On the parity of ranks of Selmer groups III

Jan Nekovář

0. Introduction

(0.0) Let F, L be number fields contained in a fixed algebraic closure \overline{Q} of Q; let M be a motive over F with coefficients in L. The L-function of M (assuming it is well-defined) is a Dirichlet series ∑_{n≥1} a_{n}n^{−s} with coefficients in L. For each embedding ι : \overline{Q} ↪ C, the complex-valued L-function

L(ιM, s) = ∑_{n≥1} ι(a_{n})n^{−s}

is absolutely convergent for Re(s) >> 0. It is expected to admit a meromorphic continuation to C and a functional equation of the form

(C_{F E})  (L · L_{∞})(ιM, s) =^{?} ε(ιM, s) (L · L_{∞})(ιM^{∗}(1), −s),

where

L_{∞}(ιM, s) = ∏_{v|∞} L_{v}(ιM, s)

is a product of appropriate Γ-factors (independent of ι) and

ε(ιM, s) = ι(ε(M)) cond(M)^{−s}, ε(M) ∈ \overline{Q}^{∗}.

(0.1) Let p be a prime number and 𝔭 | p a prime of L above p. The 𝔭-adic realization M_{𝔭} of M is a finitedimensional L_{𝔭}-vector space equipped with a continuous action of the Galois group G_{F,S} = Gal(F_{S}/F), where F_{S} ⊂ \overline{Q} is the maximal extension of F unramified outside a suitable finite set S ⊃ S_{p} ∪ S_{∞} of primes of F. According to the conjectures of Bloch and Kato [Bl-Ka] (generalized by Fontaine and Perrin-Riou [Fo-PR]),

(C_{BK}) ord_{s=0}L(ιM, s) =^{?} dim_{L_{𝔭}} H_{f}^{1}(F, M_{𝔭}^{∗}(1)) − dim_{L_{𝔭}} H^{0}(F, M_{𝔭}^{∗}(1)) = h_{f}^{1}(F, M_{𝔭}^{∗}(1)) − h^{0}(F, M_{𝔭}^{∗}(1)),

where H_{f}^{1}(F, V) ⊆ H^{1}(G_{F,S}, V) is the generalized Selmer group defined in [Bl-Ka]. (0.2) Consider the special case when the motive M is self-dual (i.e., when there exists a skew-symmetric isomorphism M ⟶^{∼} M^{∗}(1)) and pure (necessarily of weight −1). In this case H^{0}(F, M_{𝔭}) = 0 and ord_{s=0}L_{∞}(ιM, s) = 0, which means that the global ε-factor ε(M) determines the parity of ord_{s=0}L(ιM, s) (assuming the validity of (C_{F E})):

(−1)^{ord_{s=0}L(ιM,s)} = ε(M).  (0.2.1)

In this article we concentrate on the parity conjecture for Selmer groups, namely on the conjecture

(C_{BK} (mod 2))  ord_{s=0}L(ιM, s) ≡^{?} h_{f}^{1}(F, M_{𝔭}) (mod 2).

In view of (0.2.1), this conjecture can be reformulated (assuming (C_{F E})) as follows:

(−1)^{h_{f}^{1} (F,M_{𝔭})} =^{?} ε(M)  (0.2.2)

(0.3) The advantage of the formulation (0.2.2) is that the global ε-factor

ε(M) = ∏_{v} ε_{v}(M), ε_{v}(M) = ε_{v}(M_{𝔭})

is a product of local ε-factors, which can be expressed in terms of the Galois representation M_{𝔭} alone: for v ∤ p∞ (resp., v | p), ε_{v}(M) is the local ε-factor of the representation of the Weil-Deligne group of F_{v} attached to the action of Gal(\overline{F}_{v}/F_{v}) on M_{𝔭} (resp., attached to the corresponding Fontaine module D_{pst}(M_{𝔭}) over F_{v}). For v | ∞, ε_{v}(M) depends on the Hodge numbers of the de Rham realization M_{dR} of M, which can be read off from D_{dR}(M_{𝔭}) over F_{v}, for any v | p. It makes sense, therefore, to rewrite the conjecture (0.2.2) as

(−1)^{h_{f}^{1} (F,V)} =^{?} ε(V) = ∏_{v} ε_{v}(V),  (0.3.1)

for any symplectically self-dual (V ⟶^{∼} V^{∗}(1)) representation of G_{F,S} which is geometric (= potentially semistable at all primes above p) and pure (of weight −1). In the present article we consider the following question: is the conjecture (0.3.1) invariant under deformation in p-adic families of representations of G_{F,S}? In other words, if V, V′ are two representations of G_{F,S} (selfdual, geometric and pure) belonging to the same p-adic family (say, in one parameter) of representations of G_{F,S}, is it true that

(−1)^{h_{f}^{1} (F,V)}/ ε(V) =^{?} (−1)^{h_{f}^{1} (F,V′)}/ ε(V′)  ?  (0.3.2)

The main result of this article (Thm. 5.3.1) implies that (0.3.2) holds for families satisfying the Pančiškin condition at all primes v | p. The proof follows the strategy employed in [Ne 2, ch. 12] in the context of Hilbert modular forms ^{(1)} : multiplying both sides of (0.3.1) by a common sign (the contribution of the “trivial zeros”), we rewrite (0.3.1) as

(−1)^{_{h_{f}^{1}} (F,V)} =^{?} ε̃(V) = ∏_{v} ε̃_{v}(V),  (0.3.3)

where h̃_{f}^{1}(F, V) = dim_{L_{𝔭}} H̃_{f}^{1}(F, V) is the dimension of the extended Selmer group (defined in 4.2 below) and ε̃_{v}(V) = ε_{v}(V), unless v | p and the local Euler factor at v admits a “trivial zero”. The goal is to show that both sides of (0.3.3) remain constant in the family ^{(2)}.

The variation of H̃_{f}^{1}(F, V) in the family is controlled by the torsion submodule of a suitable H̃_{f}^{2}. The generalized Cassels-Tate pairing constructed in [Ne 2, ch. 10] defines a skew-symmetric form on this torsion submodule, which implies that the parity of h̃_{f}^{1}(F, V) is constant in family:

(−1)^{_{h_{f}^{1}} (F,V)} = (−1)^{_{h_{f}^{1}} (F,V′)}.

The Pančiškin condition allows us to compute explicitly the local terms ε̃_{v}(V) for all v | p, which yields

∏_{v|p∞} ε̃_{v}(V) = ∏_{v|p∞} ε̃_{v}(V′).

Finally, it follows from general principles (and the purity assumption) that

∀v ∤ p∞ ε_{v}(V) = ε_{v}(V′),

hence ε̃(V) = ε̃(V′).

(1) In [loc. cit.] we worked with automorphic ε-factors, but they coincide with the Galois-theoretical ε-factors ([Ne 2], 12.4.3, 12.5.4(iii)).

(2) Morally, ε̃(V) should be the sign in the functional equation of a p-adic L-function attached to the family.

1. Representations of the Weil-Deligne group

(1.1) The general setup ([De 1, §8], [De 2, 3.1], [Fo-PR, I.1.1-2]) (1.1.1) We use the notation of [Fo-PR, ch.I]. For a field L, denote by L^{sep} a separable closure of L and by G_{L} = Gal(L^{sep}/L) the absolute Galois group of L. Throughout this article, K will be a complete discrete valuation field of characteristic zero with finite residue field k of cardinality q = q_{k}; denote by f = f_{k} ∈ G_{k} the geometric Frobenius element (f(x) = x^{1/q}). We identify G_{k} ⟶^{∼} Ẑ via f ↦ 1 and denote by ν : G_{K} ⟶^{can}G_{k} ⟶^{∼} Ẑ the canonical surjection whose kernel Ker(ν) = I_{K} = I is the inertia group of K. The Weil group (of K) W_{K} = ν^{−1}(Z) = ∐_{n∈Z} f̃^{n}I (f̃ ∈ ν^{−1}(1)) is equipped with the topology of a disjoint union of countably many pro-finite sets. The homomorphism

| · | : W_{K} ⟶ q^{Z}, |w| = q^{−ν(w)}

corresponds to the normalized valuation | · | : K^{∗} ⟶ q^{Z} via the reciprocity isomorphism rec_{K} : K^{∗} ⟶^{∼} W_{K}^{ab} (normalized using the geometric Frobenius element). (1.1.2) Let E be a field of characteristic zero. An object of Rep_{E}(W_{K}) (= a representation of the Weil group of K over E) is a finite-dimensional E-vector space Δ equipped with a continuous homomorphism ρ = ρ_{Δ} : W_{K} ⟶ Aut_{E}(Δ) (with respect to the discrete topology on the target). As Ker(ρ) is open, ρ(I) is finite and ρ|_{I} is semi-simple. An object of Rep_{E}(′W_{K}) (= a representation of the Weil-Deligne group of K over E) is a pair (ρ, N), where ρ = ρ_{Δ} ∈ Rep_{E}(W_{K}) and N ∈ End_{E}(Δ) is a nilpotent endomorphism satisfying

∀w ∈ W_{K} ρ(w)Nρ(w)^{−1} = |w| N.

Morphisms in Rep_{E}(W_{K}) (resp., in Rep_{E}(′W_{K})) are E-linear maps commuting with the action of W_{K} (resp., with the action of W_{K} and N). We consider Rep_{E}(W_{K}) as a full subcategory of Rep_{E}(′W_{K}) via the full embedding ρ ↦ (ρ, 0). Tensor products and duals in Rep_{E}(′W_{K}) are defined in the usual way: N_{Δ⊗Δ′} = N_{Δ} ⊗ 1 + 1 ⊗ N_{Δ′}, N_{Δ^{∗}} = −(N_{Δ})^{∗}. The Tate twist of Δ ∈ Rep_{E}(′W_{K}) by an integer m ∈ Z is defined as Δ|·|^{m} = Δ⊗E|·|^{m}, where w ∈ W_{K} acts on the one-dimensional representation E|·|^{m} ∈ Rep_{E}(W_{K}) by |w|^{m}. The Frobenius semi-simplification

Δ = (ρ, N) ↦ Δ^{f−ss} = (ρ^{ss}, N)

is an exact tensor functor Rep_{E}(′W_{K}) ⟶ Rep_{E}(′W_{K}). The “forget the monodromy” functor

Δ = (ρ, N) ↦ Δ^{N−ss} = (ρ, 0)

is an exact tensor functor Rep_{E}(′W_{K}) ⟶ Rep_{E}(W_{K}). Following [Fo-PR, I.1.2.1], we put, for each Δ ∈ Rep_{E}(′W_{K}),

Δ_{g} = Δ^{ρ(I)},  Δ_{f} = Ker(N)^{ρ(I)} ⊂ Δ_{g},  P_{K}(Δ, u) = det(1 − fu | Δ_{f}) ∈ E[u].

We also set

H^{0}(Δ) = Ker (Δ_{f} ⟶^{f−1}Δ_{f}) .

(1.1.3) In the special case when E is a finite extension of Q_{p} (p ≠ char(k)) and when V ∈ Rep_{E}(G_{K}) is a representation of G_{K} over E (finite-dimensional and continuous with respect to the topology on E defined by the p-adic valuation), then V gives rise to a representation W D(V) = Δ = (ρ_{Δ}, N) ∈ Rep_{E}(′W_{K}) acting on V, which is defined as follows ([De 1, 8.4]): there exists an open subgroup J of I which acts on V unipotently, and through the map J ↪ I ↠ I(p), where I(p) is the maximal pro-p-quotient of I (isomorphic to Z_{p}). Fixing a topological generator t of I(p) and an integer a ≥ 1 such that t^{a} lies in the image of J, the nilpotent endomorphism

N = \frac{1}{a} log ρ_{V} (t^{a}) ∈ End_{E}(V)

(where ρ_{V} : G_{K} ⟶ Aut_{E}(V) denotes the action of G_{K} on V) is independent of a. Fix a lift f̃ ∈ ν^{−1}(1) ⊂ W_{K} of f and define

ρ_{Δ} : W_{K} ⟶ Aut_{E}(V)

by

ρ_{Δ}(f̃^{n}u) := ρ_{V} (f̃^{n}u) exp(−bN) (n ∈ Z, u ∈ I),

where b ∈ Z_{p} is such that the image of u in I(p) is equal to t^{b}. The pair (ρ_{Δ}, N) defines an object Δ = W D(V) of Rep_{E}(′W_{K}), the isomorphism class of which is independent of the choices of f̃and t ([De 1], Lemma 8.4.3), and which satisfies

Δ_{f} = V^{ρ_{V} (I)}, H^{0}(Δ) = V^{ρ_{V} (G_{K})}.

(1.2) Self-dual representations

(1.2.1) Definition. Let ω : W_{K} ⟶ E^{∗} be a one-dimensional object of Rep_{E}(W_{K}). We say that Δ ∈ Rep_{E}(′W_{K}) is ω-orthogonal (resp., ω-symplectic) if there exists a morphism in Rep_{E}(′W_{K}) Δ ⊗ Δ ⟶ ω which is non-degenerate (i.e., which induces an isomorphism Δ ⟶^{∼} Δ^{∗} ⊗ ω) and symmetric (resp., skewsymmetric). If ω = 1, we say that Δ is orthogonal (resp., symplectic).

(1.2.2) (1) If Δ is ω-orthogonal, then det(Δ)^{2} = ω^{dim(Δ)}.

(2) If Δ is ω-symplectic, then 2 | dim(Δ) and det(Δ) = ω^{dim(Δ)/2}. (1.2.3) Example: For m ≥ 1, define sp(m) ∈ Rep_{E}(′W_{K}) by

sp(m) = ⨁_{i=0}^{m−1} Ee_{i},  N(e_{i}) = e_{i+1},  ∀w ∈ W_{K} w(e_{i}) = |w|^{i}e_{i}.

Up to a scalar multiple, there is a unique non-degenerate morphism sp(m) ⊗ sp(m) ⟶ E| · |^{m−1} in Rep_{E}(′W_{K}), namely

sp(m) ⊗ sp(m) ⟶ E| · |^{m−1}, e_{i} ⊗ e_{j} ↦ \{ \begin{matrix}(−1)^{i}, & i + j = m − 1 \\ 0, & i + j ≠ m − 1. \end{matrix}

This morphism is | · |^{m−1}-symplectic (resp., | · |^{m−1}-orthogonal) if 2 | m (resp., if 2 ∤ m). (1.2.4) According to [De 2, 3.1.3(ii)], indecomposable f-semi-simple objects of Rep_{E}(′W_{K}) are of the form ρ ⊗ sp(m), where ρ ∈ Rep_{E}(W_{K}) is irreducible and m ≥ 1. This implies that, for each | · |-symplectic representation Δ ⟶^{∼} Δ^{∗}| · | ∈ Rep_{E}(′W_{K}), the f-semi-simplification Δ^{f−ss} is a direct sum of | · |-symplectic representations of the following type:

(1) X ⊕ X^{∗}| · | (X ∈ Rep_{E}(′W_{K})) with the standard symplectic form (x, x^{∗}) ⊗ (y, y^{∗}) ↦ x^{∗}(y) − y^{∗}(x);

(2) ρ⊗sp(m), where m ≥ 1, ρ ∈ Rep_{E}(W_{K}) is irreducible and |·|^{2−m}-symplectic (resp., |·|^{2−m}-orthogonal) if 2 ∤ m (resp., if 2 | m).

(1.3) The monodromy filtration (1.3.1) For each Δ = (ρ, N) ∈ Rep_{E}(′W_{K}), the monodromy filtration

M_{n}Δ := ∑_{i−j=n+1} ker(N^{i}) ∩ Im(N^{j}) (n ∈ Z)

is the unique increasing filtration of Δ by E-vector subspaces satisfying

⋂_{n} M_{n}Δ = 0,  ⋃_{n} M_{n}Δ = Δ,  N(M_{n}Δ) ⊆ M_{n−2}Δ,  ∀r ≥ 0 N^{r} : gr_{r}^{M}Δ ⟶^{∼} gr_{−r}^{M}Δ.

(1.3.2) Examples: (1) N = 0 ⟺ M_{−1}Δ = 0, M_{0}Δ = Δ.

(2) If N^{r} ≠ 0 = N^{r+1} (r ≥ 0), then M_{−r−1}Δ = 0, M_{−r}Δ = Im(N^{r}) ≠ 0, M_{r−1}Δ = Ker(N^{r}) ≠ Δ, M_{r}Δ = Δ.

(1.3.3) More precisely, the endomorphism N ∈ End_{E}(Δ) defines a morphism in Rep_{E}(W_{K})

N : Δ ⟶ Δ| · |^{−1},

which implies that each M_{n}Δ is a sub-object of Δ^{N−ss} in Rep_{E}(W_{K}),

N : M_{n}Δ ⟶ (M_{n−2}Δ)| · |^{−1}

and, for each r ≥ 0, the endomorphism N^{r} induces an isomorphism in Rep_{E}(W_{K})

N^{r} : gr_{r}^{M}Δ ⟶^{∼} (gr_{−r}^{M}Δ)| · |^{−r}.

(1.3.4) The monodromy filtration on the dual representation Δ^{∗} = (ρ^{∗}, −N^{∗}) satisfies M_{n}Δ^{∗} = (M_{−1−n}Δ)^{⊥} (n ∈ Z), which yields canonical isomorphisms in Rep_{E}(W_{K})

∀m ≤ n M_{n}Δ^{∗}/M_{m}Δ^{∗} ⟶^{∼} (M_{−1−m}Δ/M_{−1−n}Δ)^{∗}.

(1.3.5) If ⟨ , ⟩ : Δ ⊗ Δ ⟶ E ⊗ ω is an ω-symplectic form on Δ, then, for each r ≥ 0, the formula ⟨x, y⟩_{r} = ⟨N^{r}x, y⟩ defines an ω| · |^{−r}-symplectic (resp., ω| · |^{−r}-orthogonal) form on gr_{r}^{M}Δ ∈ Rep_{E}(W_{K}) if 2 | r (resp., if 2 ∤ r).

(1.3.6) Dimensions. The dimensions

d_{r} = d_{r}(Δ) = dim gr_{r}^{M}Δ = d_{−r} (r ∈ Z)

can be interpreted as follows. By the Jacobson-Morozov theorem, there exists a (non-unique) representation

ρ : sl(2) = sl(2, E) ⟶ End_{E}(Δ)

such that ρ(( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} )) = N. Putting H = ρ(( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )) and Δ_{m} = \{x ∈ Δ | Hx = mx\} (m ∈ Z), then

M_{n}Δ = ∑_{m≤n} Δ_{m}.

Decomposing Δ as a representation of sl(2)

Δ ⟶^{∼} ⨁_{j≥0} (S^{j}E^{2})^{⊕m_{j}(Δ)} ,

then the multiplicities m_{j} = m_{j}(Δ) are related to other numerical invariants of Δ as follows:

dim(Δ) = ∑_{j≥0}(j + 1)m_{j},  (∀r ≥ 0)  d_{−r} = ∑_{i≥0} m_{r+2i},  m_{r} = d_{−r} − d_{−r−2},

dim Im(N^{r}) = d_{r} + 2 ∑_{j>r} d_{j},  dim Ker(N^{r+1}) = d_{0} + 2 ∑_{j=1}^{r} d_{j} + d_{r+1}.

(1.3.6.1)

(1.4) Purity

(1.4.1) Definition. Let E′ be a field containing E and a ∈ Z. We say that α ∈ E′ is a q^{a}-Weil number of weight n ∈ Z if α is algebraic over Q, there exists N ∈ Z such that q^{N}α is integral over Z, and for each embedding σ : Q(α) ↪ C, the usual archimedean absolute value of σ(α) is equal to |σ(α)|_{∞} = q^{an/2}.

(1.4.2) Definition. We say that Δ ∈ Rep_{E}(′W_{K}) is strictly pure of weight n ∈ Z if Δ = ρ ∈ Rep_{E}(W_{K}) and if for each w ∈ W_{K} all eigenvalues of ρ(w) are q^{ν(w)}-Weil numbers of weight n ∈ Z.

(1.4.3) Definition. We say that Δ ∈ Rep_{E}(′W_{K}) is pure of weight n ∈ Z if, for each r ∈ Z, gr_{r}^{M}Δ ∈ Rep_{E}(W_{K}) is strictly pure of weight n + r.

(1.4.4) (1) Each representation ρ ∈ Rep_{E}(W_{K}) with finite image is strictly pure of weight 0.

(2) If Δ, Δ′ ∈ Rep_{E}(′W_{K}) are (strictly) pure of weights n and n′, respectively, then Δ ⊗ Δ′ is (strictly) pure of weight n + n′, and Δ^{∗} is (strictly) pure of weight −n.

(3) For each m ∈ Z, E| · |^{m} is strictly pure of weight −2m.

(4) For each ρ ∈ Rep_{E}(W_{K}) and m ≥ 1,

Δ = ρ ⊗ sp(m) is pure of weight n ⟺ ρ is strictly pure of weight n + m − 1
⟹ Δ_{f} = ρ^{I}| · |^{m−1} is strictly pure of weight n + 1 − m.

(5) If Δ ∈ Rep_{E}(′W_{K}) is pure of weight n < 0, then all eigenvalues of ρ(f̃) (for any f̃ ∈ ν^{−1}(1)) on Ker(N) ⊆ M_{0}Δ are q-Weil numbers of weights ≤ n < 0, hence H^{0}(Δ) = 0.

(6) If Δ ∈ Rep_{E}(′W_{K}) is pure of weight n (but not necessarily f-semi-simple), then Δ ⟶^{∼} ⨁ ρ_{j} ⊗ sp(m_{j}), where each ρ_{j} ∈ Rep_{E}(W_{K}) is strictly pure of weight n + m_{j} − 1.

(1.4.5) Definition. In the situation of 1.1.3, we say that V ∈ Rep_{E}(G_{K}) is pure of weight n ∈ Z if W D(V) ∈ Rep_{E}(′W_{K}) is pure of weight n ∈ Z in the sense of 1.4.3.

(1.5) Specialization of representations of the Weil-Deligne group (1.5.1) Let O be a discrete valuation ring containing Q; denote by E (resp., k_{O}) the field of fractions (resp., the residue field) of O. (1.5.2) An object of Rep_{O}(′W_{K}) (= a representation of the Weil-Deligne group of K over O) consists of a free O-module of finite type T, a continuous homomorphism ρ = ρ_{T} : W_{K} ⟶ Aut_{O}(T) (with respect to the discrete topology on the target) and a nilpotent endomorphism N = N_{T} ∈ End_{O}(T) satisfying

∀w ∈ W_{K} ρ(w)Nρ(w)^{−1} = |w| N.

The generic fibre (resp., the special fibre) of T is the representation T_{η} = T ⊗_{O} E ∈ Rep_{E}(′W_{K}) (resp., the representation T_{s} = T ⊗_{O} k_{O} ∈ Rep_{k_{O}}(′W_{K})). We denote by N_{η} (resp., N_{s}) the monodromy operator N_{T} ⊗ 1 on T_{η} (resp., on T_{s}). (1.5.3) For T ∈ Rep_{O}(′W_{K}), we denote by T^{∗} the representation T^{∗} = Hom_{O}(T, O) (equipped with the dual action of W_{K} and the monodromy operator N_{T^{∗}} = −(N_{T})^{∗}). Given a representation ω : W_{K} ⟶ O^{∗}, we say that T is ω-orthogonal (resp., ω-symplectic) if there exists an isomorphism j : T ⟶^{∼} T^{∗} ⊗ ω in Rep_{O}(′W_{K}) satisfying j^{∗} ⊗ ω = j (resp., j^{∗} ⊗ ω = −j).

(1.5.4) Proposition. Assume that T ∈ Rep_{O}(′W_{K}) is | · |-symplectic (hence so are T_{η} and T_{s}) and that T_{s} ∈ Rep_{k_{O}}(′W_{K}) is pure (necessarily of weight −1). Then:

(1) ∀j ≥ 0 m_{j}(T_{η}) = m_{j}(T_{s}).

(2) ∀j ≥ 0 dim_{E} Ker(N_{η}^{j}) = dim_{k_{O}} Ker(N_{s}^{j}).

(3) For each j ≥ 0, the natural injective map (Ker(N_{η}^{j}) ∩ T ) ⊗_{O} k_{O} ⟶ Ker(N_{s}^{j}) is an isomorphism.

Proof. It is enough to prove (1), since (2) is a consequence of (1) and the formulas (1.3.6.1), and (2) is equivalent to (3) for trivial reasons. We prove (1) by induction on r = min{j ≥ 0 | N_{T}^{j+1} = 0}. If r = 0, then there is nothing to prove. Assume that r ≥ 1 and that (1) holds whenever N_{T}^{r} = 0. Recall from 1.3.2(2) and 1.3.5 that

M_{−r−1}(T_{η}) = 0 ≠ M_{−r}(T_{η}) = Im(N_{η}^{r}), M_{r−1}(T_{η}) = Ker(N_{η}^{r}) ≠ T_{η} = M_{r}(T_{η}),
M_{−r−1}(T_{s}) = 0,  M_{−r}(T_{s}) = Im(N_{s}^{r}),  M_{r−1}(T_{s}) = Ker(N_{s}^{r}),  M_{r}(T_{s}) = T_{s}

and that M_{−r}(T_{η}) is | · |^{r+1}-symplectic (resp., | · |^{r+1}-orthogonal) if 2 | r (resp., if 2 ∤ r). The latter property implies that, for any eigenvalue α ∈ \overline{k_{O}} of any lift f̃ ∈ ν^{−1}(1) of f acting on (M_{−r}(T_{η}) ∩ T) ⊗_{O} k_{O} there exists another eigenvalue α′ such that αα′ = q^{−r−1}. On the other hand, (M_{−r}(T_{η})∩T)⊗_{O} k_{O} ∈ Rep_{k_{O}}(W_{K}) is a sub-object of T_{s} in Rep_{k_{O}}(′W_{K}), and all eigenvalues of f̃ on T_{s} are q-Weil numbers of weights contained in {−r − 1, −r, …, r − 1}; thus both α and α′ are q-Weil numbers of weight −r − 1. In other words, (Im(N_{η}^{r}) ∩ T) ⊗_{O} k_{O} = (M_{−r}(T_{η}) ∩ T) ⊗_{O} k_{O} is strictly pure of weight −r − 1, hence is contained in M_{−r}(T_{s}) = Im(N_{s}^{r}) = (Im(N_{T}^{r})) ⊗_{O} k_{O}. The opposite inclusion being trivial, we deduce that Im(N_{T}^{r}) is equal to Im(N_{η}^{r}) ∩ T, hence is a direct summand of T (as an O-module); it follows that

m_{r}(T_{s}) = dim_{k_{O}} Im(N_{s}^{r}) = dim_{E} Im(N_{η}^{r}) = m_{r}(T_{η}).

The representation T′ = (M_{r−1}(T_{η}) ∩ T)/(M_{−r}(T_{η}) ∩ T) ∈ Rep_{O}(′W_{K}) is also | · |-symplectic, satisfies N_{T′}^{r} = 0, and T′_{s} is pure of weight −1. By induction hypothesis, we have

∀j ≥ 0 m_{j}(T′_{s}) = m_{j}(T′_{η}).

The relations

m_{j}(T′_{?}) = \{ \begin{matrix}m_{j}(T_{?}), & j ≠ r, r − 2 \\ m_{r−2}(T_{?}) + m_{r}(T_{?}), & j = r − 2 ≥ 0 \\ 0, & otherwise \end{matrix} (? = η, s)

then imply

∀j ≥ 0 m_{j}(T_{s}) = m_{j}(T_{η}).

2. Local ε-factors

(2.1) General facts (2.1.1) Fix an algebraically closed field E′ ⊃ E. Let ψ : K ⟶ E^{′∗} be a non-trivial continuous homomorphism (with respect to the discrete topology on the target); it always exists. If ψ′ : K ⟶ E^{′∗} is another non-trivial continuous homomorphism, then there exists unique a ∈ K^{∗} such that ψ′ = ψ_{a}, where ψ_{a}(y) = ψ(ay). Denote by μ_{ψ} the unique E′-valued Haar measure on K which is self-dual with respect to ψ; then

∀a ∈ K^{∗}  μ_{ψ_{a}} = |a|^{1/2} μ_{ψ},  (2.1.1.1)

and every non-zero E′-valued Haar measure μ on K is a scalar multiple of μ_{ψ}: μ = b μ_{ψ}, for some b ∈ E^{′∗}. (2.1.2) Deligne [De 1] associated to each triple (Δ, ψ, μ), where Δ ∈ Rep_{E}(′W_{K}) and ψ, μ are as in 2.1.1, the local ε-factor ε(Δ, ψ, μ) ∈ E^{′∗} satisfying the following properties.

(2.1.2.1) ε(Δ, ψ, μ) = ε(Δ^{f−ss}, ψ, μ). (2.1.2.2) If 0 ⟶ ρ′ ⟶ ρ ⟶ ρ′′ ⟶ 0 is an exact sequence in Rep_{E}(W_{K}), then ε(ρ, ψ, μ) = ε(ρ′, ψ, μ)ε(ρ′′, ψ, μ). (2.1.2.3) ε_{0}(Δ, ψ, μ) = ε(Δ, ψ, μ) det(−f | Δ_{f}) depends only on Δ^{N−ss} ∈ Rep_{E}(W_{K}). As (Δ^{N−ss})_{f} = Δ_{g}, it

follows that

ε(Δ, ψ, μ) = ε(Δ^{N−ss}, ψ, μ) det(−f | Δ_{g}/Δ_{f}).

(2.1.2.4) ∀a ∈ K^{∗} ε(Δ, ψ_{a}, μ) = (det Δ)(a) |a|^{− dim(Δ)} ε(Δ, ψ, μ). (2.1.2.5) ∀b ∈ E^{′∗} ε(Δ, ψ, b μ) = b^{dim(Δ)} ε(Δ, ψ, μ).

(2.1.2.6) If Δ = ρ ∈ Rep_{E}(W_{K}), then ε(ρ, ψ, μ) ε(ρ^{∗}| · |, ψ_{−1}, μ^{∗}) = 1 (where μ^{∗} is the measure dual to μ with

respect to ψ).

(2.1.2.7) If Δ = ρ ∈ Rep_{E}(W_{K}), and if χ : W_{K}/I ⟶ E^{∗} is an unramified one-dimensional representation, then

ε(ρ ⊗ χ, ψ, μ) = ε(ρ, ψ, μ) χ(π)^{a(ρ)+dim(ρ)n(ψ)},

where π is a prime element of O_{K} and a(ρ) (resp., n(ψ)) is the conductor exponent of ρ (resp., of ψ).

(2.1.2.8) ([Fo-PR, I.1.2.3]) For an exact sequence in Rep_{E}(′W_{K})

(β)  0 ⟶ Δ′ ⟶ Δ ⟶ Δ′′ ⟶ 0,

set P_{K}(β) = P_{K}(Δ, u)/P_{K}(Δ′, u)P_{K}(Δ′′, u), ε(β) = ε(Δ, ψ, μ)/ε(Δ′, ψ, μ) ε(Δ′′, ψ, μ), a(β) = dim Δ′_{f} + dim Δ′′_{f} − dim Δ_{f}, and similarly for the dual exact sequence

(β^{∗}| · |)  0 ⟶ Δ^{′′∗}| · | ⟶ Δ^{∗}| · | ⟶ Δ^{′∗}| · | ⟶ 0;

then

P_{K}(β^{∗}| · |, u^{−1}) = ε(β) u^{a(β)} P_{K}(β, u).

(2.1.3) Lemma. If Δ ∈ Rep_{E}(′W_{K}), then ε(Δ, ψ, μ) ε(Δ^{∗}| · |, ψ_{−1}, μ^{∗}) = 1 (where μ^{∗} is the measure dual to μ with respect to ψ).

Proof. Thanks to (2.1.2.1-2), we can assume that Δ is f-semi-simple and indecomposable: Δ = ρ ⊗ sp(m), ρ ∈ Rep_{E}(W_{K}), m ≥ 1. In this case

Δ_{g} = ⨁_{j=0}^{m−1} ρ^{I}| · |^{j},  Δ_{g}/Δ_{f} = ⨁_{j=0}^{m−2} ρ^{I}| · |^{j},  Δ^{∗}| · | = ρ^{∗} ⊗ sp(m)| · |^{2−m}

(Δ^{∗}| · |)_{g}/(Δ^{∗}| · |)_{f} = ⨁_{j=0}^{m−2}(ρ^{∗})^{I}| · |^{2−m+j} = (Δ_{g}/Δ_{f})^{∗}

(as ρ(I) is finite, we have (ρ^{∗})^{I} = (ρ^{I})^{∗}), hence

det(−f | Δ_{g}/Δ_{f}) det(−f | (Δ^{∗}| · |)_{g}/(Δ^{∗}| · |)_{f}) = 1;

we deduce that

ε(Δ, ψ, μ) ε(Δ^{∗}| · |, ψ_{−1}, μ^{∗}) = ε(Δ^{N−ss}, ψ, μ) ε((Δ^{∗}| · |)^{N−ss}, ψ_{−1}, μ^{∗}),

which is equal to 1, by (2.1.2.6).

(2.2) | · |-symplectic representations

(2.2.1) Proposition. Let Δ ⟶^{∼} Δ^{∗}| · | ∈ Rep_{E}(′W_{K}) be | · |-symplectic. Then:

(1) ε(Δ) := ε(Δ, ψ, μ_{ψ}) does not depend on ψ.

(2) ε(Δ) = ±1; more precisely:

(3) If ρ ⟶^{∼} ρ^{∗}| · | ∈ Rep_{E}(W_{K}) is | · |-symplectic, then ε(ρ) = ±1.

(4) If Δ = X ⊕ X^{∗}| · | is as in 1.2.4(1), then ε(Δ) = ε(Δ^{N−ss}) = (det X)(−1).

(5) If Δ = ρ ⊗ sp(2n + 1) (ρ ∈ Rep_{E}(W_{K}), n ≥ 0), then ρ| · |^{n} ∈ Rep_{E}(W_{K}) is | · |-symplectic and ε(Δ) = ε(Δ^{N−ss}) = ε(ρ| · |^{n}).

(6) If Δ = ρ ⊗ sp(2n) (ρ ∈ Rep_{E}(W_{K}), n ≥ 1), then ρ| · |^{n−1} ∈ Rep_{E}(W_{K}) is orthogonal, there is an exact sequence in Rep_{E}(′W_{K})

0 ⟶ Δ^{+} ⟶ Δ ⟶ Δ^{−} ⟶ 0
Δ^{+} = ρ ⊗ sp(n)| · |^{n}, Δ^{−} = ρ ⊗ sp(n),

H^{0}(Δ^{−}) = H^{0}(ρ| · |^{n−1}) and

ε(Δ) = (−1)^{dim_{E} H^{0}(Δ^{−})}(det Δ^{+})(−1), ε(Δ^{N−ss}) = (det Δ^{+})(−1).

Proof. (1) For each a ∈ K^{∗},

ε(Δ, ψ_{a}, μ_{ψ_{a}}) = ε(Δ, ψ_{a}, |a|^{1/2}μ_{ψ})  (by (2.1.1.1))
= |a|^{dim(Δ)/2} ε(Δ, ψ_{a}, μ_{ψ})  (by (2.1.2.5))
= (det Δ)(a) |a|^{− dim(Δ)/2} ε(Δ, ψ, μ_{ψ})  (by (2.1.2.4))
= ε(Δ, ψ, μ_{ψ}).  (by 1.2.2(2))

(2) Writing Δ^{f−ss} as a direct sum of | · |-symplectic representations of the form 1.2.4(1) or 1.2.4(2), the statement follows from the explicit formulas (4)-(6) and (3), proved below.

(3) Combining (2.1.2.6), (2.1.2.4) and 1.2.2(2), we obtain

ε(ρ, ψ, μ_{ψ})^{2} = ε(ρ, ψ, μ_{ψ}) (det ρ)(−1)ε(ρ, ψ, μ_{ψ}) = ε(ρ, ψ, μ_{ψ}) ε(ρ, ψ_{−1}, μ_{ψ}) = ε(ρ, ψ, μ_{ψ}) ε(ρ^{∗}|·|, ψ_{−1}, μ_{ψ}) = 1

(4) As in the proof of (3), Lemma 2.1.3 together with (2.1.2.4) yield

ε(Δ) = ε(X, ψ, μ_{ψ}) ε(X^{∗}| · |, ψ, μ_{ψ}) = (det X)(−1) ε(X, ψ_{−1}, μ_{ψ}) ε(X^{∗}| · |, ψ, μ_{ψ}) = (det X)(−1).

Replacing X by X^{N−ss}, we obtain ε(Δ^{N−ss}) = (det X^{N−ss})(−1) = (det X)(−1) = ε(Δ).

(5) As Δ = ρ ⊗ sp(2n + 1) is | · |-symplectic, the representation ρ| · |^{n} is also | · |-symplectic, by 1.2.3-4 (in particular, det(ρ) = | · |^{(1−2n) dim(ρ)/2}). The same calculation as in the proof of Lemma 2.1.3 yields

Δ_{g}/Δ_{f} = ⨁_{j=0}^{2n−1} ρ^{I}| · |^{j},  (ρ^{I}| · |^{j})^{∗} = (ρ^{∗}| · |^{−j})^{I} = ρ^{I}| · |^{2n−1−j},  Δ_{g}/Δ_{f} = ⨁_{j=0}^{n−1} ρ^{I}| · |^{j} ⊕ (ρ^{I}| · |^{j})^{∗} ,

which implies that det (−f | Δ_{g}/Δ_{f}) = 1, hence

ε(Δ) = ε(Δ^{N−ss}) = ∏_{j=0}^{2n} ε(ρ| · |^{j}, ψ, μ_{ψ}) = ε(ρ| · |^{n}) ∏_{j=0}^{n−1} ε(ρ| · |^{j} ⊕ (ρ| · |^{j})^{∗}| · |) =^{(4)} ε(ρ| · |^{n}).

(6) As Δ = ρ ⊗ sp(2n) is | · |-symplectic, the representation ρ| · |^{n−1} is orthogonal, by 1.2.3. The same calculation as in the proof of (5) shows that

ε(Δ^{N−ss}) = ∏_{j=0}^{n−1} ε(ρ| · |^{j} ⊕ (ρ| · |^{j})^{∗}| · |) =^{(4)} ∏_{j=0}^{n−1} det(ρ| · |^{j})(−1) = (det Δ^{+})(−1)

and

Δ_{g}/Δ_{f} = ρ^{I}| · |^{n−1} ⊕ ⨁_{j=0}^{n−2} ρ^{I}| · |^{j} ⊕ (ρ^{I}| · |^{j})^{∗} , det (−f | Δ_{g}/Δ_{f}) = (−f | ρ^{I}| · |^{n−1}) .

As ρ(I) acts semi-simply, the (unramified) representation V = ρ^{I}| · |^{n−1} ∈ Rep_{E}(W_{K}) is also orthogonal; applying Lemma 2.2.2 below to u = f acting on V, we obtain

ε(Δ)/ε(Δ^{N−ss}) = det (−f | Δ_{g}/Δ_{f}) = (−1)^{dim_{E} Ker(f−1:V ⟶V)}.

Finally,

Ker(V ⟶^{f−1}V) = H^{0}(ρ| · |^{n−1}) = H^{0}(ρ ⊗ sp(n)) = H^{0}(Δ^{−}).

(2.2.2) Lemma. Let (V, q) be a non-degenerate quadratic space over a field L of characteristic not equal to 2. If u ∈ O(V, q), then

det(−u) = (−1)^{dim_{L} Ker(u−1)}, det(u) = (−1)^{dim_{L} Im(u−1)}.

Proof. The following short argument is due to J. Oesterlé. The two formulas being equivalent, it is enough to prove the second one. Let a ∈ V, q(a) ≠ 0; denote by s ∈ O^{−}(V, q) the reflection with respect to the hyperplane Ker(s − 1) = a^{⊥}. A short calculation shows that

Ker(su − 1) = \{ \begin{matrix}Ker(u − 1) ⊕ Lb, & a = (u − 1)b, b ∈ V \\ Ker(u − 1) ∩ a^{⊥} ⊊ Ker(u − 1), & a ∉ Im(u − 1), \end{matrix}

hence

dim_{L} Im(su − 1) = dim_{L} Im(u − 1) ∓ 1.  (2.2.2.1)

Writing u as a product of r ≥ 1 reflections, we deduce from (2.2.2.1), by induction, that dim_{L} Im(u − 1) ≡ r (mod 2), as claimed.

(2.2.3) Proposition. Let Δ ⟶^{∼} Δ^{∗}| · | ∈ Rep_{E}(′W_{K}) be | · |-symplectic and pure (of weight −1). Assume that there exists an exact sequence in Rep_{E}(′W_{K})

(β)  0 ⟶ Δ^{+} ⟶ Δ ⟶ Δ^{−} ⟶ 0

such that the isomorphism Δ ⟶^{∼} Δ^{∗}| · | induces isomorphisms Δ^{±} ⟶^{∼} (Δ^{∓})^{∗}| · |. Assume, in addition, that there exists a direct sum decomposition Δ = Δ_{1} ⊕ Δ_{2} in Rep_{E}(′W_{K}) compatible with the isomorphism Δ ⟶^{∼} Δ^{∗}| · | and the exact sequence (β), and such that H^{0}(Δ_{2}^{−}) = 0, while

(β_{1})  0 ⟶ Δ_{1}^{+} ⟶ Δ_{1} ⟶ Δ_{1}^{−} ⟶ 0

is a direct sum of exact sequences of the type considered in Proposition 2.2.1(6). Then

ε(Δ) = (−1)^{dim_{E} H^{0}(Δ^{−})}(det Δ^{+})(−1), ε(Δ^{N−ss}) = (det Δ^{+})(−1).

Proof. It is enough to treat separately Δ_{1} and Δ_{2}. For Δ = Δ_{1}, the statement follows from Proposition 2.2.1(6). For Δ = Δ_{2}, the assumption H^{0}(Δ^{−}) = 0 implies that P_{K}(Δ^{−}, 1) ≠ 0. As Δ is pure of weight −1 < 0, we also have H^{0}(Δ^{+}) ⊆ H^{0}(Δ) = 0, by 1.4.4(5), hence P_{K}(Δ^{+}, 1)P_{K}(Δ, 1) ≠ 0. Letting u ⟶ 1 in (2.1.2.8), we obtain ε(β) = 1, hence

ε(Δ) = ε(Δ^{+}, ψ, μ_{ψ}) ε(Δ^{−}, ψ, μ_{ψ}) = ε(Δ^{+} ⊕ (Δ^{+})^{∗}| · |) = (det Δ^{+})(−1).

Finally,

ε(Δ^{N−ss}) = ε((Δ^{+})^{N−ss}, ψ, μ_{ψ}) ε((Δ^{−})^{N−ss}, ψ, μ_{ψ}) = ε((Δ^{+})^{N−ss} ⊕ ((Δ^{+})^{N−ss})^{∗}| · |) =
= (det(Δ^{+})^{N−ss})(−1) = (det Δ^{+})(−1).

(2.2.4) Proposition. In the situation of 1.5.4, ε(T_{s}) = ε(T_{η}) ∈ {±1}.

Proof. For any O-module X, denote by red : X ⟶ X ⊗_{O} k_{O} the canonical surjection. Proposition 1.5.4 implies that

red (\frac{T ∩ (T_{η})_{g}}{T ∩ (T_{η})_{f}} ) = (T_{s})_{g}/(T_{s})_{f},

hence

red (ε(T_{η})/ε(T_{η}^{N−ss})) = red (det (−f | (T_{η})_{g}/(T_{η})_{f})) = (det (−f | (T_{s})_{g}/(T_{s})_{f}) = ε(T_{s})/ε(T_{s}^{N−ss}).

As ε(T_{η}), ε(T_{η}^{N−ss}), ε(T_{s}), ε(T_{s}^{N−ss}) ∈ {±1}, we are reduced to showing that

red (ε(T_{η}^{N−ss})) =^{?} ε(T_{s}^{N−ss}).

The following argument is based on a suggestion of T. Saito. We can replace (ρ_{T} , N_{T}) by (ρ_{T} , 0) and assume that N_{T} = 0. Furthermore, after replacing E by a finite extension, we can assume (see [De 1, 4.10]) that

T_{η}^{f−ss} = ⨁_{α} ρ_{α} ⊗ ω_{α},

where ρ_{α} ∈ Rep_{L}(W_{K}) for a subfield L ⊂ O of finite degree over Q, and ω_{α} : W_{K}/I ⟶ O^{∗} is an unramified representation of rank 1. We have

∀w ∈ W_{K} Tr(w | T_{s}) = red (Tr(w | T_{η})) ,

hence

T_{s}^{f−ss} = ⨁_{α} ρ_{α} ⊗ red(ω_{α}).

Applying (2.1.2.7) to each direct summand, we obtain

red (ε(T_{η})) = ∏_{α} red (ε(ρ_{α} ⊗ ω_{α}, ψ, μ_{ψ})) = ∏_{α} ε(ρ_{α} ⊗ red(ω_{α}), red ∘ ψ, red ∘ μ_{ψ}) = ε(T_{s}).

(2.3) The archimedean case

Let L = R or C. If H is a pure R-Hodge structure over L ([Fo-PR, III.1]) of weight −1, then

H = ⨁_{r>0} H_{r}(r)^{⊕m_{r}},

where H_{r} is a two-dimensional pure R-Hodge structure over L of Hodge type (2r − 1, 0), (0, 2r − 1). The standard formulas ([De 3, 5.3], [Fo-PR, III.1.1.10, III.1.2.7]) yield

ε(H_{r}(r)) = (−1)^{[L:R] r} × \{ \begin{matrix}1, & L = R \\ −1, & L = C. \end{matrix}

As

∀p < 0 h^{p,−1−p}(H) = m_{−p},

we obtain

ε(H) = (−1)^{[L:R] d^{−}(H)} × \{ \begin{matrix}1, & L = R \\ (−1)^{(dim_{R} H)/2}, & L = C, \end{matrix}  d^{−}(H) = ∑_{p<0} p h^{p,q}(H). (2.3.1)

3. Local p-adic Galois representations

(3.1) General facts (3.1.1) Notation. Let p be the characteristic of the residue field k of K; then q = p^{h} and K is a finite extension of Q_{p}. Denote by σ ∈ Gal(Q_{p}^{ur}/Q_{p}) ⟶^{∼} G_{F_{p}} the lift of the arithmetic Frobenius element x ↦ x^{p}. Let L be another finite extension of Q_{p}. We use the standard notation

Rep_{cris,L}(G_{K}) ⊂ Rep_{st,L}(G_{K}) ⊂ Rep_{pst,L}(G_{K}) = Rep_{dR,L}(G_{K}) ⊂ Rep_{L}(G_{K})

for Fontaine’s hiearchy of (finite-dimensional, L-linear) representations of G_{K} ([Fo]), and

D_{cris}(V) = (V ⊗_{Q_{p}} B_{cris})^{G_{K}} ,  D_{st}(V) = (V ⊗_{Q_{p}} B_{st})^{G_{K}} ,  D_{pst}(V) = \underrightarrow{lim}_{K′} (V ⊗_{Q_{p}} B_{st})^{G_{K′}} ,
D_{dR}^{i}(V) = (V ⊗_{Q_{p}} t^{i}B_{dR}^{+})^{G_{K}} ⊂ D_{dR}(V) = (V ⊗_{Q_{p}} B_{dR})^{G_{K}}

for various Fontaine’s functors (above, V ∈ Rep_{L}(G_{K}), and K′ runs through all finite extensions of K contained in \overline{K}). As in [Bl-Ka], put H^{i}(K, −) = H_{cont}^{i}(G_{K}, −) and, for ∗ = e, f, st, g,

H_{∗}^{1}(K, V) = Ker (H^{1}(K, V) ⟶ H^{1}(K, V ⊗_{Q_{p}} B_{∗})) ,  B_{e} = B_{cris}^{φ=1},  B_{f} = B_{cris},  B_{g} = B_{dR}.

If K′/K is a finite Galois extension, then

H_{∗}^{1}(K, V) = H_{∗}^{1}(K′, V)^{Gal(K′/K)},  (∗ = ∅, e, f, st, g)  (3.1.1.1)

(as both H^{1}(−, V) and H^{1}(−, V ⊗_{Q_{p}} B_{∗}) satisfy Galois descent w.r.t. the extension K′/K, and the functor of Gal(K′/K)-invariants is exact on the category of Q[Gal(K′/K)]-modules). (3.1.2) For V ∈ Rep_{dR,L}(G_{K}) and i ∈ Z, define

d_{L}^{i}(V) := dim_{L} (D_{dR}^{i}(V)/D_{dR}^{i+1}(V)) ,  d_{L}^{−}(V) := ∑_{i<0} i d_{L}^{i}(V),  d_{L}(V) := ∑_{i∈Z} i d_{L}^{i}(V).

(3.1.3) If V ∈ Rep_{pst,L}(G_{K}), then D = D_{pst}(V) is a free (Q_{p}^{ur} ⊗_{Q_{p}} L)-module of rank equal to dim_{L}(V), which is equipped (among others) with the following structure ([Fo], [Fo-PR, I.2.2]):

(1) An L-linear action ρ_{sl} : W_{K} ⟶ Aut_{L}(D), which is Q_{p}^{ur}-semi-linear in the following sense:

∀w ∈ W_{K} ∀λ ∈ Q_{p}^{ur} ∀x ∈ D ρ_{sl}(w)(λx) = f_{k}^{ν(w)}(λ) ρ_{sl}(w)(x).

(2) An L-linear, σ-semi-linear map φ : D ⟶ D commuting with ρ_{sl}(w) (for all w ∈ W_{K}):

∀w ∈ W_{K} ∀λ ∈ Q_{p}^{ur} ∀x ∈ D φ(λx) = σ(λ) φ(x).

(3) A (Q_{p}^{ur} ⊗_{Q_{p}} L)-linear nilpotent endomorphism N : D ⟶ D commuting with ρ_{sl}(w) (for all w ∈ W_{K}) and satisfying Nφ = pφN.

(4) An isomorphism of (K ⊗_{Q_{p}} L)-modules

(D ⊗_{Q_{p}^{ur}} \overline{K})^{G_{K}} ⟶^{∼} D_{dR}(V).

(3.2) Potentially semistable representations and representations of the Weil-Deligne group We recall how, for each V ∈ Rep_{pst,L}(G_{K}), the structure 3.1.3(1)-(3) can be used to define a representation of the Weil-Deligne group of K ([Fo], [Fo-PR, I.1.3.2]). (3.2.1) Fix a field E ⊃ Q_{p}^{ur} for which there exists an embedding τ : L ↪ E, and define

W D_{τ} (V) := D_{pst}(V) ⊗_{Q_{p}^{ur}⊗_{Qp} L,id⊗τ} E,

which is an E-vector space of dimension dim_{E}(W D_{τ} (V)) = dim_{L}(V). We define an E-linear action of W_{K} on W D_{τ} (V) by

ρ(w) := ρ_{sl}(w) ∘ φ^{hν(w)} ⊗ id (w ∈ W_{K})

and a monodromy operator N = N ⊗ id ∈ End_{E}(W D_{τ} (V)). This defines a representation

W D_{τ} (V) = (ρ, N) ∈ Rep_{E}(′W_{K}),

whose isomorphism class does not depend on τ. Furthermore,

W D_{τ} : Rep_{pst,L}(G_{K}) ⟶ Rep_{E}(′W_{K})

is an exact tensor functor. (3.2.2) Examples: (1) If V is potentially unramified, then W D_{τ} (V) = V ⊗_{L,τ} E ∈ Rep_{E}(W_{K}).

(2) If V is semistable, then W D(V) = D_{st}(V) ⊗_{K_{0}⊗_{Qp} L,id⊗τ} E (K_{0} = K ∩ Q_{p}^{ur}), with ρ(I) acting trivially, N = N ⊗ id and ρ(f_{k}) = φ^{h} ⊗ id. Conversely, if ρ(I) acts trivially, then V is semistable.

(3) If V = L(n) = L ⊗_{Q_{p}} Q_{p}(n) (n ∈ Z), then W D_{τ} (V) = E| · |^{n} = E ⊗ | · |^{n}.

(4) (Lubin-Tate theory) Fix a prime element π ∈ O_{K}. The reciprocity map rec_{K} : K^{∗} ⟶ G_{K}^{ab} (normalized using the geometric Frobenius element) defines a one-dimensional representation V_{π} ∈ Rep_{cris,K}(G_{K})

χ_{π} : G_{K} ⟶ G_{K}^{ab} ⟶^{∼} K̂^{∗} = π_{Z} × O_{K}^{∗} ↠ O_{K}^{∗} ↪ K^{∗},

which arises in the π-adic Tate module of any Lubin-Tate group over K associated to π. In this case

D_{pst}(V_{π}) = (Q_{p}^{ur} ⊗_{Q_{p}} K)u,  φ^{h}(u) = (1 ⊗ π)^{−1}u,  Nu = 0,  ∀w ∈ W_{K} ρ_{sl}(w)(u) = u.

If E ⊃ Q_{p}^{ur} is a field and τ : K ↪ E an embedding of fields, then W D_{τ} (V_{π}) ∈ Rep_{E}(W_{K}) is an unramified one-dimensional representation of W_{K}, on which f = f_{k} acts by τ(π)^{−1}. For K = Q_{p} and π = p we recover Example (3) for n = 1.

(3.2.3) Definition. We say that V ∈ Rep_{pst,L}(G_{K}) is pure of weight n ∈ Z if W D_{τ} (V) ∈ Rep_{E}(′W_{K}) is pure of weight n, in the sense of 1.4.3.

(3.2.4) Lemma. For each V ∈ Rep_{pst,L}(G_{K}) and each τ : L ↪ E ⊃ Q_{p}^{ur},

W D_{τ} (V)_{g}^{f_{k}=1} = D_{st}(V)^{φ=1} ⊗_{L,τ} E, H^{0}(W D_{τ} (V)) = W D_{τ} (V)_{f}^{f_{k}=1} = D_{cris}(V)^{φ=1} ⊗_{L,τ} E.

Proof. As D_{cris}(V) = D_{st}(V)^{N=0}, it is enough to prove the first equality. As both sides satisfy Galois descent with respect to finite Galois extensions K′/K, we can assume that V is semistable. In this case, 3.2.2(2) implies that

W D_{τ} (V)_{g}^{f_{k}=1} = W D_{τ} (V)^{f_{k}=1} = D_{st}(V)^{φ^{h}=1} ⊗_{K_{0}⊗_{Qp} L,id⊗τ} E (K_{0} = K ∩ Q_{p}^{ur}).

As

D_{st}(V)^{φ^{h}=1} = D_{st}(V)^{φ=1} ⊗_{Q_{p}} K_{0}

(thanks to Hilbert’s Theorem 90 for H^{1}(K_{0}/Q_{p}, GL_{n}(K_{0}))), it follows that

W D_{τ} (V)_{g}^{f_{k}=1} = D_{st}(V)^{φ=1} ⊗_{L,τ} E.

(3.2.5) Corollary. If V ∈ Rep_{pst,L}(G_{K}) is pure of weight n < 0, then D_{cris}(V)^{φ=1} = 0.

(3.2.6) Proposition. For each V ∈ Rep_{pst,L}(G_{K}),

(det_{E}(W D_{τ} (V)))(−1) = (−1)^{d_{L}(V)} (det_{L}V)(−1).

Proof. As W D_{τ} is a tensor functor and d_{L}(V) = d_{L}(det_{L}(V)), we can replace V by det_{L}(V), hence assume that dim(V) = 1; denote by χ_{V} : G_{K} ⟶ K^{∗} the character by which G_{K} acts on V. After replacing L by a finite extension, we can assume that L contains the Galois closure of K over Q_{p}. As V is potentially semistable, there exists a one-dimensional representation

χ : G_{K} ⟶ L^{∗}

with finite image and integers n_{σ} (σ : K ↪ L) such that

χ_{V} = χ ∏_{σ:K↪L}(σ ∘ χ_{π})^{−n_{σ}},

where χ_{π} : G_{K} ⟶ K^{∗} is as in 3.2.2(4). It follows from 3.2.2 that W D_{τ} (V) = (τ ∘χ)α, where α : W_{K}/I ⟶ E^{∗} is the one-dimensional unramified representation satisfying

α(f) = ∏_{σ:K↪L} τ(σ(π))^{n_{σ}}.

This implies that

(det_{E}(W D_{τ} (V)))(−1) = χ(−1),  (det_{L}V)(−1) = (−1)^{n} χ(−1),  n = ∑_{σ:K↪L} n_{σ}.

On the other hand,

d_{L}^{i}(V) = |{σ : K ↪ L | n_{σ} = i}|,

hence n = d_{L}(V).

(3.3) Representations satisfying Pančiškin’s condition We recall a few basic facts from [Ne 1].

(3.3.1) Definition. We say that V ∈ Rep_{L}(G_{K}) satisfies Pančiškin’s condition if there exists an exact sequence in Rep_{L}(G_{K})

0 ⟶ V^{+} ⟶ V ⟶ V^{−} ⟶ 0

such that V^{±} ∈ Rep_{pst,L}(G_{K}) and D_{dR}^{0}(V^{+}) = 0 = D_{dR}(V^{−})/D_{dR}^{0}(V^{−}). If this is the case, then V^{±} are uniquely determined ([Ne 1], 6.7), V ∈ Rep_{pst,L}(G_{K}) ([Ne 1], 1.28) and V^{∗}(1) also satisfies Pančiškin’s condition (with (V^{∗}(1))^{±} = (V^{∓})^{∗}(1)).

(3.3.2) Proposition. If V satisfies Pančiškin’s condition, then:

(1) H^{0}(K, V^{−}) = D_{cris}(V^{−})^{φ=1} = D_{st}(V^{−})^{φ=1}.

(2) Assume that there exists a finite Galois extension K′/K over which V becomes semistable and such that D_{cris}(V |_{G_{K′}})^{φ=1} = D_{cris}(V^{∗}(1)|_{G_{K′}})^{φ=1} = 0 (the latter condition holds, e.g., if V is pure of weight −1, by 3.2.5). Then

H_{e}^{1}(K, V) = H_{f}^{1}(K, V) = H_{st}^{1}(K, V) = H_{g}^{1}(K, V)

and there is an exact sequence

0 ⟶ H^{0}(K, V^{−}) ⟶ H^{1}(K, V^{+}) ⟶ H_{f}^{1}(K, V) ⟶ 0,

in which H^{1}(K, V^{+}) = H_{st}^{1}(K, V^{+}).

Proof. (1) This is proved in [Ne 1, 1.28(3)] under the tacit assumption that V^{−} is semistable. The general case follows by passing to a finite Galois extension over which V^{−} becomes semistable and taking Galois invariants.

(2) Over K′, the statement is proved in [Ne 1, 1.32]; the general case follows by applying (3.1.1.1).

(3.3.3) Proposition. Assume that V satisfies Pančiškin’s condition, is pure (of weight −1) and that there exists an isomorphism j : V ⟶^{∼} V^{∗}(1) in Rep_{L}(G_{K}) satisfying j^{∗}(1) = −j. Then:

(1) j induces isomorphisms V^{±} ⟶^{∼} (V^{∓})^{∗}(1).

(2) Fix an embedding of fields τ : L ↪ E ⊃ Q_{p}^{ur} and put Δ = W D_{τ} (V), Δ^{±} = W D_{τ} (V^{±}). Then Δ ∈ Rep_{E}(′W_{K}) is | · |-symplectic and the exact sequence in Rep_{E}(′W_{K})

0 ⟶ Δ^{+} ⟶ Δ ⟶ Δ^{−} ⟶ 0

satisfies the assumptions of Proposition 2.2.3.

(3) (det_{E}Δ^{+})(−1)/(det_{L}V^{+})(−1) = (−1)^{d_{L}(V^{+})} = (−1)^{d_{L}^{−}(V)}.

(4) The ε-factors of Δ and Δ^{N−ss} are equal to

ε(Δ) = (−1)^{dim_{L} H^{0}(K,V^{−})} (−1)^{d_{L}^{−}(V)} (det_{L}V^{+})(−1), ε(Δ^{N−ss}) = (−1)^{d_{L}^{−}(V)} (det_{L}V^{+})(−1).

Proof. (1) This follows from the remarks made in 3.3.1.

(2) Δ is | · |-symplectic, since W D_{τ} is a tensor functor. In order to verify the assumptions of Proposition 2.2.3, we are going to decompose Δ into several components. Firstly, the functor

Rep_{E}(′W_{K}) ⟶ Rep_{E}(′W_{K}), X ↦ X^{ρ(I)}

is exact and commutes with duals. In addition, X^{ρ(I)} is a direct summand of X, with a functorial complement X′. Secondly, for each λ ∈ \overline{E}, the minimal polynomial p_{[λ]}(T) of λ over E depends only on the G_{E}-orbit [λ] of λ. We define

Δ_{1} = ⨁_{λ∈q^{Z}} ⋃_{n≥1} Ker ((f − λ)^{n} : Δ^{ρ(I)} ⟶ Δ^{ρ(I)}) , Δ_{2} = Δ′ ⊕ ⨁_{λ∉q^{Z}} ⋃_{n≥1} Ker (p_{[λ]}(f)^{n} : Δ^{ρ(I)} ⟶ Δ^{ρ(I)}) .

The direct sum decomposition Δ = Δ_{1} ⊕Δ_{2} in Rep_{E}(′W_{K}) is compatible with the isomorphism Δ ⟶^{∼} Δ^{∗}|·| and the exact sequence

0 ⟶ Δ^{+} ⟶ Δ ⟶ Δ^{−} ⟶ 0.

By construction, every subquotient of Δ_{2} in Rep_{E}(′W_{K}) has trivial H^{0}, hence H^{0}(Δ_{2}^{−}) = 0. As Δ is pure of weight −1, it follows that

Δ_{1} = ⨁_{m≥1} σ_{m} ⊗ sp(2m),

where each σ_{m} ∈ Rep_{E}(′W_{K}) is an unramified representation of W_{K} on which q^{1−m}f acts unipotently. As V satisfies the Pančiškin condition, weak admissibility of V^{±} implies that all eigenvalues of f on Δ_{1}^{+} = Δ^{+} ∩ Δ_{1} (resp., on Δ_{1}^{−} = Δ_{1}/Δ_{1}^{+}) are of the form q^{n} with n < 0 (resp., with n ≥ 0). It follows that

Δ_{1}^{+} = ⨁_{m≥1} σ_{m} ⊗ sp(m)| · |^{m}, Δ_{1}^{−} = ⨁_{m≥1} σ_{m} ⊗ sp(m),

which proves (2).

(3) This follows from Proposition 3.2.6 applied to V^{+}.

(4) We combine Proposition 2.2.3 (which applies to Δ, thanks to (2)) with the formula (3) and the fact that H^{0}(Δ^{−}) = D_{cris}(V^{−})^{φ=1} ⊗_{L,τ} E = (D_{cris}(V^{−})^{φ=1} ∩ D_{dR}^{0}(V^{−})) ⊗_{L,τ} E = H^{0}(K, V^{−}) ⊗_{L,τ} E.

4. Global p-adic Galois representations

(4.1) Generalities

(4.1.1) Notation. Let F be a number field. For each prime l of Q, let S_{l} be the set of primes of F above l. Fix a prime number p, a finite extension L_{𝔭} of Q_{p} and a finite set S ⊃ S_{∞} ∪S_{p} of primes of F. Let F_{S} be the maximal extension of F (contained in \overline{F}) unramified outside S; put G_{F,S} = Gal(F_{S}/F). For each prime v of F fix an embedding \overline{F} ↪ \overline{F}_{v}; this defines a morphism G_{F_{v}} ⟶ G_{F} ⟶ G_{F,S}. For each Galois representation V ∈ Rep_{L_{𝔭}}(G_{F,S}) (continuous and finite-dimensional over L_{𝔭}), denote by V_{v} ∈ Rep_{L_{𝔭}}(G_{F_{v}}) the local Galois representation induced by the map G_{F_{v}} ⟶ G_{F,S}. For each v ∉ S_{∞} ∪S_{p}, denote by W D(V_{v}) ∈ Rep_{L_{𝔭}}(′W_{F_{v}}) the associated representation of the Weil-Deligne group of F_{v} (see 1.1.3). As in [Bl-Ka], we put

∀v ∉ S_{∞} ∪ S_{p} H_{f}^{1}(F_{v}, V) = H_{ur}^{1}(F_{v}, V) = Ker (H^{1}(F_{v}, V) ⟶ H^{1}(F_{v}^{ur}, V))

H_{f}^{1}(F, V) = Ker (H^{1}(G_{F,S}, V) ⟶ ⨁_{v∈S−S_{∞}} H^{1}(F_{v}, V)/H_{f}^{1}(F_{v}, V)) .

The L_{𝔭}-vector space H_{f}^{1}(F, V) does not change if we enlarge S. (4.1.2) Throughout §4, assume that V satisfies the following conditions.

(1) There exists an isomorphism j : V ⟶^{∼} V^{∗}(1) in Rep_{L_{𝔭}}(G_{F,S}) satisfying j^{∗}(1) = −j.

(2) For each v ∈ S_{p}, V_{v} ∈ Rep_{L_{𝔭}}(G_{F_{v}}) satisfies the Pančiškin condition 3.3.1:

0 ⟶ V_{v}^{+} ⟶ V_{v} ⟶ V_{v}^{−} ⟶ 0

(in particular, V_{v} ∈ Rep_{pst,L_{𝔭}}(G_{F_{v}})).

(3) For each v ∉ S_{∞} ∪ S_{p} (resp., v ∈ S_{p}), V_{v} is pure (necessarily of weight −1) in the sense of 1.4.5 (resp., in the sense of 3.2.3).

(4) For each i ∈ Z, the integer

d^{i}(V) := dim_{L_{𝔭}} (D_{dR}^{i}(V_{v})/D_{dR}^{i+1}(V_{v})) /[F_{v} : Q_{p}]

does not depend on v ∈ S_{p}. This condition is satisfied if V = M_{𝔭} is the 𝔭-adic realization of a motive (pure of weight −1) M over F with coefficients in a number field L (of which L_{𝔭} is a completion), as

d^{i}(V) = dim_{L} (F^{i}M_{dR}/F^{i+1}M_{dR})

in this case. Example: F = Q and V = (S^{2m−1}V (f))(mk − m + 1 − k/2), where m ≥ 1 and V (f) is the Galois representation (pure of weight k − 1) associated to a potentially p-ordinary Hecke eigenform f ∈ S_{k}(Γ_{0}(N)) of (even) weight k and trivial character. (4.1.3) ε-factors. We define

d^{−}(V) = ∑_{i<0} i d^{i}(V),  (4.1.3.1)

∀v ∈ S_{∞}  ε(V_{v}) = (−1)^{[F_{v}:R] d^{−}(V)} × \{ \begin{matrix}1, & F_{v} = R \\ (−1)^{dim_{L𝔭} (V)/2}, & F_{v} = C \end{matrix}  (4.1.3.2)

(in view of (2.3.1), this is the correct archimedean local ε-factor if V = M_{𝔭} is as in 4.1.2(4)) and

∀v ∉ S_{∞}  ε(V_{v}) = ε(W D(V_{v})).  (4.1.3.3)

For any prime v of F, let

ε̃(V_{v}) = ε(V_{v}) × \{ \begin{matrix}(−1)^{h^{0}(F_{v},V_{v}^{−})}, & v ∈ S_{p} \\ 1, & v ∉ S_{p}, \end{matrix}  (4.1.3.4)

where

h^{i}(F_{v}, X) = dim_{L_{𝔭}} H^{i}(F_{v}, X) (X ∈ Rep_{L_{𝔭}}(G_{F_{v}})).

Finally, define

ε(V) = ∏_{v} ε(V_{v}),  ε̃(V) = ∏_{v} ε̃(V_{v})  (4.1.3.5)

(this makes sense, as ε(V_{v}) = 1 for all but finitely many v). It follows from Proposition 3.3.3 that

∀v ∈ S_{p}  ε̃(V_{v}) = (−1)^{[F_{v}:Q_{p}] d^{−}(V)} (det V_{v}^{+})(−1) = ε(W D(V_{v})^{N−ss}),  (4.1.3.6)

hence

∏_{v∈S_{p}} ε̃(V_{v}) = (−1)^{[F:Q] d^{−}(V)} ∏_{v∈S_{p}}(det V_{v}^{+})(−1).

As

∏_{v∈S_{∞}} ε(V_{v}) = (−1)^{[F:Q] d^{−}(V)},

it follows that

∏_{v∈S_{p}∪S_{∞}} ε̃(V_{v}) = ∏_{v∈S_{p}}(det V_{v}^{+})(−1).  (4.1.3.7)

(4.2) Selmer complexes and extended Selmer groups (4.2.1) For a pro-finite group G and a representation X ∈ Rep_{L_{𝔭}}(G) (continuous, finite-dimensional), denote by C^{•}(G, X) the standard complex of (non-homogeneous) continuous cochains of G with values in X. Fix a set S_{p} ⊂ Σ ⊂ S and define, for each v ∈ S − S_{∞}, the complex

U_{v}^{+}(V) = \{ \begin{matrix}C^{•}(G_{F_{v}}, V_{v}^{+}), & v ∈ S_{p} \\ 0, & v ∈ Σ − S_{p} \\ Cur^{•}(G_{F_{v}}, V_{v}) = C^{•}(G_{F_{v}}/I_{v}, V_{v}^{I_{v}}), & v ∈ S − Σ, \end{matrix}

where I_{v} ⊂ G_{F_{v}} is the inertia group. As in ([Ne 2], 12.5.9.1), define the Selmer complex of V associated to the local conditions Δ_{Σ}(V) = (U_{v}^{+}(V))_{v∈S−S_{∞}} as

C̃f^{•}(G_{F,S}, V; Δ_{Σ}(V)) = Cone (C^{•}(G_{F,S}, V) ⊕ ⨁_{v∈S−S_{∞}} U_{v}^{+}(V) ⟶ ⨁_{v∈S−S_{∞}} C^{•}(G_{F_{v}}, V)) [−1].

(4.2.2) Proposition. (1) For each v ∉ S_{∞} ∪ S_{p}, the complexes C^{•}(G_{F_{v}}, V) and Cur^{•}(G_{F_{v}}, V) are acyclic.

(2) Up to a canonical isomorphism, the image of C̃f^{•}(G_{F,S}, V; Δ_{Σ}(V)) in the derived category D_{ft}^{b}(L_{𝔭} − Mod) does not depend on Σ and S; denote it by R̃Γ_{f}(F, V) and its cohomology by H̃_{f}^{i}(F, V) (as L_{𝔭} is a field, R̃Γ_{f}(F, V) = ⨁_{i∈Z} H̃_{f}^{i}(F, V)[−i]).

(3) There is an exact sequence

0 ⟶ ⨁_{v∈S_{p}} H^{0}(F_{v}, V_{v}^{−}) ⟶ H̃_{f}^{1}(F, V) ⟶ H_{f}^{1}(F, V) ⟶ 0.

(4) If we put h_{f}^{1}(F, V) = dim_{L_{𝔭}} H_{f}^{1}(F, V), h̃_{f}^{1}(F, V) = dim_{L_{𝔭}} H̃_{f}^{1}(F, V), then

(−1)^{h_{f}^{1} (F,V)}/ ε(V) = (−1)^{_{h_{f}^{1}} (F,V)}/ ε̃(V).

Proof. (cf. [Ne 2, 12.5.9.2]) (1) The cohomology group H^{0}(F_{v}, V) = 0 vanishes by purity (1.4.4(5)), H^{2}(F_{v}, V) ⟶^{∼} H^{0}(F_{v}, V^{∗}(1))^{∗} ⟶^{∼} H^{0}(F_{v}, V)^{∗} = 0 by duality and H^{1}(F_{v}, V) = 0 by the local Euler characteristic formula ∑_{i=0}^{2}(−1)^{i} h^{i}(F_{v}, V) = 0. Finally, dim_{L_{𝔭}} H_{ur}^{1}(F_{v}, V) = h^{0}(F_{v}, V) = 0.

(2) Independence of Σ follows from (1), independence of S is a general fact ([Ne 2], Prop. 7.8.8).

(3) It follows from (1) and [Ne 2, Lemma 9.6.3] that there is an exact sequence

0 ⟶ H̃_{f}^{0}(F, V) ⟶ H^{0}(G_{F,S}, V) ⟶ ⨁_{v∈S_{p}} H^{0}(F_{v}, V_{v}^{−}) ⟶ H̃_{f}^{1}(F, V) ⟶ H ⟶ 0,

where

H = Ker (H^{1}(G_{F,S}, V) ⟶ ⨁_{v∈S−S_{∞}} H^{1}(F_{v}, V)/Im(H^{1}(U_{v}^{+}(V)))) .

As

Im(H^{1}(U_{v}^{+}(V))) = \{ \begin{matrix}0 = H_{f}^{1}(F_{v}, V), & v ∉ S_{p} \\ H_{f}^{1}(F_{v}, V), & v ∈ S_{p} \end{matrix}

by (1) and Proposition 3.3.2(2), respectively, we deduce that H = H_{f}^{1}(F, V). Finally, H^{0}(G_{F,S}, V) = 0 by purity.

(4) This is a consequence of (3) and (4.1.3.4).

5. p-Adic families of global p-adic Galois representations

(5.1) The general setup (5.1.1) Fix a number field F, a prime number p and a finite set S ⊃ S_{p} ∪ S_{∞} of primes of F. (5.1.2) Assume that we are given the following data.

(1) A complete local noetherian domain R of dimension dim(R) = 2, whose residue field is a finite extension of F_{p} and whose fraction field ℒ is of characteristic zero.

(2) An R-module of finite type T equipped with an R-linear continuous action of G_{F,S} (with respect to the pro-finite topology of T ). Set V = T ⊗_{R} ℒ.

(3) A skew-symmetric morphism of R[G_{F,S}]-modules

( , ) : T ⊗_{R} T ⟶ R(1) = R ⊗_{Z_{p}} Z_{p}(1)

inducing an isomorphism of ℒ [G_{F,S}]-modules

V ⟶^{∼} V^{∗}(1) = Hom_{ℒ} (V, ℒ)(1).

(4) For each v ∈ S_{p} an R[G_{F_{v}}]-submodule T_{v}^{+} ⊂ T_{v} such that the isomorphism V ⟶^{∼} V^{∗}(1) induces isomorphisms of ℒ [G_{F_{v}}]-modules

V_{v}^{±} ⟶^{∼} (V_{v}^{∓})^{∗}(1) = Hom_{ℒ} (V_{v}^{∓}, ℒ)(1),

where V_{v}^{+} = T_{v}^{+} ⊗_{R} ℒ, V_{v}^{−} = V_{v}/V_{v}^{+}.

(5) A prime ideal P ∈ Spec(R) of height ht(P) = 1, which does not contain p and such that R_{P} is a discrete valuation ring. Fix a prime element ϖ_{P} of R_{P}. The residue field κ(P) = R_{P} /ϖ_{P} R_{P} is a finite extension of Q_{p}. Define

T_{P} = T ⊗_{R} R_{P} ⊂ V, V = T_{P} /ϖ_{P} T_{P} ∈ Rep_{κ(P)}(G_{F,S})

and, for each v ∈ S_{p},

(T_{P})_{v}^{+} = T_{P} ∩ V_{v}^{+},  (T_{P})_{v}^{−} = T_{P} /(T_{P})_{v}^{+},  V_{v}^{+} = (T_{P})_{v}^{+}/ϖ_{P} (T_{P})_{v}^{+} ⊂ V_{v},  V_{v}^{−} = V_{v}/V_{v}^{+}

(V_{v}^{±} ∈ Rep_{κ(P)}(G_{F_{v}})).

(6) We assume that there exists u ∈ ℒ^{∗} such that u · ( , ) induces an isomorphism of R_{P} [G_{F,S}]-modules

T_{P} ⟶^{∼} T_{P}^{∗}(1) = Hom_{R_{P}} (T_{P} , R_{P})(1).

This implies that, for each v ∈ S_{p}, u · ( , ) induces an isomorphism of R_{P} [G_{F_{v}}]-modules (T_{P})_{v}^{±} ⟶^{∼} ((T_{P})_{v}^{∓})^{∗}(1). Reducing u · ( , ) modulo P, we obtain a non-degenerate skew-symmetric morphism of κ(P)[G_{F,S}]-modules V ⊗_{κ(P)} V ⟶ κ(P)(1) which induces, for each v ∈ S_{p}, isomorphisms V_{v}^{±} ⟶^{∼} (V_{v}^{∓})^{∗}(1) in Rep_{κ(P)}(G_{F_{v}}).

(7) We assume that, for each v ∈ S_{p}, the exact sequence

0 ⟶ V_{v}^{+} ⟶ V_{v} ⟶ V_{v}^{−} ⟶ 0

satisfies the Pančiškin condition: V_{v}^{±} ∈ Rep_{pst,κ(P)}(G_{F_{v}}) and D_{dR}^{0}(V_{v}^{+}) = 0 = D_{dR}(V_{v}^{−})/D_{dR}^{0}(V_{v}^{−}).

(8) We assume that, for each v ∉ S_{∞}, V_{v} is pure of weight −1 (in the sense of 1.4.5 and 3.2.3, respectively).

(9) We assume that, for each i ∈ Z, the integer

d^{i}(V) := dim_{κ(P)} (D_{dR}^{i}(V_{v})/D_{dR}^{i+1}(V_{v})) /[F_{v} : Q_{p}]

does not depend on v ∈ S_{p}; put

d^{−}(V) = ∑_{i<0} i d^{i}(V).

(5.1.3) This implies, in particular, that V satisfies the assumptions 4.1.2(1)-(4). (5.1.4) Fix v ∉ S_{p} ∪ S_{∞}. As Aut_{R}(T ) is a pro-finite group containing a pro-p open subgroup, there exists an open subgroup J of the inertia group I = I_{v} = Gal(\overline{F}_{v}/F_{v}^{ur}) such that J acts on T through the map J ↪ I ↠ I(p), where I(p) is the maximal pro-p-quotient of I (isomorphic to Z_{p}). Fixing a topological generator t of I(p) and an integer a ≥ 1 such that t^{a} lies in the image of J, then the set of eigenvalues of t^{a} acting on V is stable under the map λ ↦ λ^{Nv}, which implies that there exists an integer c ≥ 1 divisible by a such that t^{c} acts unipotently on V. Defining

N = \frac{1}{c} log ρ_{T} (t^{c}) ∈ End_{R}(T ) ⊗ Q

(where ρ_{T} : G_{K} ⟶ Aut_{R}(T ) denotes the action of G_{F_{v}} on T ) and (fixing a lift f̃ ∈ ν^{−1}(1) ⊂ W_{K} of f)

ρ_{T} (f̃^{n}u) := ρ_{T} (f̃^{n}u) exp(−bN) ∈ Aut_{R⊗Q}(T ⊗ Q) ⊂ Aut_{R_{P}} (T_{P}) (n ∈ Z, u ∈ I),

(where b ∈ Z_{p} is such that the image of u in I(p) is equal to t^{b}), the pair (ρ_{T} , N) defines an object T = (ρ_{T} , N) of Rep_{R_{P}} (′W_{F_{v}}) in the sense of 1.5.2, the isomorphism class of which is independent of the choice of f̃([De 1], 8.4.3). By construction, the special fibre of T is isomorphic to

T_{s} ⟶^{∼} W D(V_{v}) ∈ Rep_{κ(P)}(′W_{F_{v}}).

We define

\begin{aligned}W D(V_{v}) & := T_{η} = T ⊗_{R_{P}} ℒ ∈ Rep_{ℒ} (′W_{F_{v}}) \\ ε(V_{v}) & := ε(W D(V_{v})). \end{aligned}  (5.1.4.1)

If we choose another generator of I(p), then N is multiplied by a scalar λ ∈ Z_{p}^{∗}, which does not change the isomorphism class of W D(V_{v}) ([De 1], 8.4.3).

(5.2) Selmer complexes and extended Selmer groups (5.2.1) We equip each R-module Y = T , T_{v}^{+}, T_{v}^{I_{v}} with the pro-finite topology and we denote by C^{•}(G, Y) the corresponding complex of continuous cochains (for G = G_{F,S}, G_{F_{v}}, G_{F_{v}}/I_{v}, respectively). For R′ = R_{P} , ℒ, define C^{•}(G, Y ⊗_{R} R′) = C^{•}(G, Y) ⊗_{R} R′. As in 4.2.1, fix a set S_{p} ⊂ Σ ⊂ S and define, for X = T_{P} , V, R_{X} = R_{P} , ℒ and each v ∈ S − S_{∞}, complexes of R_{X}-modules

U_{v}^{+}(X) = \{ \begin{matrix}C^{•}(G_{F_{v}}, X_{v}^{+}), & v ∈ S_{p} \\ 0, & v ∈ Σ − S_{p} \\ Cur^{•}(G_{F_{v}}, X) = C^{•}(G_{F_{v}}/I_{v}, X^{I_{v}}), & v ∈ S − Σ, \end{matrix}

and

C̃f^{•}(G_{F,S}, X; Δ_{Σ}(X)) = Cone (C^{•}(G_{F,S}, X) ⊕ ⨁_{v∈S−S_{∞}} U_{v}^{+}(X) ⟶ ⨁_{v∈S−S_{∞}} C^{•}(G_{F_{v}}, X)) [−1].

(5.2.2) Proposition. (1) For each X = T_{P} , V and each v ∉ S_{∞} ∪ S_{p}, the complexes C^{•}(G_{F_{v}}, X) and Cur^{•}(G_{F_{v}}, X) are acyclic.

(2) Up to a canonical isomorphism, the image of C̃f^{•}(G_{F,S}, X; Δ_{Σ}(X)) in D_{ft}^{b}(R_{X} − Mod) does not depend on Σ and S; denote it by R̃Γ_{f}(F, X) and its cohomology by H̃_{f}^{i}(F, X) (as ℒ is a field, R̃Γ_{f}(F, V) = ⨁_{i∈Z} H̃_{f}^{i}(F, V)[−i]).

(3) There is an exact triangle in D_{ft}^{b}(R_{P} − Mod)

R̃Γ_{f}(F, T_{P})⟶^{ϖ_{P}}R̃Γ_{f}(F, T_{P}) ⟶ R̃Γ_{f}(F, V) ⟶ R̃Γ_{f}(F, T_{P})[1]

giving rise to exact sequences

0 ⟶ H̃_{f}^{i}(F, T_{P})/ϖ_{P} H̃_{f}^{i}(F, T_{P}) ⟶ H̃_{f}^{i}(F, V) ⟶ H̃_{f}^{i+1}(F, T_{P})[ϖ_{P}] ⟶ 0,

and an isomorphism R̃Γ_{f}(F, T_{P})⊗^{L}_{R_{P}} ℒ ⟶^{∼} R̃Γ_{f}(F, V) in D_{ft}^{b}(ℒ − Mod ).

(4) There exists a skew-symmetric isomorphism in D_{ft}^{b}(R_{P} − Mod)

R̃Γ_{f}(F, T_{P}) ⟶^{∼} RHom_{R_{P}} (R̃Γ_{f}(F, T_{P}), R_{P})[−3]

inducing a skew-symmetric non-degenerate pairing

H̃_{f}^{2}(F, T_{P})_{R_{P} −tors} × H̃_{f}^{2}(F, T_{P})_{R_{P} −tors} ⟶ ℒ /R_{P} .

(5) There exists an R_{P} -module Z of finite length such that H̃_{f}^{2}(F, T_{P})_{R_{P} −tors} ⟶^{∼} Z ⊕ Z.

(6) H̃_{f}^{1}(F, T_{P}) is a free R_{P} -module of rank h̃_{f}^{1}(F, V) := dim_{ℒ} H̃_{f}^{1}(F, V).

(7) h̃_{f}^{1}(F, V) ≡ h̃_{f}^{1}(F, V) (mod 2).

Proof. (cf. [Ne 2, 12.7.13.4]) (1) It is enough to prove the statement for X = T_{P}. By ([Ne 2], Prop. 3.4.2 and 3.4.4), there is an exact sequence of complexes

0 ⟶ C^{•}(G_{F_{v}}, T_{P})⟶^{ϖ_{P}}C^{•}(G_{F_{v}}, T_{P}) ⟶ C^{•}(G_{F_{v}}, V) ⟶ 0,

which induces injections

H^{i}(G_{F_{v}}, T_{P})/ϖ_{P} H^{i}(G_{F_{v}}, T_{P}) ↪ H^{i}(F_{v}, V).

As H^{i}(F_{v}, V) = 0 by Proposition 4.2.2(1), and H^{i}(G_{F_{v}}, T_{P}) = H^{i}(G_{F_{v}}, T ) ⊗_{R} R_{P} is an R_{P} -module of finite type (by [Ne 2], Prop. 4.2.3), it follows that H^{i}(G_{F_{v}}, T_{P}) = 0. Finally, the unramified cohomology H_{ur}^{1} = H_{ur}^{1}(G_{F_{v}}, T_{P}) = T_{P}^{I_{v}}/(f_{v} − 1)T_{P}^{I_{v}} is an R_{P} -module of finite type and H_{ur}^{1}/ϖ_{P} H_{ur}^{1} is a subquotient of V^{I_{v}}/ϖ_{P} V^{I_{v}} = H_{ur}^{1}(G_{F_{v}}, V) = 0; thus H_{ur}^{1} = 0.

(2) This follows from (1), as in the proof of 4.2.2(2).

(3) According to (2), we can take Σ = S, in which case the exact triangle in question follows from the exact sequences

0 ⟶ C^{•}(G, T_{P})⟶^{ϖ_{P}}C^{•}(G, T_{P}) ⟶ C^{•}(G, V) ⟶ 0 (G = G_{F,S}, G_{F_{v}}, v ∈ S − S_{∞}).

The isomorphism R̃Γ_{f}(F, T_{P})⊗^{L}_{R_{P}} ℒ ⟶^{∼} R̃Γ_{f}(F, V) is a direct consequence of the definitions.

(4) Take again Σ = S. According to a localized version of ([Ne 2], 7.8.4.4), there exists an exact triangle in D_{ft}^{b}(R_{P} − Mod)

R̃Γ_{f}(F, T_{P})⟶^{γ}RHom_{R_{P}} (R̃Γ_{f}(F, T_{P}), R_{P})[−3] ⟶ ⨁_{v∈S−S_{∞}} Err_{v},

in which the error terms Err_{v} vanish for v ∈ S_{p} (as (T_{P})^{±} ⟶^{∼} ((T_{P})^{∓})^{∗}(1)), as well as for v ∉ S_{p} (by (1) and [Ne 2], Prop. 6.7.6(iv)). The map γ (which is an isomorphism, by the previous discussion) is skew-symmetric, by ([Ne 2], Prop. 6.6.2 and 7.7.3). The skew-symmetric non-degenerate pairing

H̃_{f}^{2}(F, T_{P})_{R_{P} −tors} × H̃_{f}^{2}(F, T_{P})_{R_{P} −tors} ⟶ ℒ /R_{P} .

is constructed from γ in ([Ne 2], Prop. 10.2.5).

(5) This follows from (4) and the structure theory of symplectic modules of finite length over discrete valuation rings (note that 2 is invertible in R_{P}).

(6) It is enough to show that H̃_{f}^{1}(F, T_{P}) has no R_{P} -torsion, which si a consequence of the exact sequence from (3) (for i = 0).

(7) In the exact sequence from (3) for i = 1, the term on the left (resp., on the right), is a κ(P)-vector space of dimension h̃_{f}^{1}(F, V), by (6) (resp., of even dimension, by (5)); thus the dimension of the middle term (= h̃_{f}^{1}(F, V)) has the same parity as h̃_{f}^{1}(F, V).

(5.3) The parity conjecture in p-adic families

(5.3.1) Theorem. Under the assumptions 5.1.2(1)-(9), the quantity

(−1)^{h_{f}^{1} (F,V)}/ ε(V) = (−1)^{_{h_{f}^{1}} (F,V)}/ ε̃(V) = (−1)^{_{h_{f}^{1}} (F,V)} ∏_{v∈S_{p}}(det V_{v}^{+})(−1) ∏_{v∉S_{p}∪S_{∞}} ε(V_{v})

depends only on V and V_{v}^{+} (v ∈ S_{p}).

Proof. We combine the equalities

(−1)^{h_{f}^{1} (F,V)}/ ε(V) = (−1)^{_{h_{f}^{1}} (F,V)}/ ε̃(V)  (Prop. 4.2.2(4))

(−1)^{_{h_{f}^{1}} (F,V)} = (−1)^{_{h_{f}^{1}} (F,V)}  (Prop. 5.2.2(7))

ε̃(V) = ∏_{v∈S_{p}∪S_{∞}} ε̃(V_{v}) ∏_{v∉S_{p}∪S_{∞}} ε(V_{v}) = ∏_{v∈S_{p}}(det V_{v}^{+})(−1) ∏_{v∉S_{∞}∪S_{p}} ε(V_{v})  (by 4.1.3.7)

∀v ∉ S_{∞} ∪ S_{p}  ε(V_{v}) = ε(V_{v})  (Prop. 2.2.4)

∀v ∈ S_{p} (det V_{v}^{+})(−1) = (det V_{v}^{+})(−1)

(both sides are equal to ±1, and the L.H.S. is the reduction of the R.H.S. modulo P).

(5.3.2) Corollary. Under the assumptions 5.1.2(1)-(4), if P, P′ ∈ Spec(R) are prime ideals satisfying 5.1.2(5)-(9), then the Galois representations V = T_{P} /P T_{P} and V′ = T_{P′}/P′T_{P′} satisfy

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

(5.3.3) Open questions. It would be of interest to generalize Corollary 5.3.2 to self-dual families of Galois representations that do not satisfy the Pančiškin condition. Is it true, in general, that

(−1)^{[F_{v}:Q_{p}] d^{−}(V)} ε(W D(V_{v})^{N−ss}) (v ∈ S_{p})

depends only on V_{v}, and that

(−1)^{h_{f}^{1} (F,V)} ∏_{v∈S_{p}} \frac{ε(W D(V_{v}))}{ε(W D(V_{v})^{N−ss})}

depends only on V? (5.3.4) Example (dihedral Iwasawa theory). Assume that F_{0} ⊂ F_{∞} are Galois extension of F such that [F_{0} : F] = 2, Γ = Gal(F_{∞}/F_{0}) ⟶^{∼} Z_{p} and Γ^{+} = Gal(F_{∞}/F) = Γ ⋊ {1, τ} is dihedral:

τ ∈ Γ^{+} − Γ,  τ^{2} = 1,  ∀g ∈ Γ τgτ^{−1} = g^{−1}.

Let V ∈ Rep_{L_{𝔭}}(G_{F,S}) be a Galois representation satisfying 4.1.2(1)-(4); fix a G_{F,S}-stable O_{𝔭}-lattice T ⊂ V (O_{𝔭} = O_{L_{𝔭}}) such that the pairing ( , )_{V} : V × V ⟶ L_{𝔭}(1) induced by j maps T × T into O_{𝔭}(1). After enlarging S if necessary, we can assume that S contains all primes that ramify in F_{0}/F; then F_{∞} ⊂ F_{S}. We define the following data of the type considered in 5.1.2:

(1) Let R = O_{𝔭}[[Γ]] be the Iwasawa algebra of Γ (isomorphic to the power series ring O_{𝔭}[[X]]). The Iwasawa algebra of Γ^{+} is a free (both left and right) R-module of rank 2:

O_{𝔭}[[Γ^{+}]] = R ⊕ Rτ = R ⊕ τR.

Denote by ι the standard O_{𝔭}-linear involution on O_{𝔭}[[Γ^{+}]] (ι(σ) = σ^{−1} for all σ ∈ Γ^{+}).

(2) Let T = T ⊗_{O_{𝔭}} O_{𝔭}[[Γ^{+}]], considered as a left R[G_{F,S}]-module with the action given by

r(x ⊗ a) = x ⊗ ra,  g(x ⊗ a) = g(x) ⊗ a(\overline{g})^{−1}  (r ∈ R, x ∈ T, a ∈ O_{𝔭}[[Γ^{+}]]),

where we have denoted by \overline{g} the image of g ∈ G_{F,S} in Γ^{+} (cf., [Ne 2], 10.3.5.3).

(3) As in ([Ne 2], 10.3.5.10), the formula

(x ⊗ (a_{1} + τa_{2}), y ⊗ (b_{1} + τb_{2})) = (x, y)_{V} (a_{1}ι(b_{2}) + ι(a_{2})b_{1})

defines a skew-symmetric R-bilinear pairing ( , ) : T × T ⟶ R(1), which induces an isomorphism

T ⊗ Q ⟶^{∼} Hom_{R}(T , R(1)) ⊗ Q

(hence satisfies 5.1.2(3)).

(4) For each v ∈ S_{p}, define T_{v}^{+} = T_{v}^{+} ⊗_{O_{𝔭}} O_{𝔭}[[Γ^{+}]] (where T_{v}^{+} = T ∩ V_{v}^{+}).

(5) Let β : Γ ⟶ L_{𝔭}(β)^{∗} be a homomorphism with finite image (where L_{𝔭}(β) is a field generated over L_{𝔭} by the values of β); then P = Ker(β : R ⟶ L_{𝔭}(β)) ∈ Spec(R) is as in 5.1.2(5), with κ(P) = L_{𝔭}(β). It follows from ([Ne 2], Lemma 10.3.5.4) that

T_{P} /ϖ_{P} T_{P} = Ind_{G_{F_{0},S}}^{G_{F,S}}(V ⊗ β),

where we have denoted by V ⊗ β ∈ Rep_{L_{𝔭}(β)}(G_{F_{0},S}) the G_{F_{0},S}-module V ⊗_{L_{𝔭}} L_{𝔭}(β) on which g ∈ G_{F_{0},S} acts by g⊗β(\overline{g}), where \overline{g} is the image of g in Γ. The discussion in ([Ne 2], 10.3.5.10) implies that 5.1.2(6) holds with u = 1. The conditions 5.1.2(7)-(9) for T_{P} /ϖ_{P} T_{P} follow from the corresponding conditions 4.1.2(2)-(4) for V.

(5.3.5) In the situation of 5.3.4, putting F_{β} = F_{∞}^{Ker(β)} and, for each L_{𝔭}[Γ]-module M,

M^{(β)} = {x ∈ M ⊗_{L_{𝔭}} L_{𝔭}(β) | ∀σ ∈ Γ σ(x) = β(σ)x},

then we have

H_{f}^{1}(F, T_{P} /ϖ_{P} T_{P}) = H_{f}^{1}(F_{0}, V ⊗ β) = (H_{f}^{1}(F_{β}, V) ⊗ β)^{Gal(F_{β}/F_{0})} = H_{f}^{1}(F_{β}, V)^{(β^{−1})},

and the action of τ induces an isomorphism of L_{𝔭}(β)-vector spaces

τ : H_{f}^{1}(F_{β}, V)^{(β^{−1})} ⟶^{∼} H_{f}^{1}(F_{β}, V)^{(β)}.

Applying Corollary 5.3.2, we obtain, for any pair of characters of finite order β, β′ : Γ ⟶ \overline{L}_{𝔭}^{∗}, that

(−1)^{h_{f}^{1} (F_{0},V ⊗β)}/ ε(F_{0}, V ⊗ β) = (−1)^{h_{f}^{1} (F_{0},V ⊗β′)}/ ε(F_{0}, V ⊗ β′).  (5.3.5.1)

In this special case one can prove Proposition 2.2.4 directly (at least if p ≠ 2) by using (2.1.2.7). It would be of interest to generalize (5.3.5.1) to more general dihedral characters, as in [Ma-Ru].

References

[Bl-Ka] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift I,

Progress in Mathematics 86, Birkhäuser, Boston, Basel, Berlin, 1990, pp. 333–400. [De 1] P. Deligne, Les constantes des équations fonctionnelles des fonctions L, in: Modular functions of one

variable II (Antwerp, 1972), Lect. Notes in Math. 349, Springer, Berlin, 1973, pp. 501–597. [De 2] P. Deligne, Formes modulaires et représentations de GL(2), in: Modular functions of one variable II

(Antwerp, 1972), Lect. Notes in Math. 349, Springer, Berlin, 1973, pp. 55–105. [De 3] P. Deligne, Valeurs de fonctions L et périodes d’intégrales, in: Automorphic forms, representations and

L-functions (Corvallis, 1977), Proc. Symposia Pure Math. 33/II, American Math. Society, Providence, Rhode Island, 1979, pp. 313–346.

[Fo] J.-M. Fontaine, Représentations ℓ-adiques potentiellement semi-stables, in: Périodes p-adiques (Bures-

sur-Yvette, 1988), Astérisque 223, Soc. Math. de France, Paris, 1994, pp. 321–347. [Fo-PR] J.-M. Fontaine, B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et

valeurs de fonctions L, in: Motives (Seattle, 1991), Proc. Symposia in Pure Math. 55/I, American Math. Society, Providence, Rhode Island, 1994, pp. 599–706.

[Ma-Ru] B. Mazur, K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Ann. of

Math., to appear. [Ne 1] J. Nekovář, On p-adic height pairings, in: Séminaire de Théorie des Nombres de Paris 1990/91, Progress

in Math. 108, (S. David, ed.), Birkhäuser, Boston, 1993, pp. 127–202. [Ne 2] J. Nekovář, Selmer complexes, Astérisque 310, Soc. Math. de France, Paris, 2006.

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