On the Parity of Ranks of Selmer Groups III

Jan Nekovář

Received: May 4, 2006 Revised: May 10, 2007

Communicated by Ulf Rehmann

Abstract. We show that the parity conjecture for Selmer groups is invariant under deformation in p-adic families of self-dual pure Galois representations satisfying Pančiškin’s condition at all primes above p.

2000 Mathematics Subject Classification: 11G40 11R23

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 finite-dimensional 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} (self-dual, 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 Lfunction 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 f̃ and 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., ωsymplectic) 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., skew-symmetric). 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), a(β) = dim Δ′_{f} + dim Δ′′_{f}− dim Δ_{f}, ε(β) = ε(Δ, ψ, μ)/ε(Δ′, ψ, μ) ε(Δ′′, ψ, μ), 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)) = 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 (nonhomogeneous) 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 nondegenerate 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}}).

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?

Documenta Mathematica 12 (2007) 243–274

(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) Rmodule 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].

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[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.

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[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 Nom-

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[Ne 2] J. Nekovář, Selmer complexes, Astérisque 310, Soc. Math. de France, Paris, 2006.

Jan Nekovář 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