^{∼} Ext_{BT_{O,φ}}^{i}(M(\underline{O}), M) = Ext_{(O−div/O_{K})}^{i}(\underline{O}, G) (i = 0, 1),

which yields, as in Corollary 2.6, a functorial O-linear isomorphism

H_{f}^{1}(K, T_{π}G) ⟶^{∼} M^{∘}/(φ_{M} − pE(u)/E(0))M.

(2.8) There is a similar description of (F − vs/O_{K}) ([B3, Thm. 3.3.7], [K2, Prop. 1.1.11]), given by a commutative diagram of exact functors

\xymatrix{M : (O − div/O_{K}) \ar[d] \ar[r] & BT_{O,φ} \ar[d] \\ M : (F − vs/O_{K}) \ar[r] & BT_{F,φ},}  (2.8.1)

where

(2.8.2) BT_{F,φ} is the F-linear category of pairs (M, φ_{M}), where M is a free (F ⊗ k)[[u]]-module of finite type equipped with an F-linear and (id ⊗ φ)-semilinear map φ_{M} : M ⟶ M such that u^{e}M ⊂ M^{∘} (where M^{∘} is the (F ⊗ k)[[u]]-submodule of M generated by φ_{M}(M));

(2.8.3) the vertical functors are given, respectively, by G ↦ G[π] and M ↦ M/πM; (2.8.4) the horizontal functors are equivalences of categories; (2.8.5) for every G ∈ (F − vs/O_{K}) there is a functorial isomorphism

M(G)[1/u] := M(G) ⊗_{k[[u]]} k((u)) ⟶^{∼} D(G_{K}(\overline{K})(−1)|_{Γ_{K∞}}),

which is a vector space over k((u)) of dimension rk_{F}(G) = dim_{F} G_{K}(\overline{K}).

(2.9) Proposition. Let G ∈ (F − vs/O_{K}), M = M(G) ∈ BT_{F,φ}. The cohomology groups H^{i}(C^{•}(M)) of the complex

C^{•}(M) := [φ_{M} − cu^{e} : M ⟶ M^{∘}] (c := p/E(0) (mod pW) ∈ k^{×})

concentrated in degrees 0 and 1 are functorially isomorphic to

H^{i}(C^{•}(M)) ⟶^{∼} Ext_{BT_{F,φ}}^{i}(M(\underline{F}), M) = Ext_{(F−vs/O_{K})}^{i}(\underline{F}, G) = H_{fl}^{i}(O_{K}, G) (i = 0, 1).

Proof. The proof of Proposition 2.5 applies, with straightforward modifications.

(2.10) Proposition (The Euler characteristic formula). Let G ∈ (F − vs/O_{K}), M = M(G) ∈ BT_{F,φ}.

(1) For every integer n > e/(p − 1) the subcomplex Cn^{•}(M) := [φ_{M} − cu^{e} : u^{n}M ⟶ u^{n+e}M] ⊂ C^{•}(M) is acyclic and C^{•}(M) is quasi-isomorphic to

C^{•}(M)/Cn^{•}(M) = [φ_{M} − cu^{e} : M/u^{n}M ⟶ M^{∘}/u^{n+e}M] .

(2) The Euler characteristic χ_{F}(O_{K}, G) := dim_{F} H_{fl}^{0}(O_{K}, G) − dim_{F} H_{fl}^{1}(O_{K}, G) is equal to

χ_{F}(O_{K}, G) = dim_{F}(M/M^{∘}) − [K : Q_{p}] rk_{F}(G).

Proof. (1) If pn > n + e, then (u^{−e}φ_{M})(u^{n+e}M) ⊂ u^{−e}u^{p(n+e)}M ⊂ u^{n+pe+1}M, the series

∑_{i≥0} −(c^{−1}u^{−e}φ_{M})^{i}((cu^{e})^{−1}m)

converges for m ∈ u^{n+e}M and defines an inverse map to φ_{M} − cu^{e} : u^{n}M ⟶ u^{n+e}M.

(2) It follows from (1) that χ_{F}(O_{K}, G) is equal to

dim_{F}(M/u^{n}M) − dim_{F}(M^{∘}/u^{n+e}M) = dim_{F}(M/M^{∘}) − dim_{F}(M/u^{e}M) = dim_{F}(M/M^{∘}) − e[k : F_{p}] rk_{F}(G).

(2.11) Corollary. If G, G′ ∈ (F − vs/O_{K}) have isomorphic generic fibres G_{K} ⟶^{∼} G′_{K}, then

dim_{F} H_{fl}^{1}(O_{K}, G) − dim_{F} H_{fl}^{1}(O_{K}, G′) = dim_{F} M(G′)/M(G′)^{∘} − dim_{F} M(G)/M(G)^{∘}.

(2.12) If G, G′ ∈ (F−vs/O_{K}) have isomorphic generic fibres, then any choice of an isomorphism f : G_{K} ⟶^{∼} G′_{K} induces an isomorphism

M(f) : M(G)[1/u] ⟶^{∼} M(G′)[1/u]  (2.12.1)

of étale φ-modules over (F ⊗ k)((u)), by (2.8.5) (conversely, any isomorphism (2.12.1) arises from an isomorphism G_{K} ⟶^{∼} G′_{K}, by [B4, Thm. 3.4.3], but we are not going to use this fact).

If we fix f and use M(f) to identify M(G)[1/u] with M(G′)[1/u], then the sum M(G) + M(G′) ⊂ M(G)[1/u] ⟶^{∼} M(G′)[1/u] is an object of BT_{F,φ}, corresponding to H ∈ (F − vs/O_{K}) which is, in the language of [R, Prop. 2.2.2], the infimum of G and G′. The inclusions M(G) ↪ M(H) = M(G) + M(G′) ↩ M(G′) correspond to morphisms G ⟶ H ⟵ G′ inducing isomorphisms of generic fibres G_{K} ⟶^{∼} H_{K} ⟵^{∼} G′_{K} compatible with f.

(2.13) Proposition. Assume that we are in the situation of 2.12.

(1) The sum of the canonical (injective – see 1.5) maps H_{fl}^{1}(O_{K}, G) ⟶ H_{fl}^{1}(O_{K}, H) ⟵ H_{fl}^{1}(O_{K}, G′) defines a surjection H_{fl}^{1}(O_{K}, G) ⊕ H_{fl}^{1}(O_{K}, G′) ⟶ H_{fl}^{1}(O_{K}, H).

(2) H_{fl}^{1}(O_{K}, G)/(H_{fl}^{1}(O_{K}, G) ∩ H_{fl}^{1}(O_{K}, G′)) ⟶^{∼} H_{fl}^{1}(O_{K}, H)/H_{fl}^{1}(O_{K}, G′), where the intersection takes

place inside H_{fl}^{1}(O_{K}, H).

(3) dim_{F} H_{fl}^{1}(O_{K}, G)/(H_{fl}^{1}(O_{K}, G) ∩ H_{fl}^{1}(O_{K}, G′)) = dim_{F} M(G′)/M(G′)^{∘} − dim_{F} M(H)/M(H)^{∘}.

Proof. We have M(H)^{∘} = M(G)^{∘} + M(G′)^{∘}, since M(H) = M(G) + M(G′). It follows that the canonical map H^{1}(C^{•}(M(G)) ⊕ H^{1}(C^{•}(M(G′)) ⟶ H^{1}(C^{•}(M(H)) is surjective, which proves (1). The statement (2) is a consequence of (1), and (3) follows from (2) and Corollary 2.11 applied to H and G′. (2.14) Decompositions of F ⊗ k and M(G). Fix embeddings of F and k into a sufficiently large finite extension of F_{p}. Let f = [k : F_{p}], f′ = [F ∩ k : F_{p}]. The map

F ⊗ k ⟶^{∼} ∏_{i=0}^{f′−1} Fk,  a ⊗ b ↦ (i ↦ aφ^{i}(b))  (2.14.1)

is an isomorphism of F-algebras under which the action of id⊗φ on the left hand side corresponds to the map (x_{0}, …, x_{f′−1}) ↦ (x_{1}, …, x_{f′−1}, φ^{v}(x_{0})) on the right hand side, where v ∈ Z satisfies v ≡ 0 (mod [F : F_{p}]) and v ≡ f′ (mod f).

For every M ∈ BT_{F,φ} we obtain from (2.14.1) a decomposition

M ⟶^{∼} ⨁_{i=0}^{f′−1} M_{i},  M_{i} := M ⊗_{F⊗k,id⊗φ^{i}} Fk,  (2.14.2)

where each M_{i} is a free Fk[[u]]-module (of rank independent of i) and φ_{M}|_{M_{i}} : M_{i} ⟶ M_{i−1} satisfies

∀m ∈ M_{i} ∀A(u) ∈ Fk[[u]]  φ_{M}(A(u)m) = (^{φ^{v_{i}}} A)(u^{p}) φ_{M}(m),  (2.14.3)

where v_{0} = v and v_{i} = 0 for i ≠ 0 (we consider the index i in (2.14.3) and in the proof of Proposition 2.15, but not in (2.14.1) and (2.14.2), as an element of Z/f′Z).

In particular, if there exists an embedding k ↪ F, then f′ = f, Fk = F, v = 0 and

M ⟶^{∼} ⨁_{\overline{σ}:k↪F} M\overline{σ},  M\overline{σ} := M ⊗_{F⊗k,id⊗\overline{σ}} F,  φ_{M}|_{M\overline{σ}} : M\overline{σ} ⟶ M_{\overline{σ}∘φ^{−1}}.  (2.14.4)

(2.15) Proposition. Let M, M′ ∈ BT_{F,φ}. If the corresponding étale φ-modules M[1/u], M′[1/u] over (F ⊗ k)((u)) are isomorphic, then

dim_{F}(M′/M^{′∘}) ≡ dim_{F}(M/M^{∘}) (mod (p − 1)[k : k ∩ F]).

Proof. A choice of bases

M = ⨁_{β} Fk[[u]] m_{β}  (m_{β} ∈ M_{i(β)}),  M′ = ⨁_{β′} Fk[[u]] m′_{β′}  (m′_{β′} ∈ M′_{i(β′)})

yields matrices Φ, Φ′ ∈ M_{r}(Fk[[u]]) ∩ GL_{r}(Fk((u))) and C ∈ GL_{r}(Fk((u))) such that

φ(m_{β}) = ∑_{α} Φ_{αβ}(u) m_{α},  φ(m′_{β′}) = ∑_{α′} Φ′_{α′β′}(u) m′_{α′},  m′_{γ′} = ∑_{γ} C_{γγ′}(u) m_{γ}.  (2.15.1)

As i(α) = i(β) − 1, i(α′) = i(β′) − 1 and i(γ) = i(γ′) if the coefficients in (2.15.1) are nonzero, these matrices have a block form

C = ( \begin{matrix}C_{0} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & C_{f′−1} \end{matrix} ) ,  Φ = ( \begin{matrix}0 & Φ_{1} & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & Φ_{f′−1} \\ Φ_{0} & 0 & ⋯ & 0 \end{matrix} )  (2.15.2)

(and similarly for Φ′), where the block C_{i} represents the identification between M′_{i}[1/u] and M_{i}[1/u], the block Φ_{i} the map φ_{M} : M_{i} ⟶ M_{i−1} and the block Φ′_{i} the map φ_{M′} : M′_{i} ⟶ M′_{i−1}.

If we compute φ(m′_{β′}) in two different ways, we obtain

φ(m′_{β′}) = ∑_{α′} Φ′_{α′β′}(u) m′_{α′} = ∑_{α′,α} C_{αα′}(u)Φ′_{α′β′}(u) m_{α},

φ(m′_{β′}) = ∑_{β} φ(C_{ββ′}(u) m_{β}) = ∑_{β} (^{φ^{v_{i(β)}}} C_{ββ′}) (u^{p}) φ(m_{β}) = ∑_{α,β} Φ_{αβ}(u) (^{φ^{v_{i(β)}}} C_{ββ′}) (u^{p}) m_{α},

hence

C(u) Φ′(u) = Φ(u) (^{φ\underline{v}}C)(u^{p}),  ^{φ\underline{v}}C = ( \begin{matrix}^{φ^{v}}C_{0} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & C_{f′−1} \end{matrix} ) ,  (2.15.3)

which can be rewritten as

C_{i−1}(u) Φ′_{i}(u) = Φ_{i}(u) (^{φ^{v_{i}}} C_{i})(u^{p}).

Theory of elementary divisors yields

dim_{Fk}(M_{i}/M_{i}^{∘}) = ord_{u}(det(Φ_{i+1}(u))), dim_{Fk}(M′_{i}/M_{i}^{′∘}) = ord_{u}(det(Φ′_{i+1}(u)))

(where M_{i}^{∘} = M^{∘} ∩ M_{i} and M_{i}^{′∘} = M^{′∘} ∩ M′_{i}), hence

dim_{Fk}(M′/M^{′∘}) − dim_{Fk}(M/M^{∘}) = ord_{u} (\frac{det(Φ′(u))}{det(Φ(u))}) = (p − 1) ord_{u}(det C(u)) ≡ 0 (mod (p − 1)),

which proves the claim.

(2.16) Theorem. Assume that G, G′ ∈ (F − vs/O_{K}) have isomorphic generic fibres; fix an isomor-

phism G_{K} ⟶^{∼} G′_{K}. The subspaces ℱ = H_{fl}^{1}(O_{K}, G) ⊂ H^{1}(K, G_{K}(\overline{K})) and ℱ′ = H_{fl}^{1}(O_{K}, G′) ⊂

H^{1}(K, G′_{K}(\overline{K})) ⟶^{∼} H^{1}(K, G_{K}(\overline{K})) then satisfy

dim_{F} ℱ′ ≡ dim_{F} ℱ ≡ dim_{F}(ℱ ∩ ℱ′) (mod (p − 1)[k : k ∩ F]).

Proof. The first congruence follows from Corollary 2.11 and Proposition 2.15 for M = M(G) and M′ = M(G′). The second congruence follows from Proposition 2.13(3) and Proposition 2.15 for M = M(G) + M(G′) = M(H) and M′ = M(G′). (2.17) Example. If μ_{p} ⊂ K, then (p − 1) | e | [K : Q_{p}] and the group schemes G = μ_{p}, G′ = \underline{Z/pZ} have isomorphic generic fibres. The corresponding cohomology groups are

ℱ = O_{K}^{×} ⊗ Z/pZ ⊂ K^{×} ⊗ Z/pZ = H^{1}(K, μ_{p}),
ℱ′ = Hom(Γ_{K}/I_{K}, Z/pZ) ⊂ Hom(Γ_{K}, Z/pZ) = H^{1}(K, Z/pZ),
ℱ ∩ ℱ′ = Z/pZ(u ⊗ 1), u ∈ O_{K}^{×}, K( ^{p} \sqrt{u})/K unramified,

hence

dim_{F_{p}} ℱ = 1 + [K : Q_{p}], dim_{F_{p}} ℱ′ = dim_{F_{p}}(ℱ ∩ ℱ′) = 1.

(2.18) Corollary. Theorem A holds in the case L = K.

Proof. As V and V′ are crystalline, the Weil-Deligne representations W D(V) and W D(V′) are unramified (and ∥ · ∥_{K}-symplectic, in the language of [N2, 1.5.3]), which implies that ε(V) = ε(V′) = 1. As p is odd, dim_{F} ℱ /(ℱ ∩ ℱ′) ≡ 0 (mod 2), by Theorem 2.16.

3. Interlude: invariants of symplectic lattices

(3.1) The goal of this section is to prove a result comparing two invariants of pairs of symplectic lattices (Proposition 3.10 below) that will be used in the proof of Theorem A. A key technical point (Lemma 3.14) was inspired by the arguments in [KMR, Lemma 2.3, Proposition 2.4]. (3.2) If Δ is a group acting on a commutative ring R by ring automorphisms, we denote by R_{τ} [Δ] = {∑_{δ∈Δ} r_{δ}δ | r_{δ} ∈ R, the sum is finite} the twisted group ring with multiplication (∑ r_{δ}δ)(∑ r′_{δ′}δ′) = ∑ r_{δ}δ(r′_{δ′})δδ′. (3.3) From now on until the end of §3, Δ will be a finite cyclic group of even order d and η : Δ ⟶ F^{×} an injective group morphism into the multiplicative group of a field F (in particular, the characteristic of F is prime to d, hence different from 2).

We let Δ act on F [[u]] ⊂ F ((u)) by δ(∑ a_{i}u^{i}) = ∑ a_{i}(η(δ)u)^{i} (so that F [[u]]^{Δ} = F [[u^{d}]] and F ((u))^{Δ} = F ((u^{d}))). The twisted group rings F [[u]]_{τ} [Δ] ⊂ F ((u))_{τ} [Δ] satisfy F [[u]]_{τ} [Δ]/(u) = F [Δ].

The Grothendieck group G_{0}(F [Δ]) = K_{0}(F [Δ]) is a free abelian group on [η^{i}] (the class of the character η^{i} : Δ ⟶ F^{×}), for i ∈ Z/dZ.

(3.4) Proposition-Definition. (1) The morphisms of abelian groups

I : K_{0}(F [Δ]) ⟶ Z/dZ, [η^{i}] ↦ i
I′ : G_{0}(F [[u]]_{τ} [Δ]) ⟶ Z/dZ, X ↦ I([X/uX]) − I([X[u] ⊗ η])

are well defined and the following relations hold: I(X^{∨}) = −I(X) (where X^{∨} denotes the dual representation) and I′(X) + I′(Y) = I′(X ∩ Y) + I′(X + Y).

(2) If X is a F [[u]]_{τ} [Δ]-module which has finite length ℓ_{F [[u]]}(X) over F [[u]], then

I′(X) = −ℓ_{F [[u]]}(X) (mod d).

Proof. (1) If 0 ⟶ X ⟶ Y ⟶ Z ⟶ 0 is an exact sequence of F [[u]]_{τ} [Δ]-modules of finite type, the snake lemma implies that

0 ⟶ X[u] ⊗ η ⟶ Y [u] ⊗ η ⟶ Z[u] ⊗ η ⟶^{∂} X/uX ⟶ Y/uY ⟶ Z/uZ ⟶ 0

is an exact sequence of F [Δ]-modules of finite type (the map ∂ is induced by multiplication by u, whence the twist by η). Therefore I′ is well defined, as claimed. The formula for I(X^{∨}) is immediate; the remaining formula follows from the exact sequence

0 ⟶ X ∩ Y ⟶ X ⊕ Y ⟶ X + Y ⟶ 0.

(2) As both I′ and ℓ_{F [[u]]} factor through the Grothendieck group of modules of finite length, it is enough to check the statement in the case when X is a simple F [[u]]_{τ} [Δ]-module. This means that X = X[u] = X/uX = F with Δ acting by η^{i}, hence I′(X) = i − (i + 1) (mod d) = −1 (mod d), as claimed.

(3.5) Definition. Let W be a F ((u))_{τ} [Δ]-module of finite type (which is equivalent to W being a finitedimensional vector space over F ((u)) equipped with a semilinear action of Δ). We say that X ⊂ W is a τ-lattice in W if it is a F [[u]]_{τ} [Δ]-module of finite type such that F ((u))X = W (in other words, if it is a Δ-stable F [[u]]-lattice in W). In this case X[u] = 0 and I′(X) = I(X/uX).

(3.6) From now on until the end of §3 we assume that we are given a nonzero W as in Definition 3.5, equipped with a nondegenerate skew-symmetric F ((u))-bilinear pairing

( , ) : W × W ⟶ F ((u))  (3.6.1)

satisfying

∀δ ∈ Δ ∀x, y ∈ W (δ(x), δ(y)) = δ((x, y))

(briefly, (3.6.1) is a Δ-equivariant symplectic form on W).

(3.7) If X ⊂ W is a τ-lattice, so is

X^{∗} = {y ∈ W | ∀x ∈ X (x, y) ∈ F [[u]]}.

If Y ⊂ W is another τ-lattice, then

(X^{∗})^{∗} = X,  (X + Y)^{∗} = X^{∗} ∩ Y^{∗},  (X ∩ Y)^{∗} = X^{∗} + Y^{∗}.  (3.7.1)

If X ⊂ Y are τ-lattices, then (3.6.1) induces a nondegenerate F [[u]]-bilinear Δ-equivariant pairing

Y/X × X^{∗}/Y^{∗} ⟶ F ((u))/F [[u]].  (3.7.2)

We say that a τ-lattice X is self-dual if X = X^{∗}. If this is the case, then (3.6.1) restricted to X ×X induces a symplectic (i.e., nondegenerate, F -bilinear and alternating) pairing

\overline{( , )} : X/uX × X/uX ⟶ F,

which is Δ-equivariant (with Δ acting trivially on F). In other words, X/uX is a (finite-dimensional) symplectic F [Δ]-module.

(3.8) The Grothendieck group G_{0}(F [Δ]_{Sp}) = K_{0}(F [Δ]_{Sp}) of such symplectic modules is isomorphic to the image

Im(K_{0}(F [Δ]) ⟶ K_{0}(F [Δ])),  A ↦ A + A^{∨},  (3.8.1)

which is a free abelian group on [η^{i}] + [η^{−i}] (0 < i < d/2), 2[1] and 2[η^{d/2}].

(3.9) Proposition-Definition. The morphism of abelian groups

J : K_{0}(F [Δ]_{Sp}) ⟶ Z/2Z, A + A^{∨} ↦ I(A) (mod 2)

is well defined.

Proof. This follows from the fact that the kernel of (3.8.1) is generated by elements of the form A − A^{∨}, for which I(A − A^{∨}) = 2I(A) ∈ 2Z.

(3.10) Proposition. If X, Y ⊂ W are self-dual τ-lattices, then

I′(X + Y) (mod 2) = I((X + Y)/u(X + Y)) (mod 2) = J(X/uX) + J(Y/uY) ∈ Z/2Z.

(3.11) Lemma. Proposition 3.10 holds if uX ⊂ Y ⊂ u^{−1}X.

Proof. The inclusions uX ⊂ Y ⊂ u^{−1}X imply that u(X + Y) ⊂ X ∩ Y ⊂ X + Y. Applying (3.7.2) to the inclusions u(X + Y) ⊂ X, X ∩ Y ⊂ X and using (3.7.1), we obtain nondegenerate Δ-equivariant F -bilinear pairings

\frac{X ∩ Y}{u(X + Y)} × \frac{u^{−1}(X ∩ Y)}{X + Y} ⟶ u^{−1}F [[u]]/F [[u]] = [η^{−1}] ⟵ \frac{X}{X ∩ Y} × \frac{Y}{X ∩ Y} .

If we multiply the first pairing by u, we obtain on (X ∩Y)/u(X +Y) a structure of a symplectic F [Δ]-module, which means that the corresponding classes in K_{0}(F [Δ]) satisfy

M = [\frac{X ∩ Y}{u(X + Y)}] = M^{∨},  N = [\frac{X}{X ∩ Y} ] = [η^{−1}]P^{∨},  P = [\frac{Y}{X ∩ Y} ] = [η^{−1}]N^{∨}.

The graded quotients of the filtrations by F [Δ]-submodules

\frac{X ∩ Y}{u(X + Y)} ⊂ \frac{X}{u(X + Y)} ⊂ \frac{X + Y}{u(X + Y)},  \frac{u(X + Y)}{uX} ⊂ \frac{X ∩ Y}{uX} ⊂ \frac{X}{uX} ,  \frac{u(X + Y)}{uY} ⊂ \frac{X ∩ Y}{uY} ⊂ \frac{Y}{uY}

can be expressed in terms of M, N and P, which yields the following relations in K_{0}(F [Δ]):

[\frac{X + Y}{u(X + Y)}] = M + N + P,  [X/uX] = M + N + [η]P,  [Y/uY] = M + P + [η]N.

Therefore

[X/uX] + [Y/uY] = [\frac{X + Y}{u(X + Y)}] + [\frac{X + Y}{u(X + Y)}]^{∨} ,

which implies that J(X/uX) + J(Y/uY) = I((X + Y)/u(X + Y)) (mod 2), as claimed.

(3.12) Lemma. If X, Y ⊂ W are self-dual τ-lattices, then there exists a chain of self-dual τ-lattices X = X_{0}, X_{1}, …, X_{n} = Y such that uX_{i} ⊂ X_{i+1} ⊂ u^{−1}X_{i} holds, for all i = 0, …, n − 1.

Proof. There exists n ≥ 0 such that u^{n}X ⊂ Y ⊂ u^{−n}X. There is nothing to prove if n = 0, 1. If n > 1, then Z = uX + (Y ∩ u^{−1}X) is a τ-lattice satisfying

Z^{∗} = (uX)^{∗} ∩ (Y ∩ u^{−1}X)^{∗} = u^{−1}X ∩ (Y + uX) = (u^{−1}X ∩ Y) + uX = Z,

thanks to (3.7.1). Moreover, uX ⊂ Z ⊂ u^{−1}X, Z ⊂ uX + Y ⊂ u^{1−n}Y and u^{n−1}Y = (u^{1−n}Y)^{∗} ⊂ Z^{∗} = Z. We define X_{1} = Z and repeat the argument with X_{1}, Y and n − 1.

(3.13) Lemma. Assume that X, Y, Z ⊂ W are self-dual τ-lattices.

(1) I′(X) = I′(Y) = 0 and I′(X + Y) = −I′(X ∩ Y).

(2) The τ-lattice U := (X + Y) ∩ Z + (X ∩ Y) is self-dual.

(3) The formula [x + y, x′ + y′] := (x, y′) (mod F [[u]]) defines a symmetric F [[u]]-bilinear Δ-equivariant pairing [ , ] : ((X + Y) ∩ Z) × ((X + Y) ∩ Z) ⟶ F ((u))/F [[u]] with kernel (X ∩ Z) + (Y ∩ Z).

Proof. (1) I′(X) = I([X/uX]), but X/uX is symplectic, hence [X/uX] = A + A^{∨} for some A ∈ K_{0}(F [Δ]), which implies that I([X/uX]) = I(A)+I(A^{∨}) = 0. The same argument applies to Y. The remaining formula follows from Proposition 3.4(1).

(2) The rules (3.7.1) imply that

U^{∗} = ((X +Y)∩Z)^{∗}∩(X ∩Y)^{∗} = ((X +Y)^{∗}+Z)∩(X +Y) = ((X ∩Y)+Z)∩(X +Y) = (X ∩Y)+Z ∩(X +Y).

(3) We apply the argument from [KMR, Lemma 2.3]. The pairing is well defined, since x ∈ X and y′ ∈ Y are unique modulo X ∩ Y = (X + Y)^{∗}. It is symmetric, since (x, y′) − (x′, y) = (x + y, x′ + y′) − (x, x′) − (y, y′) = (z, z′) − (x, x′) − (y, y′) ∈ F [[u]] (note that both z = x + y and z′ = x′ + y′ lie in Z). The submodule (X ∩ Z) + (Y ∩ Z) is contained in the kernel of [ , ], by definition. Conversely, if z = x + y lies in the kernel of [ , ], then

∀z′ = x′ + y′ ∈ (X + Y) ∩ Z ∀x′′ ∈ X ∩ Y 0 = [z, z′] = (x, y′) (mod F [[u]]) = (x, z′ + x′′) (mod F [[u]]),

which means that x ∈ U^{∗} = U. As a result, after modifying x ∈ X and y ∈ Y by an element of X ∩ Y, we can assume that x, y ∈ Z, hence z = x + y ∈ (X ∩ Z) + (Y ∩ Z).

(3.14) Lemma. If X, Y, Z ⊂ W are self-dual τ-lattices, then:

(1) I′((X + Y) ∩ Z)) ≡ I′((X ∩ Z) + (Y ∩ Z)) (mod 2).

(2) I′(X + Y) − I′(X + Z) + I′(Y + Z) (mod 2) = 0 ∈ Z/2Z.

Proof. (1) The quotient A := ((X + Y) ∩ Z)/((X ∩ Z) + (Y ∩ Z)) has finite length over F [[u]], which implies, by Proposition 3.4(2), that

I′((X + Y) ∩ Z)) − I′((X ∩ Z) + (Y ∩ Z)) = −ℓ_{F [[u]]}(A) (mod d).

We are going to deduce from the existence of a nondegenerate symmetric Δ-equivariant F [[u]]-bilinear pairing [ , ] : A × A ⟶ F ((u))/F [[u]] established in Lemma 3.13(3) that the length of A is even.

There exists an isomorphism of F [[u]]-modules A ⟶^{∼} ⨁_{i=1}^{n}(u^{−i}F [[u]]/F [[u]])^{b_{i}}. The F -vector space A[u] has a filtration

A[u] =: F^{1} ⊃ uA[u^{2}] =: F^{2} ⊃ ⋯ ⊃ u^{n−1}A =: F^{n} ⊃ 0 =: F^{n+1}

with graded quotients gr_{F}^{i} = F^{i}/F^{i+1} satisfying dim_{F} (gr_{F}^{i}) = b_{i}. For each i = 1, …, n, the pairing

[ , ]_{i} : gr_{F}^{i} × gr_{F}^{i} ⟶ u^{i−2}F [[u]]/u^{i−1}F [[u]] = [η^{i−2}]

defined by the formula

[u^{i−1}x, u^{i−1}y]_{i} := u^{2i−2} ([x, y] (mod u^{1−i}F [[u]])) (x, y ∈ A[u^{i}])

is F -bilinear, nondegenerate and Δ-equivariant. If i = 2m + 1 is odd, this implies that

[gr_{F}^{2m+1}] = ∑_{j=1}^{d/2} c_{j} ([η^{m−j}] + [η^{m−1+j}]) ∈ K_{0}(F [Δ])

for some c_{j} ∈ Z, hence b_{2m+1} ≡ 0 (mod 2). It follows that

ℓ_{F [[u]]}(A) = ∑_{i=1}^{n} ib_{i} ≡ 0 (mod 2),

as required.

(2) As in the proof of [KMR, Prop. 2.4], we use repeatedly the last formula from Proposition 3.4(1). The congruence (1) implies that

I′(X ∩ Z) + I′(Y ∩ Z) = I′((X ∩ Z) + (Y ∩ Z)) + I′(X ∩ Y ∩ Z) ≡ I′((X + Y) ∩ Z) + I′(X ∩ Y ∩ Z) (mod 2),

hence

I′(X ∩ Y) + I′(X ∩ Z) + I′(Y ∩ Z) ≡ I′(X ∩ Y) + I′((X + Y) ∩ Z) + I′(X ∩ Y ∩ Z) =
= I′(U) + 2I′(X ∩ Y ∩ Z) ≡ I′(U) ≡ 0 (mod 2).

The last congruence follows from Lemma 3.13(1), since the τ-lattice U = (X + Y) ∩ Z + (X ∩ Y) is self-dual, by Lemma 3.13(2). This finishes the proof of Lemma 3.14. Proof of Proposition 3.10. We must show that the invariant f(X, Y) = I′(X + Y) (mod 2) + J(X/uX) + J(Y/uY) ∈ Z/2Z of a pair of self-dual τ-lattices X, Y is identically zero. Lemma 3.14(2) tells us that f(X, Y) + f(Y, Z) = f(X, Z). In particular, if X = X_{0}, …, X_{n} = Y are as in Lemma 3.12, then f(X, Y) = ∑_{i=1}^{n} f(X_{i−1}, X_{i}), but f(X_{i−1}, X_{i}) = 0 for all i, by Lemma 3.11, which implies that f(X, Y) = 0.

4. Breuil–Kisin modules with totally tamely ramified descent data

(4.1) We return to the situation of 2.1 and 2.2 (in particular, p ≠ 2). Assume that L is an intermediate field Q_{p} ⊂ L ⊂ K such that K/L is a Galois extension with Galois group Δ.

(4.2) Definition [BCDT, 4.1], [Sav1, Def. 3.1]. A descent datum relative to K/L on G ∈ (p − Gr/O_{K}) is a collection of morphisms [δ] : G ⟶ ^{δ}G (δ ∈ Δ), where [id] = id, [δδ′] = (^{δ}[δ′]) ∘ [δ] and ^{δ}G is defined by a cartesian diagram

\xymatrix{^{δ}G \ar[d] \ar[r]^{can} & G \ar[d] \\ Spec(O_{K}) \ar[r]^{Spec(δ)} & Spec(O_{K}).}

Note that these conditions ensure that each [δ] is an isomorphism.

(4.3) Such a descent datum gives rise to a descent datum on the Cartier dual G^{D} and to an action of Γ_{L} on G_{K}(\overline{K}) that extends the canonical action of Γ_{K} ([BCDT, 4.1], [Sav1, 3.1]): if γ ∈ Γ_{L} and δ = γ|_{K} ∈ Δ, then the action of γ on a point Q ∈ G_{K}(\overline{K}) = G(O\overline{K}) = Hom_{(Sch/Spec(O_{K}))}(Spec(O\overline{K}), G) is given by

γ(Q) = can∘[δ^{−1}]∘Q∘Spec(γ) : Spec(O\overline{K}) ⟶^{Spec(γ)} Spec(O\overline{K}) ⟶^{Q} G(O\overline{K}) ⟶^{[δ^{−1}]} (^{δ^{−1}}G)(O\overline{K}) ⟶^{can} G(O\overline{K}). (4.3.1)

Equivalently, the generic fibre G_{K} descends to a finite flat group scheme G_{L} over Spec(L) and the action (4.3.1) is the Galois action of Γ_{L} on G_{L}(\overline{K}). (4.4) Similarly, a descent datum relative to K/L on G = (G[π^{n}])_{n≥1} ∈ (O − div/O_{K}) consists of a compatible system of descent data on all G[π^{n}]. We obtain, therefore, a diagram of exact O-linear categories and functors

\xymatrix{(O − div/O_{K})_{dd,K/L} \ar[d] \ar[r] & (O − div/O_{K}) \ar[d] \\ (O[Γ_{L}] − Mod) \ar[r] & (O[Γ_{K}] − Mod)}  (4.4.1)

in which the vertical functors are given by G ↦ T_{π}G and the horizontal ones forget the descent data. Tate’s full faithfulness theorem [Ta, Thm. 4] for the right vertical arrow implies that the diagram (4.4.1) is 2cartesian, i.e., if G ∈ (O − div/O_{K}), then an extension of the Γ_{K}-action on T_{π}G to a Γ_{L}-action defines a descent datum relative to K/L on G.

There is a similar diagram of exact F-linear categories and functors

\xymatrix{(F − vs/O_{K})_{dd,K/L} \ar[d] \ar[r] & (F − vs/O_{K}) \ar[d] \\ (F[Γ_{L}] − Mod) \ar[r] & (F[Γ_{K}] − Mod),}

which is not 2-cartesian in general.

(4.5) Proposition. For each G ∈ (O − div/O_{K})_{dd,K/L} the diagram

\xymatrix{Ext_{(O−div/O_{K})_{dd,K/L}}^{1}(\underline{O}, G) \ar[d] \ar[r] & Ext_{(O−div/O_{K})}^{1}(\underline{O}, G) \ar[d] \\ Ext_{(O[Γ_{L}]−Mod)}^{1}(O, T_{π}G) \ar[r] & Ext_{(O[Γ_{K}]−Mod)}^{1}(O, T_{π}G)}

is cartesian, both vertical arrows are injective and their respective images are equal to H_{f}^{1}(L, T_{π}G) ⊂ H^{1}(Γ_{L}, T_{π}G) and H_{f}^{1}(K, T_{π}G) ⊂ H^{1}(Γ_{K}, T_{π}G), respectively.

Proof. The statement about the right vertical arrow was discussed in 2.6. The diagram is cartesian, since (4.4.1) is 2-cartesian. This implies that the left vertical arrow is injective and its image is equal to

Ker(H^{1}(L, T_{π}G) ⟶ H^{1}(K, T_{π}G) ⟶ H^{1}(K, V_{π}G)/H_{f}^{1}(K, V_{π}G)) =

Ker(H^{1}(L, T_{π}G) ⟶ (H^{1}(K, V_{π}G)/H_{f}^{1}(K, V_{π}G))^{Δ} = H^{1}(L, V_{π}G)/H_{f}^{1}(L, V_{π}G)) = H_{f}^{1}(L, T_{π}G).

(4.6) It would be natural to try to generalise the results of §2 to the subspaces

ℱ = Im(H_{f}^{1}(L, T) ⟶ H^{1}(L, \overline{T})),  ℱ′ = Im(H_{f}^{1}(L, T′) ⟶ H^{1}(L, \overline{T}′) ⟶^{∼} H^{1}(L, \overline{T}))  (4.6.1)

(where T = T_{π}G and T′ = T_{π}G′), whenever G, G′ ∈ (O − div/O_{K})_{dd,K/L} have isomorphic \overline{T} = G[π](\overline{K}) ⟶^{∼} \overline{T}′ = G′[π](\overline{K}) (as F[Γ_{L}]-modules).

The equivalences of categories discussed in 2.3 and 2.8 were extended – in their original version due to Breuil [B3] – to the case incorporating descent data in [BCDT, 5.6]. The description of general descent data on Breuil modules is fairly complicated, especially if the extension K/L is wildly ramified. If K/L is tamely ramified and if there exists a uniformiser ϖ ∈ O_{K} such that ϖ^{e(K/L)} ∈ O_{L}, then Δ ≃ Gal(K_{∞}/L_{∞}) ≃ Gal(k((u))/k_{L}((u^{e(K/L)}))) and the theory becomes much simpler [Sav2, Prop. 3.2]. The corresponding theory for Kisin modules was discussed in this particular case in [NY, §5]. For our purposes it will be sufficient to consider the simplest possible case, namely, when the extension K/L is totally tamely ramified. (4.7) From now on until the end of §4 we assume that K/L is a totally tamely ramified Galois extension. Its Galois group Δ is then cyclic of order d dividing q − 1, where q = p^{f} = |k| is the cardinality of the common residue field of K and L. After changing the uniformiser if necessary we can – and will – assume that ϖ^{d} ∈ O_{L}. The isomorphism

η : Δ ⟶^{∼} μ_{d}(O_{L}) = μ_{d}(W) ⟶^{∼} μ_{d}(k),  η(δ) = δ(ϖ)/ϖ  (4.7.1)

is canonical (it does not depend on the choice of ϖ). The restriction of the character η ∘ rec_{L} : L^{×} ⟶ μ_{d}(k) to O_{L}^{×} factors through (O_{L}/ϖ^{d}O_{L})^{×} = k^{×} and

(η ∘ rec_{L})(−1) = (−1)^{(q−1)/d}.  (4.7.2)

If L_{∞} = ⋃_{n≥1} L((ϖ^{1/p^{n}})^{d}), then there are canonical isomorphisms

Δ ≃ Gal(K_{∞}/L_{∞}) ≃ Gal(k((u))/k((u^{d}))),

where k((u)) (resp. k((u^{d}))) is the field of norms of K_{∞}/K (resp. of L_{∞}/L) and δ(u) = η(δ)u for all δ ∈ Δ. We let, therefore, Δ act on (O ⊗_{Z_{p}} W)[[u]], O ⊗_{Z_{p}} O_{E} and (F ⊗ k)[[u]] by the following formula (cf. 3.3):

δ( ∑_{i≥0} a_{i}u^{i}) = ∑_{i≥0} a_{i}(1 ⊗ η(δ))^{i}u^{i}.

Note that (O ⊗_{Z_{p}} O_{E})^{Δ} is the p-adic completion of (O ⊗_{Z_{p}} W)[[u^{d}]][1/u^{d}]. (4.8) In this particular situation the descent data on Kisin modules have the following explicit form ([Sav2, Prop. 3.2], [NY, Def. 5.3]). An object of (BT_{O,φ})_{dd,K/L} (resp. of (BT_{F,φ})_{dd,K/L}) is an object M of BT_{O,φ} (resp. of BT_{F,φ}) equipped with maps [δ] : M ⟶ M (δ ∈ Δ) satisfying

(4.8.1) [δ](m + m′) = [δ](m) + [δ](m′); (4.8.2) [id] = id and [δδ′] = [δ] ∘ [δ′]; (4.8.3) [δ] ∘ φ_{M} = φ_{M} ∘ [δ]; (4.8.4) [δ] ((∑_{i≥0} a_{i}u^{i})m) = (∑_{i≥0} a_{i}(1⊗η(δ)^{i})u^{i})[δ](m), for all m ∈ M and a_{i} ∈ O⊗_{Z_{p}} W (resp. a_{i} ∈ F⊗k).

As pointed out by the referee, this semilinearity condition was initially stated incorrectly in [NY], where the authors considered only the special case when ∑_{i≥0} a_{i}u^{i} = a_{j}u^{j} is a monomial.

(4.9) The equivalences of categories discussed in 2.4, 2.7 and 2.8 extend as follows. The diagram (2.8.1) has an analogue (compatible with (2.8.1) via the functors “forget the descent data”)

\xymatrix{M : (O − div/O_{K})_{dd,K/L} \ar[d] \ar[r] & (BT_{O,φ})_{dd,K/L} \ar[d] \\ M : (F − vs/O_{K})_{dd,K/L} \ar[r] & (BT_{F,φ})_{dd,K/L},}

in which the horizontal functors are equivalences of categories and the vertical functors are as in (2.8.3). The properties from 2.4 have the following analogues. If G ∈ (O − div/O_{K})_{dd,K/L}, then the following statements hold.

(4.9.1) M(G) ⊗_{(O⊗_{Zp} W)[[u]]} O_{E} is an étale φ-module over O ⊗_{Z_{p}} O_{E} equipped with a semilinear action of Δ, which implies that it comes, by extension of scalars, from an étale φ-module (M(G) ⊗_{(O⊗_{Zp} W)[[u]]} O_{E})^{Δ} over (O ⊗_{Z_{p}} O_{E})^{Δ} attached to L_{∞}/L. Moreover, there is a functorial isomorphism

(M(G) ⊗_{(O⊗_{Zp} W)[[u]]} O_{E})^{Δ} ⟶^{∼} D_{L}((T_{π}G)(−1)|_{L_{∞}}),

where D_{L} denotes Fontaine’s functor attached to L_{∞}/L. (4.9.2) The K-linear continuous representation V_{π}G = (T_{π}G)(−1)⊗_{O}K of Γ_{L} becomes crystalline (with Hodge–

Tate weights contained in {−1, 0}) after restriction to Γ_{K}. Its filtered (φ, Gal(K/L)) = (φ, Δ)-module D := D_{cris}((V_{π}G)(−1)|_{Γ_{K}}) is functorially isomorphic to (M(G)/uM(G))[1/p], with filtration on D_{dR} as in (2.4.2).

(4.9.3) In particular, D is a free K ⊗_{Q_{p}} K_{0}-module equipped with an id ⊗ φ-semilinear bijection φ_{D} : D ⟶ D and a K ⊗_{Q_{p}} K_{0}-linear action of Δ commuting with φ_{D}. The Weil-Deligne representation W D((V_{π}G)(−1)) of the Weil group W_{L} of L attached to (V_{π}G)(−1) is defined as follows ([Fo2], [FoPR, I.1.3.2]). Fix a sufficiently large finite extension K′ of K and an embedding σ : K_{0} ↪ K′. The K′-vector space D_{σ} := D_{K⊗_{Qp} K_{0},id⊗σ} K′ is then equipped with a K′-linear action of the Weil group W_{L}, given by the formula

w(d ⊗ λ) := (φ_{D}^{[K_{0}:Q_{p}]n(w)} ∘ w|_{K})(d) ⊗ λ,

where n : W_{L} ⟶ W_{L}/I_{L} ⟶^{∼} Z sends the geometric Frobenius element Fr_{geom} ∈ W_{L}/I_{L} of L to 1 ∈ Z (note that the action of the inertia group I_{L} on D_{σ} factors through I_{L}/I_{K} = Δ). By definition, W D((V_{π}G)(−1)) is the isomorphism class of the K′[W_{L}]-module D_{σ}. It does not depend on the choice of σ, since φ_{D} ⊗ id : D_{σ} ⟶ D_{σφ^{−1}} is an isomorphism of K′[W_{L}]-modules.

(4.9.4) If L = Q_{p} and K = Q_{p}(μ_{p}), then e = d = p−1, k = F_{p}, f = 1 and η = id : Δ = (Z/pZ)^{×} ⟶^{∼} Aut(F_{p}). If O = Z_{p}, F = F_{p}, K = Q_{p} and 0 ≤ a < p − 1, then M = Z_{p}[[u]]m, φ(m) = m, δ(m) = η(δ)^{−a}m is an object M ∈ (BT_{O,φ})_{dd,K/L} corresponding to G = (μ_{p^{n}})_{n≥1} with an appropriate descent datum. In this case M^{Δ} = u^{a}Z_{p}[[u^{p−1}]], (M(G) ⊗_{(O⊗_{Zp} W)[[u]]} O_{E})^{Δ} = u^{a}O_{E}^{Δ} and the Γ_{L}-module T = T_{p}G is isomorphic to Z_{p}(1) ⊗ η^{−a}.

(4.10) The discussion in 2.5–2.13 has straightforward analogues in our situation, since p ∤ |Δ|. If G ∈ (O − div/O_{K})_{dd,K/L} and M = M(G), then T := T_{π}G and \overline{T} := T/πT = G[π](\overline{K}) are Γ_{L}-modules and the restriction maps H^{i}(Γ_{L}, X) ⟶ H^{i}(Γ_{K}, X)^{Δ} (X = T, \overline{T}) are isomorphisms. This implies that H_{f}^{1}(L, T) ≃ H_{f}^{1}(K, T)^{Δ} and

ℱ := Im (H_{f}^{1}(L, T) ⟶ H^{1}(L, \overline{T})) ≃ Im (H_{f}^{1}(K, T) ⟶ H^{1}(K, \overline{T}))^{Δ} = H_{fl}^{1}(O_{K}, G[π])^{Δ}.  (4.10.1)

Let \overline{M} := M/πM = M(G[π]) ∈ (F − vs/O_{K})_{dd,K/L} and c := p/E(0) (mod p)W ∈ k^{×}. The complexes

C^{•}(M) = [φ_{M} − pE(u)/E(0) : M ⟶ M^{∘}] , C^{•}(\overline{M}) = [φ\overline{M} − cu^{e} : \overline{M} ⟶ \overline{M}^{∘}]

are complexes of (O ⊗_{Z_{p}} W)[Δ]-modules and (F ⊗ k)[Δ]-modules, respectively, since Δ acts trivially on E(u) = E_{L}(u^{d}), where E_{L} is the minimal polynomial of ϖ^{d} over K_{0}.

(4.11) Proposition. The functorial morphisms

H^{i}(C^{•}(M)) ⟶^{∼} Ext_{BT_{O,φ}}^{i}(M(\underline{O}), M) ⟶ H^{i}(K, T), H^{i}(C^{•}(\overline{M})) ⟶^{∼} Ext_{BT_{F,φ}}^{i}(M(\underline{F}), \overline{M}) ⟶ H^{i}(K, \overline{T})

(i = 0, 1) from (2.7) and Proposition 2.9 are Δ-equivariant. They induce functorial isomorphisms

H^{0}(C^{•}(M)^{Δ}) = H^{0}(C^{•}(M))^{Δ} ⟶^{∼} H^{0}(K, T)^{Δ}, H^{0}(C^{•}(\overline{M})^{Δ}) = H^{0}(C^{•}(\overline{M}))^{Δ} ⟶^{∼} H^{0}(K, \overline{T})^{Δ},

H^{1}(C^{•}(M)^{Δ}) = H^{1}(C^{•}(M))^{Δ} ⟶^{∼} H_{f}^{1}(K, T)^{Δ}, H^{1}(C^{•}(\overline{M})^{Δ}) = H^{1}(C^{•}(\overline{M}))^{Δ} ⟶^{∼} ℱ .

Proof. The Δ-equivariance follows from the functoriality aspects of Propositions 2.5 and 2.9. The statement of this proposition can be reinterpreted as follows.

(4.12) Proposition. There are functorial isomorphisms

H^{i}(C^{•}(M)^{Δ}) ⟶^{∼} Ext_{(BT_{O,φ})_{dd,K/L}}^{i}(M(\underline{O}), M), H^{i}(C^{•}(\overline{M})^{Δ}) ⟶^{∼} Ext_{(BT_{F,φ})_{dd,K/L}}^{i}(M(\underline{F}), \overline{M})

(i = 0, 1) compatible with those from (2.7) and Proposition 2.9 via the functor “forget the descent data”.

Proof. As p ∤ |Δ|, the splitting s in the proof of Proposition 2.5 can be chosen to be Δ-equivariant, which implies that s(1) ∈ N^{Δ} and m_{s} = (φ_{N} − pE(u)/E(0))s(1) ∈ M^{Δ}, since E(u) = E_{L}(u^{d}) is Δ-invariant.

(4.13) Proposition (The Euler characteristic formula). Let G ∈ (F − vs/O_{K})_{dd,K/L}, M := M(G) ∈ (BT_{F,φ})_{dd,K/L} and C^{•}(M)^{Δ} := [φ_{M} − cu^{e} : M^{Δ} ⟶ (M^{∘})^{Δ}].

(1) There are functorial isomorphisms H^{i}(C^{•}(M)^{Δ}) ⟶^{∼} H_{fl}^{i}(O_{K}, G)^{Δ}.

(2) For every integer n > e/(p − 1) divisible by d the subcomplex Cn^{•}(M)^{Δ} := [φ_{M} − cu^{e} : u^{n}M^{Δ} ⟶ u^{n+e}M^{Δ}] ⊂ C^{•}(M)^{Δ} is acyclic and C^{•}(M)^{Δ} is quasi-isomorphic to

C^{•}(M)^{Δ}/Cn^{•}(M)^{Δ} = [φ_{M} − cu^{e} : (M/u^{n}M)^{Δ} ⟶ (M^{∘}/u^{n+e}M)^{Δ}] .

(3) The Euler characteristic χ_{F}(O_{L}, G) := dim_{F} H_{fl}^{0}(O_{K}, G)^{Δ} − dim_{F} H_{fl}^{1}(O_{K}, G)^{Δ} is equal to

χ_{F}(O_{L}, G) = dim_{F}(M/M^{∘})^{Δ} − [L : Q_{p}] rk_{F}(G).

Proof. (1) Combine (4.10.1) with Proposition 4.11. Part (2) is a direct consequence of Proposition 2.10(1). Part (3) follows from (2) as in the proof of Proposition 2.10(2):

−χ_{F}(O_{L}, G) + dim_{F}(M/M^{∘})^{Δ} = dim_{F}(M/u^{e}M)^{Δ} = ed^{−1} dim_{F}(M/u^{d}M)^{Δ}.

For every integer j ≥ 0 and every i = 0, …, d − 1, multiplication by u^{i} induces an isomorphism of F ⊗ kmodules

u^{i} : (u^{j}M/u^{j+1}M)^{(η^{−i})} := {x ∈ u^{j}M/u^{j+1}M | ∀δ ∈ Δ [δ](x) = (id ⊗ η(δ)^{−i})(x)} ≃ (u^{j+i}M/u^{j+i+1}M)^{Δ} ,

which implies that

dim_{F}(u^{j}M/u^{j+d}M)^{Δ} = ∑_{i=0}^{d−1} dim_{F} (u^{j}M/u^{j+1}M)^{(η^{−i})} = dim_{F}(u^{j}M/u^{j+1}M) = [k : F_{p}] rk_{F}(G)

and ed^{−1} dim_{F}(M/u^{d}M)^{Δ} = ed^{−1}[k : F_{p}] rk_{F}(G) = [L : Q_{p}] rk_{F}(G).

The referee pointed out that this argument shows that every Δ-eigenspace M^{(η^{−i})} (i ∈ Z/dZ) is free of rank rk_{(F⊗k)[[u]]}(M) = rk_{F}(G) over (F ⊗ k)[[u]]^{Δ} = (F ⊗ k)[[u^{d}]].

(4.14) Corollary. If G, G′ ∈ (F − vs/O_{K})_{dd,K/L} have isomorphic generic fibres G_{K} ⟶^{∼} G′_{K} (as F-vector space schemes over Spec(K) equipped with descent data relative to K/L, i.e., as F-vector space schemes over Spec(L)), then

dim_{F} H_{fl}^{1}(O_{K}, G)^{Δ} − dim_{F} H_{fl}^{1}(O_{K}, G′)^{Δ} = dim_{F}(M(G′)/M(G′)^{∘})^{Δ} − dim_{F}(M(G)/M(G)^{∘})^{Δ}.

(4.15) Under the asumptions of Corollary 4.14, a choice of an isomorphism f : G_{K} ⟶^{∼} G′_{K} induces a Δ-equivariant isomorphism M(G)[1/u] ⟶^{∼} M(G′)[1/u] of étale φ-modules over (F⊗k)((u)). As in 2.12, the sum M(G) + M(G′) ⊂ M(G)[1/u] is an object of (BT_{F,φ})_{dd,K/L} corresponding to H ∈ (F − vs/O_{K})_{dd,K/L} equipped with morphisms G ⟶ H ⟵ G′ inducing isomorphisms G_{K} ⟶^{∼} H_{K} ⟵^{∼} G′_{K} compatible with f.

(4.16) Proposition. Assume that we are in the situation of 4.15.

(1) The sum of the canonical injective maps ℱ = H_{fl}^{1}(O_{K}, G)^{Δ} ⟶ H_{fl}^{1}(O_{K}, H)^{Δ} ⟵ H_{fl}^{1}(O_{K}, G′)^{Δ} = ℱ′ defines a surjection ℱ ⊕ ℱ′ ⟶ H_{fl}^{1}(O_{K}, H)^{Δ}.

(2) ℱ /(ℱ ∩ ℱ′) ⟶^{∼} H_{fl}^{1}(O_{K}, H)^{Δ}/ℱ′, where the intersection takes place inside H_{fl}^{1}(O_{K}, H)^{Δ}.

(3) dim_{F} ℱ /(ℱ ∩ ℱ′) = dim_{F}(M(G′)/M(G′)^{∘})^{Δ} − dim_{F}(M(H)/M(H)^{∘})^{Δ}.

Proof. Taking Δ-invariants is exact, since p ∤ |Δ|, so the claims (1) and (2) follow from Proposition 2.13. The statement (3) follows from (2) and Corollary 4.14. (4.17) Decomposition of M (the case k ↪ F). From now on until the end of §4 we assume that there exists an embedding σ_{0} : k ↪ F. This condition is not very restrictive, since one can always replace F by a finite extension F′, O by O′ = O ⊗_{W (F)} W (F′) and G ∈ (O −div/O_{K})_{dd,K/L} (resp. G ∈ (F−vs/O_{K})_{dd,K/L}) by O′ ⊗_{O} G (resp. by F′ ⊗_{F} G). As in (2.14.4), any M ∈ (BT_{F,φ})_{dd,K/L} decomposes as

M ⟶^{∼} ⨁_{i∈Z/fZ} M_{i}, M_{i} := M ⊗_{F⊗k,id⊗σ_{i}} F, σ_{i} := σ_{0} ∘ φ^{i} : k ↪ F, φ_{M}|_{M_{i}} : M_{i} ⟶ M_{i−1}, (4.17.1)

where M_{i} is a free F[[u]]-module of rank rk_{F}(M), for each i ∈ Z/fZ (f = [k : F_{p}]). The relation (4.8.4) implies that

∀δ ∈ Δ ∀A(u) ∈ F[[u]] ∀m ∈ M_{i}  [δ](A(u)m) = A(η_{i}(δ)u)[δ](m),  (4.17.2)

where

η_{i} := σ_{i} ∘ η = η_{i−1}^{p} : Δ ⟶^{∼} μ_{d}(F).  (4.17.3)

In particular, M_{i} is Δ-stable and M_{i}/uM_{i} is an F[Δ]-module, of dimension rk_{F}(M) as an F-vector space.

(4.18) Proposition-Definition. (1) Every M ∈ (BT_{F,φ})_{dd,K/L} is of the form M = ⨁_{β} F[[u]]m_{β}, where m_{β} ∈ M_{i(β)} and, for all δ ∈ Δ, [δ]m_{β} = η_{i(β)}(δ)^{−a(β)}m_{β}, a(β) ∈ {0, 1, …, d − 1}. Such a basis {m_{β}} will be called Δ-adapted. Define b(β) ∈ {0, 1, …, d − 1} by b(β) ≡ p a(β) (mod d).

(2) For each i ∈ Z/fZ the class [M_{i}/uM_{i}] ∈ K_{0}(F[Δ]) is equal to ∑_{i(β)=i}[η_{i}^{−a(β)}].

(3) For each i ∈ Z/fZ the multisets 𝒜_{i}(M) := {a(β) | i(β) = i} and ℬ_{i}(M) := {b(β) | i(β) = i} are independent of the chosen Δ-adapted basis. We define a_{i}(M) := ∑_{a∈𝒜_{i}(M)} a and b_{i}(M) := ∑_{b∈ℬ_{i}(M)} b.

Proof. We only need to prove (1). Fix i ∈ Z/fZ. There exists a basis \{\overline{m}_{β}\} of M_{i}/uM_{i} over F on which Δ acts diagonally, by a morphism D : Δ ⟶ ∏_{β=1}^{t} F^{×} ⊂ GL_{t}(F), where t = dim_{F} M_{i}/uM_{i}. Choose representatives n_{β} in M_{i} of all \overline{m}_{β}; they form a basis of M_{i} over F[[u]]. The action of Δ on \{n_{β}\} is given by a 1-cocycle A ∈ Z^{1}(Δ, GL_{t}(F[[u]])) such that A (mod u) = D. For n ≥ 1 let U_{n} = Ker(GL_{t}(F[[u]]) ⟶ GL_{t}(F[[u]]/u^{n})). We have A_{1} = AD^{−1} ∈ Z^{1}(Δ, U_{1}) for a new action of Δ on U_{1}, given by δ ∗ B = D(δ) [δ](B) D(δ)^{−1}. As U_{1} = \underleftarrow{lim}_{n} U_{1}/U_{n} and H^{1}(Δ, U_{n}/U_{n+1}) = H^{1}(Δ, M_{t}(F)) = 0 (since p ∤ |Δ|), the cohomology set H^{1}(Δ, U_{1})

is trivial, which means that there exists B ∈ U_{1} such that A_{1}(δ) = B^{−1} (δ ∗ B). After changing {n_{β}} by the matrix B^{−1} we obtain a basis {m_{β}} of M_{i} over F[[u]] on which Δ acts by D. (4.19) The Frobenius matrix. Let M ∈ (BT_{F,φ})_{dd,K/L}. The action of φ = φ_{M} : M ⟶ M in any Δ-adapted basis {m_{β}} of M can be written as

φ(m_{β}) = ∑_{α} Φ_{αβ}(u) m_{α},  (4.19.1)

where i(α) = i(β) − 1 and Φ_{αβ}(u) ∈ F[[u]]. The formulas (4.8.3) and (4.17.2) then yield, for each δ ∈ Δ,

(φ ∘ [δ])m_{β} = φ(η_{i}(δ)^{−a(β)}m_{β}) = η_{i}(δ)^{−a(β)} ∑_{α} Φ_{αβ}(u) m_{α} = ([δ] ∘ φ)m_{β} =

= ∑_{α}[δ](Φ_{αβ}(u) m_{α}) = ∑_{α} Φ_{αβ}(η_{i−1}(δ)u) [δ]m_{α} = ∑_{α} Φ_{αβ}(η_{i−1}(δ)u) η_{i−1}(δ)^{−a(α)}m_{α},

where i = i(β) = i(α) + 1. This implies, thanks to (4.17.3), that

∀ζ ∈ μ_{d}(F)  Φ_{αβ}(ζu) = ζ^{a(α)−pa(β)}Φ_{αβ}(u).  (4.19.2)

Recall that b(β) was defined by the conditions

b(β) ≡ pa(β) (mod d),  0 ≤ b(β) < d;  (4.19.3)

we can rewrite (4.19.2) as

Φ̃_{αβ}(u) := u^{−a(α)} Φ_{αβ}(u) u^{b(β)} ∈ F((u^{d})).  (4.19.4)

By the definition of a Δ-adapted basis, M^{Δ} is a free F[[u^{d}]]-module with basis {m̃_{β} = u^{a(β)}m_{β}}. The formula (4.19.4) implies that

F[[u]]φ(m_{β}) ∩ M^{Δ} = F[[u^{d}]]u^{b(β)}φ(m_{β}),

hence (M^{∘})^{Δ} is the F[[u^{d}]]-submodule of M^{Δ} (freely) generated by the elements

u^{b(β)}φ(m_{β}) = ∑_{α} Φ̃_{αβ}(u) m̃_{β}.

It follows that the matrix Φ̃(u) := (Φ̃_{αβ}(u)) ∈ GL_{r}(F((u^{d}))) lies in M_{r}(F[[u^{d}]]) and that

dim_{F}(M/M^{∘})^{Δ} = ord_{u^{d}}(det Φ̃(u)).  (4.19.5)

We can reformulate (4.19.4) in a matrix form as

Φ̃ = A^{−1}ΦB ∈ GL_{r}(F((u^{d}))),  (4.19.6)

where A (resp. B) is the diagonal matrix with entries A_{ββ} = u^{a(β)} (resp. B_{ββ} = u^{b(β)}). As in (2.15.2), these matrices have the following block form:

A = ( \begin{matrix}A_{0} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & A_{f−1} \end{matrix} ) ,  Φ = ( \begin{matrix}0 & Φ_{1} & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & Φ_{f−1} \\ Φ_{0} & 0 & ⋯ & 0 \end{matrix} )  (4.19.7)

(and similarly for B and Φ̃), where Φ_{i} is the matrix of φ|_{M_{i}} : M_{i} ⟶ M_{i−1}. The formulas (4.19.5-6) can then be refined as

Φ̃_{i} = A_{i−1}^{−1}Φ_{i}B_{i}  (4.19.8)

and

dim_{F}(M_{i−1}/M_{i−1}^{∘})^{Δ} = ord_{u^{d}}(det Φ̃_{i}(u)) = \frac{r_{i}(M) + b_{i}(M) − a_{i−1}(M)}{d},  (4.19.9)

where

r_{i}(M) := ord_{u}(det Φ_{i}(u)) ≡ a_{i−1}(M) − b_{i}(M) ≡ a_{i−1}(M) − p a_{i}(M) (mod d).  (4.19.10)

In (4.19.8)-(4.19.10) above we consider the index i as an element of Z/fZ.

The following result is a variant of Proposition 2.15. However, only the calculations from its proof (but not the statement itself) will be used in the proof of the general case of Theorem A.

(4.20) Proposition. If M, M′ ∈ (BT_{F,φ})_{dd,K/L} and if the corresponding étale φ-modules M[1/u], M′[1/u] over (F ⊗ k)((u)) ≃ ∏_{i∈Z/fZ} F((u)) are Δ-equivariantly isomorphic, then

dim_{F}(M′/M^{′∘})^{Δ} − dim_{F}(M/M^{∘})^{Δ} ≡ ∑_{i∈Z/fZ} (\frac{b_{i}(M′) − p a_{i}(M′)}{d} − \frac{b_{i}(M) − p a_{i}(M)}{d}) (mod (p − 1)).

Proof. As in 4.19, choose Δ-adapted bases {m_{β}} and {m′_{β′}} of M and M′, respectively. We obtain matrices Φ, A, B, Φ̃ satisfying (4.19.1-8) and analogous matrices Φ′, A′, B′, Φ̃′ for M′. We have

φ(m′_{β′}) = ∑_{α′} Φ′_{α′β′}(u) m′_{α′}, m′_{β′} = ∑_{β} C_{ββ′}(u) m_{β},

where C_{ββ′}(u) ∈ F((u)) are the matrix elements of a matrix C = C(u) = (C_{ββ′}(u)) representing an isomorphism of étale φ-modules M[1/u] ⟶^{∼} M′[1/u]. Note that i(β′) = i(β) whenever C_{ββ′}(u) ≠ 0. Applying [δ] (δ ∈ Δ) to the identities above and using (4.8.3) and (4.17.2), we obtain

∑_{β} C_{ββ′}(η_{i}(δ)u) η_{i}(δ)^{−a(β)}m_{β} = [δ]m′_{β′} = η_{i}(δ)^{−a(β′)}m′_{β′} = ∑_{β} η_{i}(δ)^{−a(β′)}C_{ββ′}(u) m_{β},

where i = i(β′) = i(β), hence

u^{−a(β)} C_{ββ′}(u) u^{a(β′)} ∈ F((u^{d})),

which can be restated in a matrix form as

C̃(u) := A^{−1}C(u)A′ ∈ GL_{r}(F((u^{d}))).  (4.20.1)

As in (2.15.3) we have Φ′ = C^{−1} Φ φ(C) = C(u)^{−1} Φ C(u^{p}), which can be rewritten, using the matrices U(u^{d}) := A(u^{p})^{−1}B(u), U′(u^{d}) := A′(u^{p})^{−1}B′(u) ∈ GL_{r}(F((u^{d}))) and (4.19.6), as

Φ̃′ = C̃(u)^{−1} Φ̃ U(u^{d})^{−1} C̃(u^{p}) U′(u^{d}).

Combined with (4.19.5) this yields

dim_{F}(M′/M^{′∘})^{Δ} − dim_{F}(M/M^{∘})^{Δ} = (p − 1) ord_{u^{d}}(det C̃(u)) + ord_{u^{d}} (\frac{det(U′(u^{d}))}{det(U(u^{d}))}) ,

which proves the claim, since the first term on the right hand side is divisible by p − 1 and

ord_{u}(det(U(u^{d}))) = ∑_{i∈Z/fZ}(b_{i}(M) − p a_{i}(M)).

(4.21) Proposition (The Euler characteristic formula). If M ∈ (BT_{F,φ})_{dd,K/L}, then χ_{F}(M) = dim_{F} H^{0}(C^{•}(M)^{Δ}) − dim_{F} H^{1}(C^{•}(M)^{Δ}) is equal to

χ_{F}(M) = ( ∑_{i∈Z/fZ} \frac{r_{i}(M) + b_{i}(M) − a_{i−1}(M)}{d}) − \frac{ef}{d} rk_{F}(M),

in the notation of 4.18 and 4.19.

Proof. Combine Proposition 4.13(3) with (4.19.9).

(4.22) Definition. We say that M ∈ (BT_{F,φ})_{dd,K/L} is Δ-balanced if the F[Δ]-modules M_{i}/uM_{i} (i ∈ Z/fZ) are all isomorphic (which is equivalent to the class [M_{i}/uM_{i}] ∈ K_{0}(F[Δ]) being independent of i).

(4.23) Proposition. (1) M ∈ (BT_{F,φ})_{dd,K/L} is Δ-balanced if and only if 𝒜_{i−1}(M) = ℬ_{i}(M) for all i ∈ Z/fZ.

(2) If M ∈ (BT_{F,φ})_{dd,K/L} is Δ-balanced, then

∑_{i∈Z/fZ} \frac{p a_{i}(M) − b_{i}(M)}{d} = \frac{p − 1}{d} ∑_{i∈Z/fZ} a_{i}(M) ≡ \frac{p^{f} − 1}{d} a_{i_{0}}(M) (mod (p − 1)),

for any i_{0} ∈ Z/fZ.

(3) If N ∈ (BT_{O,φ})_{dd,K/L}, then M = N/πN ∈ (BT_{F,φ})_{dd,K/L} is Δ-balanced. More precisely, N = M(G) for some G ∈ (O − div/O_{K})_{dd,K/L} and [M_{i}/uM_{i}] corresponds to [(N_{i}/uN_{i})[1/p]] = [W D((V_{π}G)(−1))|_{Δ}] = [W D((V_{π}G))|_{Δ}] ∈ K_{0}(K[Δ]) under the canonical isomorphism K_{0}(F[Δ]) ≃ K_{0}(K[Δ]).

Proof. (1) According to Proposition 4.18(2) and (4.17.3), [M_{i}/uM_{i}] = ∑_{a∈𝒜_{i}(M)} [η_{i}^{−a}] = ∑_{a∈𝒜_{i}(M)} [η_{i−1}^{−pa}] = ∑_{b∈ℬ_{i}(M)} [η_{i−1}^{−b}], which implies that [M_{i}/uM_{i}] = [M_{i−1}/uM_{i−1}] is equivalent to 𝒜_{i−1}(M) = ℬ_{i}(M).

(2) It follows from (1) that ∑_{i} b_{i}(M) = ∑_{i} a_{i}(M) and a_{i}(M) ≡ pa_{i+1}(M) (mod d), hence ∑_{i} a_{i}(M) ≡ (1 + p + ⋯ + p^{f−1})a_{i_{0}}(M) (mod d).

(3) We have [M_{i}/uM_{i}] = [(N_{i}/uN_{i})[1/p]], by the definition of the isomorphism K_{0}(F[Δ]) ≃ K_{0}(K[Δ]) [Se1, §15.5]. The equality [(N_{i}/uN_{i})[1/p]] = [W D((V_{π}G)(−1))|_{Δ}] follows from (4.9.3).

(4.24) Proposition-Definition (Cartier duality). (1) For M ∈ (BT_{O,φ})_{dd,K/L} we define its dual M^{D} ∈ (BT_{O,φ})_{dd,K/L} as follows. As an (O ⊗_{Z_{p}} W)[[u]]-module, M^{D} := Hom_{(O⊗_{Zp} W)[[u]]}(M, (O ⊗_{Z_{p}} W)[[u]]), and we require the tautological pairing ⟨ , ⟩ : M^{D} × M ⟶ (O ⊗_{Z_{p}} W)[[u]] to be Δ-equivariant and satisfy

⟨φ(m^{D}), φ(m)⟩ = (pE(u)/E(0)) φ(⟨m^{D}, m⟩),

where φ on (O ⊗_{Z_{p}} W)[[u]] is given by φ(1) = 1.

(2) There exist functorial isomorphisms M(G)^{D} ⟶^{∼} M(G^{D}), for all G ∈ (O − div/O_{K})_{dd,K/L}.

(3) If G ∈ (O−div/O_{K})_{dd,K/L} and if T := T_{π}G is equipped with an isomorphism of O[Γ_{L}]-modules j : T ⟶^{∼} T^{∗}(1) = Hom_{O}(T, O)(1), then j is induced by a unique isomorphism j_{G} : G ⟶^{∼} G^{D} in (O − div/O_{K})_{dd,K/L}, which in turn gives rise to a perfect Δ-equivariant (O ⊗_{Z_{p}} W)[[u]]-bilinear pairing ( , ) : M(G) × M(G) ⟶ (O ⊗_{Z_{p}} W)[[u]] defined by (x, y) = ⟨j_{G}(x), y⟩. This pairing satisfies

(φ(m), φ(m′)) = (pE(u)/E(0)) φ((m, m′)).

If, in addition, j is skew-symmetric (i.e., j^{∗} = −j), then j_{G}^{D} = −j_{G} and ( , ) is skew-symmetric, too.

(4) For M ∈ (BT_{F,φ})_{dd,K/L} we define its dual M^{D} ∈ (BT_{F,φ})_{dd,K/L} as follows: as an (F ⊗ k)[[u]]-module, M^{D} := Hom_{(F⊗k)[[u]]}(M, (F ⊗ k)[[u]]), with the tautological pairing ⟨ , ⟩ : M^{D} × M ⟶ (F ⊗ k)[[u]] being Δ-equivariant and satisfying

⟨φ(m^{D}), φ(m)⟩ = cu^{e} φ(⟨m^{D}, m⟩),

where c ∈ k^{×} is as in Proposition 2.9.

(5) There exist functorial isomorphisms M(H)^{D} ⟶^{∼} M(H^{D}), for all H ∈ (F − vs/O_{K})_{dd,K/L}.

Proof. (1) Firstly, we must check that the formula for ⟨φ(m^{D}), φ(m)⟩ does, indeed, define a map φ : M^{D} ⟶ M^{D} (which is then unique). If m ∈ M, then we can write (pE(u)/E(0))m = ∑_{i=1}^{n} f_{i} φ(m_{i}), f_{i} ∈ (O ⊗_{Z_{p}} W)[[u]] and m_{i} ∈ M, which implies that

⟨φ(m^{D}), m⟩ = ∑_{i=1}^{n} f_{i} φ(⟨m^{D}, m_{i}⟩),

which proves the claim. Secondly, we must check that, for each m^{D} ∈ M^{D}, there exist finitely many elements g_{l} ∈ (O ⊗_{Z_{p}} W)[[u]] and m_{l}^{D} ∈ M^{D} such that (pE(u)/E(0))m^{D} = ∑_{l} g_{l} φ(m_{l}^{D}). This equality is equivalent to

∀m ∈ M ⟨(pE(u)/E(0))m^{D}, φ(m)⟩ = ∑_{l} g_{l} ⟨φ(m_{l}^{D}), φ(m)⟩,

hence to

∀m ∈ M ⟨m^{D}, φ(m)⟩ = ∑_{l} g_{l} φ(⟨m_{l}^{D}, m⟩).

Fix a basis {e_{i}} of M over (O ⊗_{Z_{p}} W)[[u]]; let {e_{i}^{D}} be the dual basis of M^{D}. It is enough to treat the case m^{D} = e_{i}^{D}. If we write φ(e_{j}) = ∑_{l} f_{jl} e_{l} (f_{jl} ∈ (O ⊗_{Z_{p}} W)[[u]]), then

∀m ∈ M ⟨e_{i}^{D}, φ(m)⟩ = ∑_{l} f_{li} φ(⟨e_{l}^{D}, m⟩),

as required. This finishes the proof that M^{D} is a well-defined object of (BT_{O,φ})_{dd,K/L}.

(2) Write M = M(G) and M^{D} = M(G′), for G, G′ ∈ (O − div/O_{K})_{dd,K/L}. We need to exhibit a functorial isomorphism between G′ and G^{D}.

Note that the perfect pairing in (1) can be written in a more canonical φ-equivariant form as

⟨ , ⟩ : M^{D} × M ⟶ M(\underline{Z}_{p}),

since M(\underline{Z}_{p}) = W [[u]](−1), φ(1) = pE(u)/E(0) and M(μ_{p^{∞}}) = W [[u]], φ(1) = 1.

It follows from (4.9.2) and the fact that D_{cris} and D_{dR} are tensor functors that ⟨ , ⟩ induces a perfect pairing

D_{cris}((V_{π}G′)(−1)|_{Γ_{K}}) ⊗_{O⊗_{Zp} W} D_{cris}((V_{π}G)(−1)|_{Γ_{K}}) ⟶ D_{cris}(K(−1)|_{Γ_{K}})

of filtered (φ, Δ)-modules with coefficients in K ⊗_{Q_{p}} K_{0}, hence a perfect pairing of K[Γ_{L}]-modules

(V_{π}G′)(−1) ⊗_{K} (V_{π}G)(−1) ⟶ K(−1).

Similarly, (4.9.1) implies that ⟨ , ⟩ induces a perfect pairing

(T_{π}G′)(−1)|_{Γ_{L∞}} ⊗_{O} (T_{π}G)(−1)|_{Γ_{L∞}} ⟶ O(−1)

of O[Γ_{L_{∞}}]-modules, compatible with the previous pairing. As a result, it induces an isomorphism of O[Γ_{L}]- modules

T_{π}G′ ⟶^{∼} Hom_{O[Γ_{L}]}(T_{π}(G), O(1)) = T_{π}(G^{D}).

Tate’s full faithfulness theorem implies that this isomorphism is induced by a (unique) isomorphism G′ ⟶^{∼} G^{D} in (O − div/O_{K})_{dd,K/L}, which yields an isomorphism M(G)^{D} ⟶^{∼} M(G^{D}), functorial by construction.

(3) The existence and uniqueness of j_{G} follow from Tate’s full faithfulness theorem. The remaining statements are direct consequences of (1) and (2).

(4) The same arguments as in (1) show that M^{D} is a well-defined object of (BT_{F,φ})_{dd,K/L}.

(5) There exist G ∈ (BT_{O,φ})_{dd,K/L} and an exact sequence 0 ⟶ H ⟶ G[π]. Denote by G′ ∈ (BT_{O,φ})_{dd,K/L} the quotient of G by H: there exists a morphism β : G ⟶ G′ which induces an exact sequence of group schemes

0 ⟶ H ⟶ G[π^{n}] ⟶ G′[π^{n}]

and an exact sequence of O[Γ_{L}]-modules

0 ⟶ T_{π}(G) ⟶ T_{π}(G′) ⟶ H(\overline{L}) ⟶ 0.

In order to ease the notation we are going to write R := (O ⊗_{Z_{p}} W)[[u]].

It follows from [K1, §2.3] that the exact sequence

0 ⟶ M(H) ⟶ M(G[π]) ⟶^{M(β_{1})} M(G′[π])

can also be obtained by applying the snake lemma to the multiplication by π on the exact sequence of R-modules (equipped with actions of φ and Δ)

0 ⟶ M(G) ⟶^{M(β)} M(G′) ⟶ M(H) ⟶ 0.

After applying RHom_{R}(−, R) to the latter sequence, we get exact sequences

0 ⟶ M(G′)^{D} ⟶ M(G)^{D} ⟶ N ⟶ 0

and (via the snake lemma, as before)

0 ⟶ N ⟶ M(G′)^{D}/πM(G′)^{D} ⟶ M(G)^{D}/πM(G)^{D},

the second of which is isomorphic, thanks to (2), to

0 ⟶ N ⟶ M(G′[π]^{D}) ⟶^{M(β_{1}^{D})} M(G[π]^{D}).

However,

N = Ext_{R}^{1}(M(H), R) ⟶^{∼} Hom_{R}(M(H), R/π) = M(H)^{D}

and all of the maps above are compatible with the actions of φ and Δ, which implies that

M(H)^{D} ⟶^{∼} M(Ker(G′[π]^{D} ⟶^{β_{1}^{D}} G[π]^{D})) = M(H^{D}),

as claimed (this isomorphism is functorial, by construction).

(4.25) Proposition. If M ∈ (BT_{F,φ})_{dd,K/L} and if M ⊕ M^{D} is Δ-balanced, then:

(1) ∑_{i=0}^{f−1}(a_{i}(M) − b_{i+1}(M)) p^{i} ≡ 0 (mod 2d).

(2) χ_{F}(M) ≡ \frac{1}{d} ∑_{i=0}^{f−1}(r_{i+1}(M) − e rk_{F}(M)) p^{i} (mod 2).

Proof. (1) For each i ∈ Z/fZ consider the following multisets: 𝒜_{i}^{∗}(M) = {a ∈ 𝒜_{i}(M) | a ≠ 0}, ℬ_{i}^{∗}(M) = {b ∈ ℬ_{i}(M) | b ≠ 0}, 𝒜_{i}(M^{D}) = 𝒜_{i}(M)_{−} = {a_{−} | a ∈ 𝒜_{i}(M)}, ℬ_{i}(M^{D}) = ℬ_{i}(M)_{−} = {b_{−} | b ∈ ℬ_{i}(M)}, where a_{−} = d − a if a ≠ 0 (resp. a_{−} = 0 if a = 0).

According to Proposition 4.23, the assumption that M ⊕ M^{D} is Δ-balanced is equivalent to an equality of multisets

∀i ∈ Z/fZ 𝒜_{i}(M ⊕ M^{D}) = 𝒜_{i}(M) ∪^{·} 𝒜_{i}(M)_{−} = ℬ_{i+1}(M ⊕ M^{D}) = ℬ_{i+1}(M) ∪^{·} ℬ_{i+1}(M)_{−},

which is, in turn, equivalent to

∀i ∈ Z/fZ 𝒜_{i}^{∗}(M) ∪^{·} 𝒜_{i}^{∗}(M)_{−} = ℬ_{i+1}^{∗}(M) ∪^{·} ℬ_{i+1}^{∗}(M)_{−}.

It follows that there exist bijections

𝒜_{0}^{∗}(M) ⟶^{∼} 𝒜_{i}^{∗}(M), a ↦ a[i]

and signs λ(a, i) ∈ {±1} (a ∈ 𝒜_{0}^{∗}(M), i ∈ Z/fZ) such that

p^{i} a[i] ≡ λ(a, i) a (mod d).

If we let μ(a, i) = λ(a, i + 1)λ(a, i)^{−1} ∈ {±1} and

b_{a}[i + 1] = \{ \begin{matrix}a[i] & a ∈ 𝒜_{0}^{∗}(M), μ(a, i) = 1, \\ d − a[i] & a ∈ 𝒜_{0}^{∗}(M), μ(a, i) = −1, \end{matrix}

then

ℬ_{i+1}^{∗}(M) = {b_{a}[i + 1] | a ∈ 𝒜_{0}^{∗}(M)}.

Fix a ∈ 𝒜_{0}^{∗}(M). The equality ∏_{i∈Z/fZ} μ(a, i) = 1 implies that

∑_{{i∈Z/fZ \atop μ(a,i)=−1}} a[i] p^{i} ≡ a ∑_{{i∈Z/fZ \atop μ(a,i)=−1}} λ(a, i) ≡ 0 (mod d),

hence

∑_{i∈Z/fZ}(a[i] − b_{a}[i + 1]) p^{i} ≡ ∑_{{i∈Z/fZ \atop μ(a,i)=−1}}(2a[i] − d) p^{i} ≡ d ∑_{{i∈Z/fZ \atop μ(a,i)=−1}} 1 ≡ 0 (mod 2d).

Taking the sum over all a ∈ 𝒜_{0}^{∗}(M) yields (1).

(2) Write r_{i+1}(M) = a_{i}(M) − b_{i+1}(M) + dt_{i} (t_{i} ∈ Z), for all i ∈ Z/fZ. We have just proved that

0 ≡ ∑_{i=0}^{f−1}(r_{i+1}(M) − dt_{i}) p^{i} ≡ ∑_{i=0}^{f−1} r_{i+1}(M) p^{i} + d ∑_{i=0}^{f−1} t_{i} (mod 2d),

which implies, thanks to Proposition 4.21, that

χ_{F}(M) = ∑_{i=0}^{f−1}(t_{i} − e rk_{F}(M)/d) ≡ \frac{1}{d} ∑_{i=0}^{f−1}(r_{i+1}(M) − e rk_{F}(M)) p^{i} (mod 2).

(4.26) Proposition (The inertia character, [NY, Cor. 5.7]). If M = M(G) ∈ (BT_{F,φ})_{dd,K/L} for G ∈ (F − vs/O_{K})_{dd,K/L}, then the action of I_{L} on T = ∧^{rk_{F}(G)}G_{L}(\overline{K}) is given by the character

det_{G_{L}} : I_{L} ⟶ F^{×},  ∀λ ∈ μ_{p^{f} −1}(O_{L}) = μ_{p^{f} −1}(k)  det_{G_{L}}(rec_{L}(λ)) = σ_{0}(λ)^{− \frac{p^{f} −1}{d} a_{0}(M)−y_{0}},

where y_{0} = \frac{1}{d} ∑_{i=0}^{f−1}(r_{i+1}(M) − e rk_{F}(G)) p^{i}. In particular,

det_{G_{L}}(rec_{L}(−1)) = (−1)^{\frac{p^{f} −1}{d} a_{0}(M)}(−1)^{y_{0}}.

Proof. If rk_{F}(G) = 1, this is a special case of [NY, Cor. 5.7], which generalises a result of Savitt [Sav2, Cor. 2.6], which is, in turn, a generalisation – in the case of finite extensions of Q_{p} – of an earlier result of Raynaud [R, Thm. 3.4.1]. However, the notation and conventions of [NY] differ from ours, which is why we repeat the details of the calculation in our context.

The character ψ = det_{G_{L}} is tame; its composition with rec_{L} : O_{L}^{×} ⟶ Im(I_{L} ⟶ Γ_{L}^{ab}) factors through O_{L}^{×} ⟶ k^{×} = μ_{p^{f} −1}(k). It is sufficient to describe the action of I_{L_{∞}} on the Tate twist

T (−rk_{F}(G))|_{Γ_{L∞}} = (N^{Δ} ⊗_{(F⊗k)[[u^{d}]]} (F ⊗ k_{E^{ur}}))^{φ=id} ,  (4.26.1)

where k_{E^{ur}} ≃ k((u^{d}))^{sep} and N := ∧^{rk_{F}(G)}M(G)[1/u]. The calculations in 4.19 imply that N = ⨁_{i∈Z/fZ} N_{i}, where N_{i} = F((u)) n′_{i} and n′_{i} is the wedge product of all m_{β} such that i(β) = i. The action of δ ∈ Δ is given by [δ]n′_{i} = η_{i}(δ)^{−a_{i}(M)}n′_{i} and the Frobenius acts as φ(n′_{i}) = u^{r_{i}(M)}A_{i}(u^{d}) n′_{i−1} for some A_{i}(u^{d}) ∈ F[[u^{d}]] such that A_{i}(0) ≠ 0.

There exist power series B_{i}(u^{d}) ∈ 1 + u^{d}F[[u^{d}]] such that the new basis elements n_{i} = B_{i}(u^{d})n′_{i} of the N_{i} satisfy (cf. [Sav2, Thm. 3.5], [NY, Thm. 5.5])

φ(n_{i}) = c_{i−1}u^{r_{i}(M)}n_{i−1}

for some c_{i−1} = A_{i}(u^{d})B_{i}(u^{pd})B_{i−1}(u^{d})^{−1} ∈ F^{×} (this follows from the fact that the map

1 + u^{d}F[[u^{d}]] ⟶ 1 + u^{d}F[[u^{d}]], B(u^{d}) ↦ B(u^{p^{f} d})/B(u^{d})

is surjective). As a result, N^{Δ} = ⨁_{i∈Z/fZ} F((u^{d}))ñ_{i}, where ñ_{i} = u^{a_{i}(M)}n_{i} and

φ(ñ_{i}) = c_{i−1}u^{s_{i}} ñ_{i−1}, s_{i} = pa_{i}(M) + r_{i}(M) − a_{i−1}(M) ∈ dZ.

As explained in [NY, Lemma 5.6], there exist α_{i} ∈ \overline{k}^{×} such that α_{i}^{p}c_{i−1} = α_{i−1} for all i ∈ Z/fZ.

We are looking for integers x_{i} ∈ Z (i ∈ Z/fZ) for which the elements β_{i} = ((u^{d})^{1/(p^{f} −1)})^{x_{i}} ∈ F((u^{d}))^{sep} satisfy

∀i ∈ Z/fZ φ(α_{i}β_{i}ñ_{i}) = α_{i−1}β_{i−1}ñ_{i−1},

which is equivalent to

∀i ∈ Z/fZ  \frac{p^{f} − 1}{d} s_{i} = x_{i−1} − px_{i}.  (4.26.2)

If these relations are satisfied, then (4.26.1) implies that T |_{Γ_{L∞}} = F · αβñ, where αβñ = (α_{i}β_{i}ñ_{i})_{i}. The solution of the system (4.26.2) is given by

−x_{0} = ∑_{i=0}^{f−1} \frac{s_{i+1}}{d} p^{i} = ∑_{i=0}^{f−1} \frac{r_{i+1}(M)}{d} p^{i} + \frac{p^{f} − 1}{d} a_{0}(M), x_{i−1} = \frac{p^{f} − 1}{d} s_{i} + px_{i}.

For g ∈ Γ_{L_{∞}} and p ∤ m, the actions of g on ϖ_{L}^{1/m} = (ϖ^{d})^{1/m} and (u^{d})^{1/m} are compatible as follows ([NY, §5.1]): g(ϖ_{L}^{1/m})/ϖ_{L}^{1/m} = g((u^{d})^{1/m})/(u^{d})^{1/m}.

On the other hand, rec_{L}(λ)(ϖ_{L}^{1/(p^{f} −1)})/ϖ_{L}^{1/(p^{f} −1)} = λ for all λ ∈ μ_{p^{f} −1}(O_{L}), by [Se3, §1.5, Prop. 3] (with a sign change, due to our normalisation of the reciprocity map). Putting everything together, we deduce that I_{L} acts on (4.26.1) by the character

rec_{L}(λ)(αβñ)/αβñ = (rec_{L}(λ)(β_{i})/β_{i})_{i} = (σ_{i}(λ)^{x_{i}})_{i} ∈ F^{×} ⊂ (F ⊗ \overline{k})^{×}

(note that σ_{i}(λ)^{x_{i}} = σ_{0}(λ)^{x_{0}}). As it acts on F(1) by the character rec_{L}(λ) ↦ λ^{z}, where z = e(1 + p + ⋯ + p^{f−1})/d, the statement of the proposition follows from the fact that x_{0} + z rk_{F}(G) = −y_{0} − (p^{f} − 1)a_{0}(M)/d.

(4.27) The reader may be puzzled by the fact that the right hand side of the formula for det_{G_{L}}(rec_{L}(λ)) in Proposition 4.26 seemingly depends on the choice of σ_{0} : k ↪ F. However, this is not the case. Indeed, if we replace σ_{0} by σ_{0}^{new} = σ_{1} = σ_{0} ∘ φ, then σ_{0}^{new}(λ) = σ_{0}(λ)^{p} and η_{i}^{new} = η_{i+1} = η_{i}^{p}, which implies that a_{i}^{new}(M) = a_{i+1}(M) and r_{i}^{new}(M) = r_{i+1}(M). Therefore

p y_{0}^{new} = \frac{1}{d} ∑_{i=0}^{f−1}(r_{i+2}(M) − e rk_{F}(G)) p^{i+1} = \frac{1}{d} ∑_{j=1}^{f}(r_{j+1}(M) − e rk_{F}(G)) p^{j} = y_{0} + \frac{p^{f} − 1}{d}(r_{1}(M) − e rk_{F}(G)).

After multiplying the congruence

pa_{1}(M) + (r_{1}(M) − e rk_{F}(G)) ≡ a_{0}(M) (mod d)

by (p^{f} − 1)/d, we obtain

p (\frac{p^{f} − 1}{d}a_{0}^{new}(M) + y_{0}^{new}) ≡ \frac{p^{f} − 1}{d}a_{0}(M) + y_{0} (mod (p^{f} − 1)),

which implies that

∀λ ∈ μ_{p^{f} −1}(O_{L}) σ_{0}^{new}(λ)^{\frac{p^{f} −1}{d} a_{0}^{new}(M)+y_{0}^{new}} = σ_{0}(λ)^{\frac{p^{f} −1}{d} a_{0}(M)+y_{0}}.

(4.28) Proposition. Let H ∈ (F−vs/O_{K})_{dd,K/L}, M = M(H) ∈ (BT_{F,φ})_{dd,K/L}. If M ⊕M^{D} is Δ-balanced, then

det_{H_{L}}(rec_{L}(−1)) = (−1)^{\frac{p^{f} −1}{d} a_{0}(M)}(−1)^{χ_{F}(O_{L},H)} = ((η ∘ rec_{L})(−1))^{a_{0}(M)} (−1)^{χ_{F}(O_{L},H)}.

Proof. Combine Proposition 4.26 with Proposition 4.25(2), Proposition 4.13(1) and (4.7.2).

(4.29) Proposition. Let G ∈ (O − div/O_{K})_{dd,K/L}. If there exists a skew-symmetric isomorphism G ≃ G^{D} and an exact sequence 0 ⟶ H ⟶ G[π] ⟶ H^{D} ⟶ 0 in (F − vs/O_{K})_{dd,K/L}, then

(−1)^{χ_{F}(O_{L},H)} = det_{H_{L}}(rec_{L}(−1)) ε(V_{π}G).

Proof. If [M(H)_{0}/uM(H)_{0}] = ∑_{j=0}^{d} c_{j}[η_{0}^{−j}] ∈ K_{0}(F[Δ]) (c_{j} ∈ N), then a_{0}(M(H)) = ∑_{j} jc_{j} and the class [W D(V_{π}G)|_{I_{L}}] ∈ K_{0}(K[Δ]_{Sp}) ≃ K_{0}(F[Δ]_{Sp}) is equal to [M(G[π])_{0}/uM(G[π])_{0}] = ∑_{j=0}^{d} c_{j}([η_{0}^{−j}] + [η_{0}^{j}]), by Proposition 4.23(3). The formula (5.5.1) from the proof of Theorem 5.5 below states that ε(V_{π}G) =

ε(W D(V_{π}G)) = (−1)^{\frac{p^{f} −1}{d} a_{0}(M(H))}; we conclude by applying Proposition 4.28.

5. Proof of Theorem A

(5.1) Let Q_{p} ⊂ K ⊃ O ⟶ O/πO = F (p ≠ 2) be as in 1.1. Let L be a finite extension of Q_{p} with ring of integers O_{L}, residue field k_{L} and absolute Galois group Γ_{L} = Gal(\overline{L}/L). Assume that T and T′ are free O-modules of finite rank equipped with a continuous O-linear action of Γ_{L}; set V = T ⊗_{O} K and V′ = T′ ⊗_{O} K.

(5.2) Definition. If K ⊂ \overline{L} is a finite extension of L, we say that V |_{Γ_{K}} is a Barsotti–Tate representation if the following equivalent conditions are satisfied (the equivalence between (2) and (3) was proved in [B3, Thm. 5.3.2]).

(1) There exists G ∈ (O − div/O_{K}) such that T |_{Γ_{K}} = T_{π}G;

(2) there exists G ∈ (O − div/O_{K}) such that V |_{Γ_{K}} = V_{π}G;

(3) V |_{Γ_{K}} is a crystalline representation of Γ_{K}, with Hodge–Tate weights contained in {0, 1}.

(5.3) Proposition. Let K, K′ ⊂ \overline{L} be finite extensions of L.

(1) If V |_{Γ_{K}} is Barsotti–Tate and K^{ur} ⊂ K^{′ur}, then V |_{Γ_{K′}} is Barsotti–Tate.

(2) If K/L and K′/L are tamely ramified, then K^{ur} ⊂ K^{′ur} if and only if e(K/L) | e(K′/L).

(3) If K/L is an abelian tamely ramified extension, then, for every uniformiser ϖ_{L} ∈ O_{L}, K′ = L(ϖ_{L}^{1/|k_{L}^{×}|}) is a totally tamely ramified (cyclic) extension of L and e(K/L) | e(K′/L) = |k_{L}^{×}|.

Proof. (1) The condition 5.2(3) depends only on V |_{I_{K}}. The statements (2) and (3) are straightforward. (5.4) Assume that ⟨ , ⟩ : T × T ⟶ O(1) and ⟨ , ⟩′ : T′ × T′ ⟶ O(1) are Γ_{L}-equivariant skewsymmetric O-bilinear pairings inducing isomorphisms T ⟶^{∼} T^{∗}(1) = Hom_{O}(T, O(1)), T′ ⟶^{∼} T^{′∗}(1), and that there exists an isomorphism of F[Γ_{L}]-modules \overline{T} = T/πT ⟶^{∼} \overline{T}′ = T′/πT′ compatible with the pairings \overline{⟨ , ⟩} : \overline{T} × \overline{T} ⟶ F(1) and \overline{⟨ , ⟩}′ : \overline{T}′ × \overline{T}′ ⟶ F(1) induced by ⟨ , ⟩ and ⟨ , ⟩′.

Fix such an isomorphism and use it to identify H^{1}(L, \overline{T}) = H^{1}(Γ_{L}, \overline{T}) ⟶^{∼} H^{1}(Γ_{L}, \overline{T}′). The arithmetic local constant of Mazur and Rubin ([MR, Def. 4.5], [N5, (2.2.1)]) attached to the above data is defined as

δ_{L}(T, T′) = δ_{L}(T′, T) = dim_{F}(ℱ /ℱ ∩ ℱ′) (mod 2) ∈ Z/2Z,

where

ℱ = Im (H_{f}^{1}(L, T) ⟶ H^{1}(L, \overline{T})) ⊂ H^{1}(L, \overline{T}), ℱ′ = Im (H_{f}^{1}(L, T′) ⟶ H^{1}(L, \overline{T}′)) ⊂ H^{1}(L, \overline{T}′).

(5.5) Theorem (= Theorem A). If, in the situation of 5.4, there exist abelian tamely ramified finite extensions K/L and K′/L such that V |_{Γ_{K}} and V′|_{Γ_{K′}} are Barsotti–Tate representations, then

(−1)^{δ_{L}(T,T′)} = ε(W D(V))/ε(W D(V′)) = ε(V)/ε(V′).

Proof. According to Proposition 5.3 there exists a totally tamely ramified (cyclic) extension K/L such that V |_{Γ_{K}} and V′|_{Γ_{K}} are Barsotti–Tate. The extension L ⊂ K = L(ϖ) (ϖ^{d} = ϖ_{L}) is of the form considered in 4.7–4.29, with k_{L} = k = O_{K}/ϖO_{K} and d = e(K/L) | (p^{f} −1), where f = [k : F_{p}]. We can – and will – assume that d is even. As in (4.7.1), there is a canonical isomorphism η : Δ = Gal(K/L) ⟶^{∼} μ_{d}(O_{L}) = μ_{d}(k).

After replacing F by a suitable finite extension F′, O by O′ = O ⊗_{W (F)} W (F′), T by T ⊗_{O} O′ and T′ by T′ ⊗_{O} O′, we can – and will – assume that there exists an embedding σ_{0} : k ↪ F.

By assumption, there exist G, G′ ∈ (O − div/O_{K})_{dd,K/L} such that T_{π}G = T and T_{π}G′ = T′. As in (4.17.1), the Breuil–Kisin modules with descent data M = M(G[π]), M′ = M(G′[π]) ∈ (BT_{F,φ})_{dd,K/L} decompose into M = ⨁_{i} M_{i}, M′ = ⨁_{i} M′_{i} (i ∈ Z/fZ). As in 4.15, the fixed isomorphism \overline{T} = G[π]_{L}(\overline{K}) ⟶^{∼} \overline{T}′ = G′[π]_{L}(\overline{K}) of F[Γ_{L}]-modules gives rise to a Δ-equivariant isomorphism M[1/u] ⟶^{∼} M′[1/u] of étale φ-modules over (F ⊗ k)((u)) ⟶^{∼} ∏_{i} F((u)). According to Proposition 4.16,

ℱ /(ℱ ∩ ℱ′) ⟶^{∼} H_{fl}^{1}(O_{K}, H)^{Δ}/H_{fl}^{1}(O_{K}, G′[π])^{Δ},

where H ∈ (F − vs/O_{K})_{dd,K/L} and M(H) = M + M′, hence, in the notation of Proposition 4.21,

δ_{L}(T, T′) = χ_{F}(M′) − χ_{F}(M + M′) (mod 2).

According to Proposition 4.24(3), the pairings ⟨ , ⟩ and ⟨ , ⟩′ are induced by skew-symmetric isomorphisms G ⟶^{∼} G^{D} and G′ ⟶^{∼} G^{′D}. The corresponding isomorphisms G[π] ⟶^{∼} G[π]^{D} and G′[π] ⟶^{∼} G[π]^{′D} give rise to Δ-equivariant skew-symmetric (F ⊗ k)[[u]]-bilinear perfect pairings

M × M ⟶ (F ⊗ k)[[u]] ⟵ M′ × M′

compatible with the isomorphism M[1/u] ⟶^{∼} M′[1/u]. In other words, for each i ∈ Z/fZ, M_{i} and M′_{i} are self-dual τ-lattices in M_{i}[1/u] ⟶^{∼} M′_{i}[1/u], in the language of 3.7 (for F = F and η_{i} = σ_{i} ∘ τ : Δ ⟶^{∼} μ_{d}(F)).

As M = M(G)/πM(G) and M′ = M(G′)/πM(G′), Proposition 4.23(3) tells us that M and M′ are Δ-balanced. The exact sequence

0 ⟶ M ∩ M′ ⟶ M ⊕ M′ ⟶ M + M′ ⟶ 0

can be identified with

0 ⟶ M(H^{D}) ⟶ M ⊕ M′ ⟶ M(H) ⟶ 0,

since

∀i ∈ Z/fZ M(H^{D})_{i} = M(H)_{i}^{∗} = (M_{i} + M′_{i})^{∗} = M_{i}^{∗} ∩ M_{i}^{′∗} = M_{i} ∩ M′_{i}.

It follows that M(H) ⊕ M(H^{D}) is Δ-balanced as well. Applying Proposition 4.28 to G′[π] and H (and using the fact that G′[π]_{L} = H_{L}), we obtain

(−1)^{δ_{L}(T,T′)} = (−1)^{χ_{F}(M+M′)}/(−1)^{χ_{F}(M′)} = (η_{0}^{a_{0}(M+M′)−a_{0}(M′)}) (rec_{L}(−1)).

The Weil-Deligne representation W D(V) = W D(V_{π}G) has trivial monodromy N = 0 and the action of I_{L} factors through the cyclic group I_{L}/I_{K} = Δ. It follows that that the Frobenius semisimplification W D(V)^{f−ss} is a direct sum of one-dimensional representations of W_{L}. As it is ∥ · ∥_{L}-symplectic in the sense of [N2, 1.5.3], it is of the form

W D(V)^{f−ss} = ⨁_{α}(χ_{α} ⊕ χ_{α}^{−1}∥ · ∥_{L}), χ_{α} : W_{L} ⟶ \overline{K}^{×},

which implies that

ε(V) = ε(W D(V)) = ∏_{α} χ_{α}(rec_{L}(−1)) = ∏_{α}(χ_{α}|_{I_{L}})(rec_{L}(−1)).

The restriction W D(V)|_{I_{L}} = W D(V (−1))|_{I_{L}} is a symplectic K[Δ]-module and its class in the group K_{0}(K[Δ]_{Sp}) ⟶^{∼} K_{0}(F[Δ]_{Sp}) (see 3.8) is equal to the class of M_{0}/uM_{0}, by Proposition 4.23(3). The same arguments apply to V′ and M′; in particular, a_{0}(M′) = −I′(M′_{0}/uM′_{0}) = 0 ∈ Z/dZ, by Lemma 3.13(1).

If we write

[M_{0}/uM_{0}] = ∑_{j=0}^{d/2} c_{j}([η_{0}^{j}] + [η_{0}^{−j}]), [M′_{0}/uM′_{0}] = ∑_{j=0}^{d/2} c′_{j}([η_{0}^{j}] + [η_{0}^{−j}]) ∈ K_{0}(F[Δ]_{Sp})

(with c_{j}, c′_{j} ∈ Z), then

ε(V) = η_{0}(rec_{L}(−1))^{∑_{j} jc_{j}} = (−1)^{\frac{p^{f} −1}{d} ∑_{j} jc_{j}},  ε(V′) = (−1)^{\frac{p^{f} −1}{d} ∑_{j} jc′_{j}}.  (5.5.1)

Proposition 3.10, which is the main result of §3, tells us that

−a_{0}(M + M′) = I′(M_{0} + M′_{0}) ≡ J(M_{0}/uM_{0}) + J(M′_{0}/uM′_{0}) = ∑_{j} jc_{j} + ∑_{j} jc′_{j} (mod 2),

which implies the desired formula

(−1)^{δ_{L}(T,T′)} = (−1)^{\frac{p^{f} −1}{d} a_{0}(M+M′)} = ε(V)/ε(V′).

6. Proof of Theorem B

(6.1) Let Q_{p} ⊂ K ⊃ O ⟶ O/πO = F be as in 1.1 (p ≠ 2). (6.2) From now on until the end of §6 we consider the following global situation. Let L be a number field, S a finite set of primes of L containing S_{∞} ∪ S_{p} = \{v | ∞\} ∪ \{v | p\}, L_{S}/L the maximal subextension of \overline{L}/L unramified outside S, Γ_{L,S} = Gal(L_{S}/L), S_{f} = S ∖ S_{∞}. (6.3) Let T be a free O-module of finite rank equipped with a continuous O-linear action of Γ_{L,S}; set V := T ⊗_{O} K. Assume that, for each v ∈ S_{p}, V_{v} := V |_{Γ_{Lv}} is a de Rham representation of Γ_{L_{v}} = Gal(\overline{L}_{v}/L_{v}). For each v ∈ S_{f}, the Bloch–Kato subspace H_{f}^{1}(L_{v}, V) ⊂ H^{1}(L_{v}, V) = H^{1}(Γ_{L_{v}}, V) [BK, (3.7.1),(3.7.2)] (equal to H_{ur}^{1}(L_{v}, V) = H^{1}(Γ_{L_{v}}/I_{v}, V^{I_{v}}) if v ∉ S_{p}) defines subspaces H_{f}^{1}(L_{v}, T) := Ker(H^{1}(L_{v}, T) ⟶ H^{1}(L_{v}, V)/H_{f}^{1}(L_{v}, V)) ⊂ H^{1}(L_{v}, T) and H_{f}^{1}(L_{v}, V/T) := Im(H_{f}^{1}(L_{v}, V) ⟶ H^{1}(L_{v}, V/T)) ⊂ H^{1}(L_{v}, V/T). The corresponding Bloch–Kato Selmer groups and generalised Tate–Šafarevič groups ([BK, Def. 5.1, (5.13)], [Fl, p. 114–115], [FoPR, II.1.3.1, II.5.3.4]) are defined as

H_{f}^{1}(L, X) := Ker(H^{1}(Γ_{L,S}, X) ⟶ ⨁_{v∈S_{f}} H^{1}(L_{v}, X)/H_{f}^{1}(L_{v}, X)) (X = T, V, V/T),

�(T) := H_{f}^{1}(L, V/T)/Im(H_{f}^{1}(L, V)) = H_{f}^{1}(L, V/T)/H_{f}^{1}(L, V/T)_{div}

(these groups are independent of S, since H_{f}^{1}(L_{v}, X) = H_{ur}^{1}(L_{v}, X) if v ∉ S_{p} and V = V^{I_{v}}). Fontaine and Perrin-Riou [FoPR, II.1.3.1, II.5.3.6] also defined

H_{f}^{0}(L, X) := H^{0}(Γ_{L,S}, X) (X = T, V),
H_{f}^{i}(L, T) := Hom_{O}(H_{f}^{3−i}(L, V^{∗}(1)/T^{∗}(1)), K/O) (i = 2, 3),
H_{f}^{i}(L, V) := Hom_{K}(H_{f}^{3−i}(L, V^{∗}(1)), K) (i = 2, 3),

where T^{∗}(1) := Hom_{O}(T, O)(1) and V^{∗}(1) := T^{∗}(1)⊗_{O}K = Hom_{K}(V, K)(1); then H_{f}^{i}(L, V) = H_{f}^{i}(L, T)⊗_{O}K for all i = 0, 1, 2, 3 and H_{f}^{1}(L, T)_{tors} = H^{0}(Γ_{L,S}, V/T)/Im(H^{0}(Γ_{L,S}, V)) = H^{0}(L, V/T)/H^{0}(L, V/T)_{div}. Strictly speaking, the groups H_{f}^{i}(L, X) were denoted by H̃_{f}^{i}(L, X) in [FoPR], but we prefer to reserve the notation H̃_{f}^{i} for the extended Selmer groups defined in [N1, 6.1.2].

Flach [Fl, Thm. 1] (see also [FoPR, II.5.4.2]) constructed a nondegenerate pairing of O-modules of finite length

�(T) × �(T^{∗}(1)) ⟶ K/O,  (6.3.1)

which coincides (perhaps up to a sign) with the classical Cassels–Tate pairing between the groups �(T) = �(A/L)[p^{∞}]/�(A/L)[p^{∞}]_{div} and �(T^{∗}(1)) = �(Â/L)[p^{∞}]/�(Â/L)[p^{∞}]_{div} if O = Z_{p}, T = T_{p}(A) and T^{∗}(1) = T_{p}(Â), for an abelian variety A over L. (6.4) One can also define the Selmer group (again independent of S)

H_{ℱ}^{1} (L, \overline{T}) := Ker(H^{1}(Γ_{L,S}, \overline{T}) ⟶ ⨁_{v∈S_{f}} H^{1}(L_{v}, \overline{T})/ℱ_{v})

for the F[Γ_{L,S}]-module \overline{T} = T/πT and the Selmer structure [MR, Def. 1.2] ℱ = (ℱ_{v})_{v∈S_{f}}, where

ℱ_{v} := Im(H_{f}^{1}(L_{v}, T) ⟶ H^{1}(L_{v}, \overline{T})).  (6.4.1)

According to [N5, (1.1.1)],

dim_{F} H_{ℱ}^{1} (L, \overline{T}) − dim_{F} H^{0}(L, \overline{T}) = dim_{F} H_{f}^{1}(L, V/T)[π] − dim_{K} H^{0}(L, V) = χ_{f}(L, V) + dim_{F} �(T)[π], (6.4.2) where

χ_{f}(L, V) := dim_{K} H_{f}^{1}(L, V) − dim_{K} H^{0}(L, V) = h_{f}^{1}(L, V) − h^{0}(L, V).

(6.5) If ⟨ , ⟩ : V × V ⟶ K(1) is a skew-symmetric pairing inducing an isomorphism of O[Γ_{L,S}]-modules T ⟶^{∼} T^{∗}(1), then the composition of (6.3.1) with �(T) ⟶^{∼} �(T^{∗}(1)) is a skew-symmetric ([Fl, Thm. 2], [FoPR, II.5.4.3]) nondegenerate pairing

�(T) × �(T) ⟶ K/O  (6.5.1)

of O-modules of finite length, which implies that �(T) = Y ⊕ Y for some Lagrangian (maximal isotropic with respect to (6.5.1)) submodule Y ⊂ �(T) (recall that p ≠ 2). The formula (6.4.2) then yields [N5, (1.1.3)]

dim_{F} H_{ℱ}^{1} (L, \overline{T}) − dim_{F} H^{0}(L, \overline{T}) ≡ χ_{f}(L, V) (mod 2).  (6.5.2)

Moreover, each ℱ_{v} is a Lagrangian subspace (see [BK, Prop. 3.8] for v ∈ S_{p}) of the quadratic space (H^{1}(L_{v}, \overline{T}), ∪). Recall that the cup product attached to \overline{⟨ , ⟩} : \overline{T} × \overline{T} ⟶ F(1)

∪ : H^{1}(L_{v}, \overline{T}) × H^{1}(L_{v}, \overline{T}) ⟶ H^{2}(L_{v}, F(1)) ≃ F

is nondegenerate, by Tate local duality. In other words, ℱ is a self-dual Selmer structure on \overline{T}.

For each v ∈ S_{f} ∖S_{p} (resp. v ∈ S_{p}) the representation W D(V_{v}) of the Weil-Deligne group of L_{v} atached to V_{v} = V |_{Γ_{Lv}} (resp. to D_{pst}(V_{v})) is ∥ · ∥_{L_{v}}-symplectic, in the language of [N2, 1.5.3]. The local epsilon constant

ε_{v}(V) := ε(W D(V_{v}), ψ, dx_{ψ}) ∈ {±1}

(where dx_{ψ} is a Haar measure on L_{v} self-dual with respect to an additive character ψ ≠ 1 of L_{v}) does not depend on ψ [N2, Prop. 2.2.1]. (6.6) Assume that T′ is another free O-module of finite rank equipped with a continuous O-linear action of Γ_{L,S}, ⟨ , ⟩′ : V′ × V′ ⟶ K(1) (V′ := T′ ⊗_{O} K) is a pairing inducing an isomorphism of Γ_{L,S}-modules T′ ⟶^{∼} T^{′∗}(1) and \overline{T} ⟶^{∼} \overline{T}′ = T′/πT′ is an isomorphism of F[Γ_{L,S}]-modules compatible with the pairings \overline{⟨ , ⟩} and \overline{⟨ , ⟩}′ : \overline{T}′ × \overline{T}′ ⟶ F(1).

Fix such an isomorphism and use it to identify the quadratic spaces (H^{1}(L_{v}, \overline{T}), ∪) ⟶^{∼} (H^{1}(L_{v}, \overline{T}′), ∪), for all v ∈ S_{f}. We obtain the arithmetic local constants of Mazur and Rubin ([MR, Def. 4.5], [N5, (2.2.1)])

δ_{v}(T, T′) = δ_{v}(T′, T) := dim_{F} ℱ_{v}/(ℱ_{v} ∩ ℱ′_{v}) (mod 2) ∈ Z/2Z,

where ℱ′_{v} ⊂ H^{1}(L_{v}, \overline{T}′) is defined as in (6.4.1). The ranks of the Selmer groups H_{ℱ}^{1} (L, \overline{T}) ⊂ H^{1}(Γ_{L,S}, \overline{T}) and H_{ℱ′}^{1} (L, \overline{T}′) ⊂ H^{1}(Γ_{L,S}, \overline{T}′) are related by a formula of Mazur and Rubin [MR, Thm. 1.4]

dim_{F} H_{ℱ}^{1} (L, \overline{T}) − dim_{F} H_{ℱ′}^{1} (L, \overline{T}′) ≡ ∑_{v∈S_{f}} δ_{v}(T, T′) (mod 2),  (6.6.1)

which implies, thanks to (6.5.2), that

χ_{f}(L, V) − χ_{f}(L, V′) ≡ ∑_{v∈S_{f}} δ_{v}(T, T′) (mod 2).  (6.6.2)

(6.7) Theorem (= Theorem B). If, in the situation of 6.6, there exist finite abelian extensions K(v)/L_{v} and K(v)′/L_{v} (for all v ∈ S_{p}) such that the representations V_{Γ_{K(v)}} and V′_{Γ_{K(v)′}} are Barsotti–Tate, then

(−1)^{χ_{f} (L,V)}/ε(V) = (−1)^{χ_{f} (L,V′)}/ε(V′),

where ε(V) := ∏_{v∈S} ε_{v}(V), ε(V′) := ∏_{v∈S} ε_{v}(V′) and ε_{v}(V) = ε_{v}(V′) := (−1)^{dim_{K}(V)/2} for all v ∈ S_{∞}.

Proof. Firstly, we observe that χ_{f}(L, V) (mod 2) ∈ Z/2Z and ε_{L}(V) = ε(V) (and the analogous expressions for V′) do not change if we replace L by an abelian extension L′/L of odd degree. Indeed, the set of nontrivial characters of Gal(L′/L) can be written as a disjoint union of pairs {α, α^{−1}} and

ε_{L′}(V)/ε_{L}(V) = ∏_{α} ε_{L}(V ⊗ α)ε_{L}(V ⊗ α^{−1}),

but ε_{L}(V ⊗ α)ε_{L}(V ⊗ α^{−1}) = 1. Similarly,

χ_{f}(L′, V) − χ_{f}(L, V) = ∑_{α}(χ_{f}(L, V ⊗ α) + χ_{f}(L, V ⊗ α^{−1})),

but χ_{f}(L, V ⊗ α) = χ_{f}(L, V ⊗ α^{−1}), by [N1, Prop. 12.5.9.5(iv)].

Secondly, we can replace K(v) by K(v)K(v)′ and assume that K(v) = K(v)′, for all v ∈ S_{p}. Let M(v) be an intermediate field L_{v} ⊂ M(v) ⊂ K(v) such that [M(v) : L_{v}] = p^{m_{v}} and p ∤ [K(v) : M(v)]. There exist integers r, n ≥ 1 and characters β_{i}^{(v)} : Gal(M(v)/L_{v}) ⟶ μ_{p^{n}} such that (β_{1}^{(v)}, …, β_{r}^{(v)}) : Gal(M(v)/L_{v}) ↪ μ_{p^{n}}^{⊕r} is injective, for all v ∈ S_{p}. According to [AT, ch. 10, Thm. 5] (see also [NSW, Cor. 9.2.3]) there exist global characters β_{1}, …, β_{r} : Gal(L^{ab}/L) ⟶ μ_{p^{n}} such that β_{i}|_{Γ_{Lv}} = β_{i}^{(v)}, for all i = 1, …, r and v ∈ S_{p}. The fixed field L′ = (L^{ab})^{G} of G = ⋂_{i=1}^{r} Ker(β_{i}) is a finite abelian extension of L such that [L′ : L] = p^{m} and L′_{v′} ⊃ M(v), for all v′ | v ∈ S_{p}.

Thirdly, we can replace L by L′ (thanks to the first step) and assume that K(v) and K(v)′ are tamely ramified abelian extensions of L_{v}, for all v ∈ S_{p}, by the second step. We have

∀v ∈ S_{f} (−1)^{δ_{v}(T,T′)} = ε_{v}(V)/ε_{v}(V′),

by Theorem 5.5 (= Theorem A) if v ∈ S_{p} (resp. by [N6, Thm. 2.17] if v ∉ S_{p}). Applying (6.6.2), we obtain

(−1)^{χ_{f} (L,V)−χ_{f} (L,V′)} = ∏_{v∈S_{f}} ε_{v}(V)/ε_{v}(V′) = ∏_{v∈S} ε_{v}(V)/ε_{v}(V′) = ε(V)/ε(V′)

(the middle equality follows from the fact that ε_{v}(V) = ε_{v}(V′) for all v | ∞, by definition). Theorem is proved.

7. Tamagawa numbers and isogenies

(7.1) Let K, O, π, F, L, S, T and V be as in 6.1–6.3. (7.2) The formalism of Fontaine and Perrin-Riou ([FoPR, II.4.2, II.5.3]). Consider the following determinant objects, which are invertible modules over K or O (we ignore the issue of signs, which plays no role in what follows).

det_{K} RΓ_{f}(L, V) := ⨂_{i=0}^{3} (det_{K}H_{f}^{i}(L, V))^{(−1)^{i}} ⊃ det_{O} RΓ_{f}(L, T) := ⨂_{i=0}^{3} (det_{O}H_{f}^{i}(L, T))^{(−1)^{i}}

Δ̃′_{f}(T) := det_{O} RΓ_{f}(L, T) ⊗_{O} det_{O} H^{0}(L ⊗ R, T)^{−1},

Δ′_{f}(T) := ⨂_{i=0}^{1}(det_{O}H_{f}^{i}(L, T) ⊗ det_{O}H_{f}^{i}(L, T^{∗}(1)))^{(−1)^{i}} ⊗_{O} det_{O} H^{0}(L ⊗ R, T)^{−1},

Δ′_{f}(V) := det_{K} RΓ_{f}(L, V) ⊗_{K} det_{K} H^{0}(L ⊗ R, V)^{−1} ⊃ Δ̃′_{f}(T), Δ′_{f}(T).

According to [FoPR, II.5.3.8(ii)],

Δ̃′_{f}(T) = Δ′_{f}(T) ⊗_{O} det_{O}�(T^{∗}(1))

(recall that det_{O}(M) = π^{−ℓ_{O}(M)}O ⊂ K, for every O-module M of finite length). (7.3) The Tamagawa factors [FoPR, I.4]. The tangent space at v ∈ S_{p} is defined [BK, Def. 3.10] as

t_{V,v} := D_{dR}(V_{v})/F il^{0} = H^{0}(Γ_{L_{v}}, V_{v} ⊗_{Q_{p}} B_{dR}/B_{dR}^{+}).

Let ω_{v} be a basis of det_{K}(t_{V,v}); then ω = ⊗_{v∈S_{p}} ω_{v} is a basis of det_{K}(t_{V}) = ⊗_{v∈S_{p}} det_{K}(t_{V,v}). For v ∈ S_{f} ∖S_{p} we let det_{K}(t_{V,v}) = K and ω_{v} = 1.

For every v ∈ S_{f} the determinants

det_{K} RΓ_{f}(L_{v}, V) := ⨂_{i=0}^{1} (det_{K}H_{f}^{i}(L_{v}, V))^{(−1)^{i}} ⊃ det_{O} RΓ_{f}(L_{v}, T) := ⨂_{i=0}^{1} (det_{O}H_{f}^{i}(L_{v}, T))^{(−1)^{i}}

are canonically (up to a sign) isomorphic to

det_{K}(t_{V,v})^{−1} = K · ω_{v}^{−1} ⊃ O · Tam_{ω_{v}}^{0}(T) ω_{v}^{−1},

where

Tam_{ω_{v}}^{0}(T) ∈ K^{×}/O^{×}

is the local Tamagawa factor [FoPR, I.4.1.2]. If v ∈ S_{f} ∖ S_{p}, then

Tam_{ω_{v}}^{0}(T) O = det_{O} (H^{1}(I_{v}, T)_{tors}^{Fr(v)=id})^{−1},  (7.3.1)

by [FoPR, I.4.2.2(ii)]. We denote by

Tam_{ω}^{0}(T) := ∏_{v∈S_{f}} Tam_{ω_{v}}^{0}(T)

the (finite part of the) global Tamagawa factor.

(7.4) The fundamental line Δ_{f}(V). Let ω′_{T} be any generator of the invertible O-module Δ̃′_{f}(T). According to [FoPR, II.5.3.6, II.5.3.8], the invertible O-module

Δ′_{f}(T) ⊗_{O} det_{O}�(T^{∗}(1)) ⊗_{O} Tam_{ω}^{0}(T)^{−1}ω = Δ̃′_{f}(T) ⊗_{O} Tam_{ω}^{0}(T)^{−1}ω ⊂ Δ_{f}(V) := Δ′_{f}(V) ⊗_{K} det_{K}(t_{V})

(7.4.1)

is equal to Δ̃′_{f}(T) ⊗ ω |ω′_{T} ⊗ ω|_{EP,V} = O · (ω′_{T} ⊗ ω) |ω′_{T} ⊗ ω|_{EP,V}, where |·|_{EP,V} is an O-linear version of the Euler–Poincaré norm on Δ_{f}(V) defined in [FoPR, II.4.1.9] (in the context considered here the norm has values in K^{×}/O^{×}). This norm depends only on V, which implies that the invertible O-module (7.4.1) does not depend on T, nor on ω. Equivalently, for fixed ω,

⨂_{i=0}^{1}(det_{O}H_{f}^{i}(L, T) ⊗ det_{O}H_{f}^{i}(L, T^{∗}(1)))^{(−1)^{i}} ⊗_{O} det_{O} H^{0}(L ⊗ R, T)^{−1}π^{−ℓ_{O}(�(T^{∗}(1)))} Tam_{ω}^{0}(T)^{−1} (7.4.2)

does not depend on T. Note that

⨂_{i=0}^{1}(det_{O}H_{f}^{i}(L, T))^{(−1)^{i}} = π^{ℓ_{O}(H_{f}^{1}(L,T)_{tors})} det_{O}H^{0}(L, T) ⊗_{O} det_{O}(H_{f}^{1}(L, T)/tors)^{−1}  (7.4.3)

(and similarly for T^{∗}(1)). (7.5) The self-dual case. If T ⟶^{∼} T^{∗}(1) is self-dual with respect to a skew-symmetric pairing ⟨ , ⟩ : V × V ⟶ K(1) as in 6.5, then H_{f}^{1}(L, T)_{tors} = H_{f}^{1}(L, T^{∗}(1))_{tors} and ℓ_{O}(�(T^{∗}(1))) ≡ 0 (mod 2), which implies that

⨂_{i=0}^{1}(det_{O}(H_{f}^{i}(L, T)/tors) ⊗ det_{O}(H_{f}^{i}(L, T^{∗}(1))/tors))^{(−1)^{i}} ⊗_{O} det_{O} H^{0}(L ⊗ R, T)^{−1}Tam_{ω}^{0}(T)^{−1} (mod K^{×2})

(7.5.1)

does not depend on T. (7.6) A formula of Cassels. If T′ ⊃ T is a Γ_{L,S}-stable O-lattice in V such that T^{′∗}(1) = {x ∈ V | ⟨x, T′⟩ ∈ O(1)} = aT′ for some a ∈ O ∖ {0}, then we can apply (7.5.1) to T′ (and the pairing a^{−1}⟨ , ⟩). Comparing the expressions for T and T′, we deduce that

a^{χ_{f} (L,V)}Tam_{ω}^{0}(T)/Tam_{ω}^{0}(T′) (mod K^{×2}) = π^{ℓ_{O}(H^{0}(L⊗R,T′)/H^{0}(L⊗R,T))} (mod K^{×2}) ∈ K^{×}/O^{×}K^{×2}.
(7.6.1)

The statement (7.4.2) is an abstract version of a formula of Cassels [C1, Thm. 1.1] comparing the global Tamagawa factors of two isogeneous eliptic curves. Its consequence (7.6.1) (cf. [C1, Thm. 1.5]) gives a formula for χ_{f}(L, V) (mod 2) ∈ Z/2Z, provided that ord_{π}(a) ≡ 1 (mod 2). (7.7) From now on until the end of §7 assume that we are in the situation of 7.6 with a = π. In concrete terms, πT′ ⊂ T and H_{L} := πT′/πT ⊂ T/πT = \overline{T} is a Γ_{L,S}-stable Lagrangian subspace with respect to the pairing \overline{⟨ , ⟩} : \overline{T} × \overline{T} ⟶ F(1). Moreover, there are exact sequences of O[Γ_{L,S}]-modules

0 ⟶ T ⟶ T′ ⟶ H_{L} ⟶ 0,  0 ⟶ H_{L} ⟶ \overline{T} ⟶ H′_{L} ⟶ 0  (7.7.1)

and an isomorphism H′_{L} ⟶^{∼} H_{L}^{∗}(1) = Hom_{F}(H_{L}, F)(1). Conversely, if H_{L} ⊂ \overline{T} is a Γ_{L,S}-stable Lagrangian subspace, then its inverse image under T ⟶ \overline{T} is of the form πT′, with T′ ⊃ T as above.

For v ∈ S_{f} define h_{v} ∈ Z (independent of ω_{v}) by

π^{h_{v}}O^{×} := Tam_{ω_{v}}^{0}(T′)/Tam_{ω_{v}}^{0}(T).

The exact sequence

0 ⟶ H^{0}(L_{v}, T) ⟶ H^{0}(L_{v}, T′) ⟶ H^{0}(L_{v}, H_{L}) ⟶ H_{f}^{1}(L_{v}, T) ⟶^{α} H_{f}^{1}(L_{v}, T′)  (7.7.2)

induced by (7.7.1) implies that

h_{v} = ℓ_{O}(Coker(α)) − ℓ_{O}(H^{0}(L_{v}, H_{L})).

For v ∈ S_{∞} define

h_{v} := ℓ_{O}(H^{0}(L_{v}, T′)/H^{0}(L_{v}, T)),  ε_{v}(V) = (−1)^{dim_{K}(V)/2}.  (7.7.3)

(7.8) Denote by

det_{H_{L}} : Γ_{L,S}^{ab} ⟶ F^{×}

the Galois action on ∧^{r}H_{L}, where r = dim_{F}(H_{L}) = dim_{K}(V)/2. Conjecturally (cf. [BS], [CFKS, Thm. 2.7]),

(−1)^{h_{v}} det_{H_{L}}(rec_{L_{v}}(−1)) =^{?} ε_{v}(V)  (7.8.1)

holds, for all v ∈ S. If this is the case, then (7.6.1) yields

(−1)^{χ_{f} (L,V)} = ∏_{v∈S} ε_{v}(V) det_{H_{L}}(rec_{L_{v}}(−1)) = ∏_{v∈S} ε_{v}(V) = ε(V).  (7.8.2)

(7.9) Proposition. Assume that we are in the situation of 7.7.

(1) If L_{v} ≃ C, then h_{v} = r = dim_{K}(V)/2, det_{H_{L}}(rec_{L_{v}}(−1)) = 1 and (7.8.1) holds.

(2) If L_{v} ≃ R, let r_{v}^{±} := dim_{F} Ker(c_{v} ∓ id : H_{L}(\overline{L}_{v}) ⟶ H_{L}(\overline{L}_{v})) be the multiplicity of the eigenvalue ±1 for the action of the complex conjugation c_{v} ∈ Gal(\overline{L}_{v}/L_{v}) on H_{L}(\overline{L}_{v}). In this case h_{v} = r_{v}^{+} = r − r_{v}^{−}, det_{H_{L}}(rec_{L_{v}}(−1)) = (−1)^{r_{v}^{−}} and (7.8.1) holds.

(3) If v ∈ S_{f}, dim_{K}(V) = 2 and the monodromy operator N on W D(V_{v}) is nonzero, then (7.8.1) holds.

(4) If v ∈ S_{f} ∖ S_{p} and if the inertia group I_{v} = I_{L_{v}} acts on V through a finite quotient of order prime to p, then Tam_{ω_{v}}^{0}(T′) = Tam_{ω_{v}}^{0}(T) = 1, h_{v} = 0, ε_{v}(V) = det_{H_{L}}(rec_{L_{v}}(−1)) and (7.8.1) holds.

(5) If v ∈ S_{p} and if there exists an abelian tamely ramified extension K(v)/L_{v} such that V |_{Γ_{K(v)}} is a Barsotti–Tate representation, then (7.8.1) holds.

Proof. The statements (1) and (2) follow from (7.7.3), by definition.

(3) Nontriviality of N together with ∧^{2} ≃ O(1) imply that T_{v} = T |_{Γ_{Lv}} is reducible and that there is an exact sequence of O[Γ_{L_{v}}]-modules

0 ⟶ T_{v}^{+} ⟶ T_{v} ⟶ T_{v}^{−} ⟶ 0,  T_{v}^{+} ≃ O(1) ⊗ μ,  T_{v}^{−} ≃ O ⊗ μ,  μ : Γ_{L_{v}} ⟶ {±1},

and the extension class

[T_{v}] ∈ Ext_{O[Γ_{Lv}]}^{1}(O, O(1)) = (\underleftarrow{lim}_{n} L_{v}^{×} ⊗ Z/p^{n}Z) ⊗_{Z_{p}} O =: L_{v}^{×} ˆ⊗ O

does not lie in O_{L_{v}}^{×} ˆ⊗ O. In other words,

a(T_{v}) := (ord_{v} ⊗ id)([T_{v}]) ∈ O \ {0}

(and similarly for T′_{v} and a(T′_{v})).

This implies that H^{0}(L_{v}, V) = 0, hence H^{2}(L_{v}, V) ≃ H^{0}(L_{v}, V^{∗}(1))^{∗} ≃ H^{0}(L_{v}, V)^{∗} = 0 and

dim_{K} H_{f}^{1}(L_{v}, V) = \frac{1}{2} dim_{K} H^{1}(L_{v}, V) = δ_{v} · [L_{v} : Q_{p}], δ_{v} := \{ \begin{matrix}1, & v | p, \\ 0, & v ∤ p, \end{matrix}

by the local Euler characteristic formula. Let V_{v}^{±} := T_{v}^{±} ⊗_{O} K = T_{v}^{′±} ⊗_{O} K.

The local epsilon factor is given by the standard formula ([N1, Lemma 12.3.13(vi)])

ε_{v}(V) = μ(rec_{L_{v}}(−1)) · \{ \begin{matrix}−1, & μ = 1, \\ 1, & μ ≠ 1. \end{matrix}

We are now going to compute h_{v}. The vanishing of H^{0}(L_{v}, V) implies that

h_{v} = ℓ_{O}(Coker(α)) − ℓ_{O}(Ker(α)),

where the map α : H_{f}^{1}(L_{v}, T) ⟶ H_{f}^{1}(L_{v}, T′) from (7.7.2) is induced by the inclusion u : T ↪ T′. Multiplication by π on T factors as T ⟶^{u} T′ ⟶^{u′} T. Denote by α′ : H_{f}^{1}(L_{v}, T′) ⟶ H_{f}^{1}(L_{v}, T) the map induced by u′ and let

h′_{v} := ℓ_{O}(Coker(α′)) − ℓ_{O}(Ker(α′)).

Again, α′α is given by multiplication by π on H_{f}^{1}(L_{v}, T), which implies that

h_{v} + h′_{v} = ℓ_{O}(Coker(α′α)) − ℓ_{O}(Ker(α′α)) = dim_{K} H_{f}^{1}(L_{v}, V) = δ_{v} · [L_{v} : Q_{p}].

As Hom_{O[Γ_{Lv}]}(O(1) ⊗ μ, O ⊗ μ) = 0, the inclusion u : T ↪ T′ induces morphisms of O[Γ_{L_{v}}]-modules u^{±} : T_{v}^{±} ⟶ T_{v}^{′±} satisfying

Ker(u^{±}) = 0, ℓ_{O}(Coker(u^{+})) + ℓ_{O}(Coker(u^{−})) = ℓ_{O}(Coker(u)) = rk_{F}(H_{L}) = 1.

We say that the inclusion u : T ↪ T′ is generic if H_{L} ≠ T_{v}^{+}/πT_{v}^{+} (which is equivalent to u^{+} being an isomorphism). If H_{L} = T_{v}^{+}/πT_{v}^{+} (which is equivalent to u^{−} being an isomorphism) we say that u is special. Note that u is generic if and only if u′ : T′ ↪ T is special.

Returning to the computation of h_{v}, assume first that v ∤ p. The wild inertia subgroup I_{v}^{w} ⊂ I_{v} acts trivially on V, since p ≠ 2; it follows that H^{1}(I_{v}, T) ≃ T_{I_{v}}(−1) (and similarly for T′).

If μ is ramified, then T_{I_{v}} = T′_{I_{v}} = 0, hence Tam_{ω_{v}}^{0}(T′) = Tam_{ω_{v}}^{0}(T) = 1 and h_{v} = 0.

If μ is unramified, then there is an exact sequence of Γ_{L_{v}}/I_{v}-modules

0 ⟶ T_{v}^{+}/a(T)T_{v}^{+} ⟶ T_{I_{v}} ⟶ T_{v}^{−} ⟶ 0,

which implies that

H^{1}(I_{v}, T)_{tors}^{Fr(v)=id} = ((O/a(T)O) ⊗ μ)^{Fr(v)=id} = \{ \begin{matrix}O/a(T)O, & μ(Fr(v)) = 1, \\ 0, & μ(Fr(v)) = −1 \end{matrix}

(and similarly for T′).

To sum up the case v ∤ p,

h_{v} = \{ \begin{matrix}ord_{π}(a(T′_{v})/a(T_{v})), & μ = 1, \\ 0, & μ ≠ 1. \end{matrix}

If μ = 1, then T_{v}^{+} = O(1), T_{v}^{−} = O and [T_{v}] is the image of 1 ∈ H^{0}(L_{v}, T_{v}^{−}) under the boundary map H^{0}(L_{v}, T_{v}^{−}) ⟶ H^{1}(L_{v}, T_{v}^{+}). If, in addition, u is special, then u^{−} is an isomorphism and ord_{π}(a(T′_{v})) = ord_{π}(a(T_{v})) + 1 (for example, if O = Z_{p} and if T_{v} is the Tate module of an elliptic curve over L_{v} with Tate parameter q ∈ L_{v}^{×}, then u arises from the isogeny L_{v}^{×}/q^{Z} ⟶ L_{v}^{×}/q^{pZ} given by z ↦ z^{p}). Combined with the previous discussion, this yields, for v ∤ p and μ = 1,

h_{v} = \{ \begin{matrix}−1, & H_{L} ≠ T_{v}^{+}/πT_{v}^{+}, \\ 1, & H_{L} = T_{v}^{+}/πT_{v}^{+}. \end{matrix}

Assume now that v | p.

The vanishing of

Ker(H^{1}(L_{v}, T_{v}^{−}) ⟶ H^{1}(L_{v}, V_{v}^{−})) = H^{1}(L_{v}, T_{v}^{−})_{tors} ≃ H^{0}(L_{v}, T_{v}^{−} ⊗ K/O)/div

combined with the fact that

H_{f}^{1}(L_{v}, V) = Ker(H^{1}(L_{v}, V) ⟶ H^{1}(L_{v}, V^{−}))

([N2, Prop. 3.3.2(2)]) imply – together with analogous statements for T′_{v} – that there is a commutative diagram with exact rows

0 ⟶ H^{0}(L_{v}, T_{v}^{−}) ⟶ H^{1}(L_{v}, T_{v}^{+}) ⟶ H_{f}^{1}(L_{v}, T) ⟶ 0
↓^{u_{∗}^{−}}  ↓^{u_{∗}^{+}}  ↓^{α}
0 ⟶ H^{0}(L_{v}, T_{v}^{′−}) ⟶ H^{1}(L_{v}, T_{v}^{′+}) ⟶ H_{f}^{1}(L_{v}, T′) ⟶ 0

If u is generic, then the map u_{∗}^{+} is an isomorphism, hence

h_{v} = −ℓ_{O}(Coker(u_{∗}^{−})) = \{ \begin{matrix}−1, & μ = 1, \\ 0, & μ ≠ 1. \end{matrix}

If u is special, then u′ is generic and the previous formula holds for h′_{v}, which implies that

h_{v} = δ_{v} · [L_{v} : Q_{p}] + \{ \begin{matrix}1, & μ = 1, \\ 0, & μ ≠ 1. \end{matrix}

We are now ready to conclude the proof of (7.8.1) in the case (3) of Proposition 7.9 (no longer distinguishing the cases v ∤ p and v | p).

If u is generic, then the F[Γ_{L_{v}}]-module H_{L}(\overline{L}_{v}) is isomorphic to T_{v}^{−}/πT_{v}^{−} = F ⊗ μ, which implies that the left hand side of (7.8.1) is equal to

(−1)^{h_{v}}μ(rec_{L_{v}}(−1)),

which is, in turn, equal to ε_{v}(V).

If u is special, then the F[Γ_{L_{v}}]-module H_{L}(\overline{L}_{v}) is isomorphic to T_{v}^{+}/πT_{v}^{+} = F(1) ⊗ μ, hence det_{H_{L}} = μ\overline{χ}_{cycl,L_{v},p}. The formula (1.1.1) then implies that the left hand side of (7.8.1) is given by

(−1)^{h_{v}}μ(rec_{L_{v}}(−1))(−1)^{δ_{v}·[L_{v}:Q_{p}]} = (−1)^{h′_{v}}μ(rec_{L_{v}}(−1)),

which is again equal to ε_{v}(V).

(4) The arguments from [CFKS, 2.17] apply. The assumptions imply that H^{1}(I_{v}, T)_{tors} = H^{1}(I_{v}, T′)_{tors} = 0, hence Tam_{ω_{v}}^{0}(T′) = Tam_{ω_{v}}^{0}(T) = 1 and h_{v} = 0. According to [CFKS, Lemma 2.14(4)] (and its proof) there exists a free O-module U of rank dim_{K}(V)/2 equipped with a continuous action of W_{L_{v}} such that the semisimplification of U/πU is isomorphic to the semisimplification of H_{L}|_{W_{Lv}}. According to [D, Thm. 6.5], modified ε-constants ε_{0} respect congruences modulo π, hence

ε_{v}(V) det(−Fr(v) | V^{I_{v}}) ≡ ε(U ⊕ U^{∗}(1)) det(−Fr(v) | (U ⊕ U^{∗}(1))^{I_{v}}) (mod π).

The congruences

ε(U ⊕ U^{∗}(1)) (mod π) = det_{U}(rec_{L_{v}}(−1)) (mod π) = det_{H_{L}}(rec_{L_{v}}(−1)) ∈ {±1} ⊂ F^{×},
det(−Fr(v) | V^{I_{v}}) (mod π) = det(−Fr(v) | \overline{T}^{I_{v}}) = det(−Fr(v) | (U ⊕ U^{∗}(1))^{I_{v}}) (mod π) ∈ F^{×}

imply that ε_{v}(V) ≡ det_{H_{L}}(rec_{L_{v}}(−1)) (mod π), hence ε_{v}(V) = det_{H_{L}}(rec_{L_{v}}(−1)), since 1 ≠ −1 ∈ F^{×}.

(5) Thanks to Proposition 5.3 we can assume that the extension K(v)/L_{v} is totally tamely ramified. By assumption, there exist G, G′ ∈ (O−div/O_{K(v)})_{dd,K(v)/L_{v}} such that T = T_{π}G and T′ = T_{π}G′; then \overline{T} = G[π] and there exists an exact sequence

0 ⟶ H ⟶ G[π] ⟶ H^{D} ⟶ 0

in H ∈ (F−vs/O_{K(v)})_{dd,K(v)/L_{v}} whose generic fibre is given by the right exact sequence in (7.7.1). Moreover, there are exact sequences of finite flat group schemes over O_{K(v)} (with descent data relative to L_{v}/K(v))

0 ⟶ H ⟶ G[π^{n}] ⟶ G′[π^{n}]

whose generic fibres give rise, in the limit, to the left exact sequence in (7.7.1). The discussion in 1.4 and 4.10 implies that the exact sequence (7.7.2) extends to an exact sequence

0 ⟶ H^{0}(L_{v}, T) ⟶ H^{0}(L_{v}, T′) ⟶ H_{fl}^{0}(O_{K(v)}, H)^{Δ} ⟶ H_{f}^{1}(L_{v}, T) ⟶
H_{f}^{1}(L_{v}, T′) ⟶ H_{fl}^{1}(O_{K(v)}, H)^{Δ} ⟶ 0,

where Δ = Gal(K(v)/L_{v}). As a result,

h_{v} = dim_{F} H_{fl}^{1}(O_{K(v)}, H)^{Δ} − dim_{F} H_{fl}^{0}(O_{K(v)}, H)^{Δ} = −χ_{F}(O_{L_{v}}, H),

in the notation of Proposition 4.13. The formula (7.8.1) was proved in this case in Proposition 4.29.

(7.10) Theorem (= Theorem F). If, in the situation of 7.7, the following conditions are satisfied:

(i) for each v ∈ S_{f} ∖ S_{p} either there exist a finite abelian extension L(v)/L_{v} and a finite Galois extension K(v)/L(v) such that p ∤ [K(v) : L(v)] and T |_{Γ_{K(v)}} is unramified, or dim_{K}(V) = 2 and W D(V_{v}) has nontrivial monodromy; (ii) for each v ∈ S_{p} either there exists a finite abelian extension K(v)/L_{v} such that T |_{Γ_{K(v)}} is a Barsotti–Tate representation, or dim_{K}(V) = 2 and W D(V_{v}) has nontrivial monodromy; then

(−1)^{χ_{f} (L,V)} = ε(V).

Proof. After replacing L by a suitable abelian extension of p-power degree (using the arguments from the proof of Theorem 6.7) we can assume that p ∤ [L(v) : L_{v}] in (i) and K(v)/L_{v} is tamely ramified in (ii). The formula (7.8.1) holds for all v ∈ S_{∞} (by Proposition 7.9(1),(2)), all v ∈ S_{f} ∖ S_{p} (by Proposition 7.9(3),(4)) and all v ∈ S_{p}, by Proposition 7.9(3),(5). We conclude by (7.8.2).

8. The parity conjecture for Selmer groups of Hilbert modular forms of weight two

(8.1) Let F be a totally real number field and f a cuspidal Hilbert modular newform over F of level 𝔫, parallel weight 2 and trivial character. The field K_{f} generated over Q by the Hecke eigenvalues λ_{f}(v) = λ_{f}(T (v)) (v ∤ 𝔫) of f is a totally real finite extension of Q. For every prime number p and every prime 𝔭 | p in K_{f} one can attach to f an absolutely irreducible continuous two-dimensional representation V_{𝔭}(f) of Γ_{F} = Gal(\overline{F} /F) with coefficients in any fixed finite extension K of (K_{f})_{𝔭}, characterised by the fact that

∀v ∤ p𝔫  det(1 − X Fr_{geom}(v) | V_{𝔭}(f)) = 1 − λ_{f}(v)X + (Nv)X^{2}.  (8.1.1)

The Tate twist V := V_{𝔭}(f)(1) admits a nondegenerate Γ_{F} -equivariant skew-symmetric pairing ⟨ , ⟩ : V × V ⟶ K(1) and any Γ_{F} -stable O-lattice T ⊂ V (O = O_{K}) becomes self-dual T ⟶^{∼} T^{∗}(1) if we multiply the pairing by a suitable scalar in K^{∗}. Note that we use a cohomological, rather than a homological, normalisation of V_{𝔭}(f): in the case when F = Q and f = ∑_{n≥1} a_{n}q^{n} (a_{n} ∈ Q, a_{1} = 1) corresponds to the isogeny class of an elliptic curve E defined over Q, then K_{f} = Q, 𝔭 = p, λ_{f}(ℓ) = a_{ℓ} for all prime numbers ℓ ∤ N_{E}, V_{𝔭}(f) := H_{et}^{1}(E\overline{Q}, Q_{p}) = V_{p}(E)^{∗} ≃ V_{p}(E)(−1) (for K = Q_{p}) and V = V_{p}(E) = T_{p}(E) ⊗_{Z_{p}} Q_{p}.

Denote by π(f) the automorphic representation of P GL_{2}(A_{F}) attached to f ([N1, §12.3]); the corresponding Euler factors are related by

∀v ∤ ∞p𝔫  L_{v}(π(f), s + 1/2) = L_{v}(f, s + 1) = L_{v}(V_{𝔭}(f), s + 1) = L_{v}(V, s)  (8.1.2)

(which boils down to (8.1.1) if v ∤ p𝔫). The archimedean factors are given by

∀v | ∞  L_{v}(π(f), s − 1/2) = L_{v}(f, s) = 2(2π)^{−s}Γ(s),  ε_{v}(π(f), 1/2) = −1.

(8.2) More precisely, the following local compatibilities are satisfied, for all primes v ∤ ∞ of F ([Ca, Thm. A], [T2, Thm. 1.6], [Sa, Thm. 1], [B2, Thm. 1], [Sk, Thm. 1]). Let V_{𝔭}(f)_{v} := V_{𝔭}(f)|_{Γ_{Fv}} and V_{v} := V_{𝔭}(f)_{v}(1).

(8.2.1) If v | p, then V_{v} is a de Rham representation of Γ_{F_{v}}, with Hodge–Tate weights contained in {0, 1}. (8.2.2) The Frobenius-semisimplification W D(V_{𝔭}(f)_{v})^{f−ss} of the Weil-Deligne representation attached to

V_{𝔭}(f)_{v} if v ∤ p (resp. to D_{pst}(V_{𝔭}(f)_{v}) if v | p) corresponds to π(f)_{v} under the local Langlands correspondence ([BH, Thm. 35.1]).

(8.2.3) If v | p and v ∤ 𝔫, then π(f)_{v} is an unramified principal series representation; the Galois representation

V_{v} is crystalline with Hodge–Tate weights contained in {0, 1}, hence Barsotti–Tate [B3, Thm. 5.3.2]. (8.2.4) If π(f)_{v} = i_{B}^{G}(χ_{1}, χ_{2}) (χ_{i} : F_{v}^{×} ⟶ \overline{Q}^{×}) is a principal series representation (normalised induction from

the upper-triangular Borel subgroup B to G = GL_{2}(F_{v})), then W D(V_{v})^{f−ss} = ψ_{1} ⊕ ψ_{2} (ψ_{i} : W_{F_{v}}^{ab} ⟶ \overline{Q}^{×}), where χ_{i}| · |^{1/2} = ψ_{i} ∘ rec_{F_{v}}. This implies – thanks to (8.2.3) and cyclic base change for Hilbert modular forms if v | p ([Sk, §2.2]) – that there exists an abelian extension K(v)/F_{v} such that the representation V_{v}|_{Γ_{K(v)}} is unramified if v ∤ p (resp. Barsotti–Tate if v | p). If the finite order characters ψ_{i}|_{I_{v}} have orders prime to p, then one can choose K(v) such that p ∤ [K(v) : F_{v}].

(8.2.5) ε_{v}(V) = ε(V_{𝔭}(f)_{v}(1)) = ε(π(f)_{v}, 1/2) = ε_{v}(π(f), 1/2). (8.2.6) W D(V_{𝔭}(f)_{v}) is pure of weight 1 [Bl], hence H^{0}(F′, V_{𝔭}(f)(1)) = 0, for all finite extensions F′/F.

(8.3) Digression on representations of GL_{2}. Let ℓ be a prime number and E a finite extension of Q_{ℓ}, with ring of integers O_{E} and residue field k_{E}.

Let G = GL_{2}(E), K = GL_{2}(O_{E}) ⊃ K_{1} = Ker(K ⟶ \overline{G} = GL_{2}(k_{E})). Denote by B ⊂ G and \overline{B} ⊂ \overline{G} the standard upper-triangular Borel subgroups.

Let Π be a nonzero smooth admissible complex representation of G. For every open compact subgroup U ⊂ G the restriction Π|_{U} is a direct sum of irreducible finite-dimensional smooth representations of U (each of which factors through U/U′, for some open subgroup U′ ⊂ U). We say that a nonzero irreducible finite-dimensional smooth representation of U is a type of Π if it occurs in Π|_{U}.

(8.4) Proposition. The irreducible smooth representations Π of G of dimension dim(Π) > 1 satisfying Π^{K} = 0 ≠ Π^{K_{1}} are the following ones.

(1) A principal series representation Π = i_{B}^{G}(χ_{1}, χ_{2}) (normalised induction), where the restriction to O_{E}^{×} of

each χ_{i} factors as χ_{i}|_{O_{E}^{×}} : O_{E}^{×} ⟶ k_{E}^{×} ⟶^{\overline{χ}_{i}} C^{×}, and \overline{χ}_{1} ≠ 1 or \overline{χ}_{2} ≠ 1 (in other words, Π is a tame principal series representation).

(2) A twisted Steinberg representation Π = St ⊗ χ, where χ|_{O_{E}^{×}} : O_{E}^{×} ⟶ k_{E}^{×} ⟶^{\overline{χ}} C^{×}.

(3) A level 0 cuspidal representation [BH, Prop. 12.6]. The group K/K_{1} = \overline{G} acts on Π^{K_{1}} as follows.

(1) Π^{K_{1}} = I\overline{G} ; \overline{B}(\overline{χ}_{1}, \overline{χ}_{2}).

(1a) If \overline{χ}_{1} = \overline{χ}_{2} (which is equivalent to Π ⊗ χ_{1}^{−1} being an unramified principal series representation), then Π^{K_{1}} = (\overline{χ}_{1} ∘ det) ⊕ (\overline{St} ⊗ \overline{χ}_{1}), where \overline{St} = C[P^{1}(k_{E})]/C is an irreducible representation of \overline{G} of dimension |k_{E}|; (1b) if \overline{χ}_{1} ≠ \overline{χ}_{2} (in other words, if no twist Π ⊗ χ is an unramified principal series representation), then Π^{K_{1}} is an irreducible principal series representation of \overline{G}.

(2) Π^{K_{1}} = \overline{St} ⊗ \overline{χ}.

(3) Π^{K_{1}} is an irreducible cuspidal representation of \overline{G}.

Proof. Combine [BH, 6.3 Cor. 1] with [BH, Thm. 11.5, Prop. 12.6, Prop. 14.3, Thm. 14.5].

(8.5) Corollary. Let Π′ be an irreducible smooth representation of G of dimension dim(Π′) > 1. If Π = i_{B}^{G}(χ_{1}, χ_{2}) is a tame principal series representation as in Proposition 8.4(1) such that the type I\overline{G} ; \overline{B}(\overline{χ}_{1}, \overline{χ}_{2}) in the case 8.4(1a) (resp. the type (\overline{χ}_{1} ∘ det) in the case 8.4(1b)) occurs in Π′|_{K}, then Π′ = i_{B}^{G}(χ′_{1}, χ′_{2}) with χ′_{i}|_{O_{E}^{×}} = χ_{i}|_{O_{E}^{×}} (i = 1, 2). In other words, Π′ is also a tame principal series representation.

Proof. This follows directly from Proposition 8.4. (8.6) The conjectures of Bloch and Kato ([BK, Conj. 5.3], [FoPR, III.4.2.2]) predict that the dimension χ_{f}(F, V_{𝔭}(f)(1)) = h_{f}^{1}(F, V_{𝔭}(f)(1)) := dim_{K} H_{f}^{1}(F, V_{𝔭}(f)(1)) should be equal to

h_{f}^{1}(F, V_{𝔭}(f)(1)) =^{?} ord_{s=1}L(f, s) = ord_{s=1/2} L(π(f), s)

(the second equality follows from the fact that ord_{s=1/2} L_{v}(π(f), s) = 0 for all v | ∞). The (mod 2) version of this conjecture

h_{f}^{1}(F, V_{𝔭}(f)(1)) ≡^{?} ord_{s=1}L(f, s) = ord_{s=1/2} L(π(f), s) (mod 2)

is equivalent to

(−1)^{h_{f}^{1} (F,V_{𝔭}(f)(1))} =^{?} (−1)^{ord_{s=1/2} L(π(f),s)} = ε(π(f), 1/2) = ∏_{v} ε_{v}(V_{𝔭}(f)(1))  (8.6.1)

(by the functional equation for L(π(f), s)), if we define ε_{v}(V_{𝔭}(f)(1)) := ε(π(f)_{v}, 1/2) = −1 for all v | ∞.

(8.7) Theorem (= Theorem C). If p ≠ 2, then

dim_{K} H_{f}^{1}(F, V_{𝔭}(f)(1)) ≡ ord_{s=1}L(f, s) (mod 2)

holds, for every cuspidal Hilbert modular newform f over F of parallel weight 2 and trivial character.

Proof. This was proved in [N5, Thm. 1.4] if 2 ∤ [F : Q] or if π(f)_{v} is not a principal series representation for some finite prime v of F. We reduce the general case to this particular instance using Theorem B (= Theorem 6.7) and the level raising techniques that were employed in [N5, Thm. 3.5] to deduce the congruence in Theorem 8.7 from [N5, Thm. 1.4] in most non CM cases and many CM cases. However, the argument below will be more direct in the sense that it will not appeal to [N4, Thm. B(c)], the proof of which is fairly involved.

Firstly, throughout the proof we are free to replace K by a finite extension. Secondly, we can assume that 2 | [F : Q] and that π(f)_{v} is a principal series representation, for each v ∤ ∞. The arguments used in the proof of Theorem 6.7 imply, thanks to (8.2.4), that there exists an abelian extension F′/F of p-power degree (necessarily totally real) such that V_{𝔭}(f)(1) will become unramified (resp. Barsotti–Tate) over a finite abelian extension K(w)/F′_{w} of degree prime to p, for each prime w ∤ ∞p (resp. w | p) of F′. We can replace F by F′ and f by its (necessarily cuspidal) base change to F′.

After this reduction step the representation V = V_{𝔭}(f)(1) of Γ_{F,S} (where S = {v | ∞p𝔫}) will satisfy conditions (i) and (ii) of Theorem 7.10 (with L = F and all finite abelian extensions K(v)/L_{v} in Theorem 7.10(ii) being tamely ramified). On the automorphic side, the “tamely ramified” condition corresponds to the fact that, for each v | p, the representation π(f)_{v} of GL_{2}(F_{v}) is a tame principal series representation, i.e., is of the form described in Proposition 8.4(1).

If the residual representation \overline{T} := T/πT is reducible, then (8.6.1) holds, by Theorem 7.10.

It remains to treat the case when \overline{T} is irreducible (hence absolutely irreducible, since the complex conjugation at any infinite prime of F acts on \overline{T} with two distinct eigenvalues ±1 ∈ F).

According to [BDJ, Lemma 2.9 and its proof], there exist

(8.7.1) a prime v_{0} ∤ ∞p𝔫 of F; (8.7.2) a cuspidal Hilbert newform f′ of weight 2 of level dividing 𝔫v_{0} and divisible by v_{0}; (8.7.3) a prime 𝔭′ | 𝔭 in K_{f′}; (8.7.4) a finite extension K of Q_{p} equipped with fixed embeddings (K_{f})_{𝔭} ↪ K ↩ (K_{f′})_{𝔭′}

with the following properties:

(8.7.5) The form f′ has trivial character (by taking ψ = 1 in the proof of [BDJ, Lemma 2.9]). (8.7.6) The automorphic representation π(f′) of P GL_{2}(A_{F}) attached to f′ has the following property. For each

v | p, the representation π(f′)_{v} of GL_{2}(F_{v}) contains the following type of K_{v,1} = Ker(GL_{2}(O_{F_{v}}) ⟶ GL_{2}(k(v))) (where k(v) is the residue field of v): the type I\overline{G} ; \overline{B}(\overline{χ}_{1}, \overline{χ}_{2}) if π(f)_{v} is as in the case (1a) of Proposition 8.4 (resp. the type (\overline{χ}_{1} ∘ det) if π(f)_{v} is as in the case (1b) of Proposition 8.4).

(8.7.7) The two-dimensional Galois representations V = V_{𝔭}(f)(1) and V′ = V_{𝔭′}(f′)(1) of Γ_{F} with coefficients in K admit Γ_{F} -stable O_{K}-lattices T ⊂ V and T′ ⊂ V′ such that the residual representations \overline{T} = T/πT and \overline{T}′ = T′/πT′ of Γ_{F} have isomorphic semisimplifications.

According to Corollary 8.5, condition (8.7.6) implies that

(8.7.8) for each v | p, π(f′)_{v} is as in Proposition 8.4(1).

As a result, the representation V′ becomes Barsotti–Tate over a tamely ramified abelian extension of F_{v}, for all v | p.

After rescaling the pairings V × V ⟶ K(1) and V′ × V′ ⟶ K(1) we can assume that they induce isomorphisms T ⟶^{∼} T^{∗}(1) and T′ ⟶^{∼} T^{′∗}(1). Condition (8.7.7) combined with the absolute irreducibility of \overline{T} imply that, after rescaling the pairing ⟨ , ⟩ : T × T ⟶ O′(1) by an element of O^{×}, there exists an isomorphism \overline{T} ⟶^{∼} \overline{T}′ compatible with the pairings on T and T′.

Therefore the representations T and T′ of Γ_{F,S∪{v_{0}}} satisfy the assumptions of Theorem 6.7, hence

(−1)^{χ_{f} (F,V)}/ε(V) = (−1)^{χ_{f} (F,V′)}/ε(V′).

The right hand side is equal to 1, by [N5, Thm. 1.4] (since π(f′)_{v_{0}} is an unramified twist of the Steinberg representation), which means that (8.6.1) holds, as claimed. Theorem 8.7 is proved. (8.8) As pointed out by the referee of the present article, the level raising argument in the proof of [N5, Thm. 3.5] (which combined [T1, Thm. 1] with [DS, Lemme 6.11]) needs to be made more precise to ensure that the form g in the proof of [loc. cit.] has trivial character. As above, this follows from the proof of [BDJ, Lemma 2.9]. Equivalently, one can replace the space of functions on D^{×}\D̂^{×}/U in the proof of [T1, Thm. 1] (for the trivial local system L_{k} = C) by the space of functions on D^{×}\D̂^{×}/F̂^{×}U (cf. [N4, §1.1, 1.2]).

(8.9) Theorem (= Theorem D). Let A be an abelian variety with real multiplication by O_{M} (where M is a totally real number field of degree [M : Q] = dim(A)) defined over a totally real number field F. If 𝔭 | p in M and p ≠ 2, then

dim_{M_{𝔭}} H_{f}^{1}(F, V_{𝔭}(A)) = rk_{O_{M}} A(F) + cork_{O_{M𝔭}} �(A/F)[𝔭^{∞}] ≡ ord_{s=1}L(ιA/F, s) (mod 2)

holds, for every ι : M ↪ R.

Proof. As explained in [N1, 12.11.6], [N5, 4.2] and [N6, 4.13], this is a consequence of Theorem 8.7, potential modularity of A [BLGGT, Cor. 5.4.2, Cor. 5.4.3] and Solomon’s induction theorem [CR, Thm. 15.10].

(8.10) Theorem (= Theorem E). Let E be an elliptic curve defined over a totally real number field F. If p ≠ 2, then

dim_{Q_{p}} H_{f}^{1}(F, V_{p}(E)) = rk_{Z}E(F) + cork_{Z_{p}}�(E/F)[p^{∞}] ≡ ord_{s=1}L(E/F, s) (mod 2).

Proof. This is a special case of Theorem 8.9 when M = Q. Potential modularity of E is proved in [W, Thm. A.1].

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Université Pierre et Marie Curie (Paris 6) Institut de Mathématiques de Jussieu Théorie des Nombres, Case 247 4, place Jussieu F-75252 Paris cedex 05 FRANCE