Infinitesimal Deformations and the ℓ-Invariant

To Andrei Alexandrovich Suslin, for his 60th birthday

Denis Benois

Received: November 11, 2009 Revised: January 22, 2010

Abstract. We give a formula for the generalized Greenberg’s ℓ-invariant which was constructed in [Ben2] in terms of derivatives of eigenvalues of Frobenius.

2000 Mathematics Subject Classification: 11R23, 11F80, 11S25, 11G40, 14F30 Keywords and Phrases: p-adic representation, (φ, Γ)-module, L-function

Introduction

0.1. Let M be a pure motive over ℚ with coefficients in a number field E. Assume that the L-function L(M, s) is well defined. Fixinig an embedding ι : E ↪ ℂ we can consider it as a complex-valued Dirichlet series L(M, s) =

∑^{∞} ; n=0a_{n}n^{−s} which converges for s ≫ 0 and is expected to admit a meromorphic

continuation to ℂ with a functional equation of the form

Γ(M, s) L(M, s) = ε(M, s) Γ(M^{∗}(1), −s) L(M^{∗}(1), −s)

where Γ(M, s) is the product of some Γ-factors and the ε-factor has the form ε(M, s) = ab^{s}. Assume that M is critical and that L(M, 0) ≠ 0. Fix a finite place λ|p of E and assume that the λ-adic realization M_{λ} of M is semistable in the sense of Fontaine [Fo3]. The (φ, N)-module D_{st}(M_{λ}) associated to M_{λ} is a finite dimensional E_{λ}-vector space equipped with an exhaustive decreasing filtration Fil^{i}D_{st}(M_{λ}), a E_{λ}-linear bijective frobenius φ : D_{st}(M_{λ}) ⟶ D_{st}(M_{λ}) and a nilpotent monodromy operator N such that N φ = p φ N. We say that a (φ, N)-submodule D of D_{st}(M_{λ}) is regular if

D_{st}(M_{λ}) = D ⊕ Fil^{0}D_{st}(M_{λ})

as E_{λ}-vector spaces. The theory of Perrin-Riou [PR] suggests that to any regular D one can associate a p-adic L-function L_{p}(M, D, s) interpolating rational parts of special values of L(M, s). In particular, the interpolation formula at s = 0 should have the form

L_{p}(M, D, 0) = E(M, D) \frac{L(M, 0)}{Ω_{∞}(M)}

where Ω_{∞}(M) is the Deligne period of M and E(M, D) is a certain product of Euler-like factors. Therefore one can expect that L_{p}(M, D, 0) = 0 if and only if E(M, D) = 0 and in this case one says that L_{p}(M, D, s) has a trivial zero at s = 0.

0.2. According to the conjectures of Bloch and Kato [BK], the E_{λ}-adic representation M_{λ} should have the following properties:

C1) The Selmer groups H_{f}^{1}(M_{λ}) and H_{f}^{1}(M_{λ}^{∗}(1)) are zero. C2) H^{0}(M_{λ}) = H^{0}(M_{λ}^{∗}(1)) = 0 where we write H^{∗} for the global Galois cohomology. Moreover one expects that C3) φ : D_{st}(M_{λ}) ⟶ D_{st}(M_{λ}) is semisimple (semisimplicity conjecture). We also make the following assumption which is a direct generalization of the hypothesis U) from [G]. C4) The (φ, Γ)-module D_{rig}^{†}(M_{λ}) has no saturated subquotients of the form U_{m,n} where U_{m,n} is the unique crystalline (φ, Γ)-module sitting in a non split exact sequence

0 ⟶ R_{L}(|x|x^{m}) ⟶ U_{m,n} ⟶ R_{L}(x^{−n}) ⟶ 0, L = E_{λ}

(see §1 for unexplained notations).

In [Ben2], we extended the theory of Greenberg [G] to L-adic pseudo geometric representations which are semistable at p and satisfy C1-4). Namely to any regular D ⊂ D_{st}(V) of a reasonably behaved representation V we associated an integer e ⩾ 0 and an element L(V, D) ∈ L which can be seen as a vast generalization of the L-invariants constructed in [Mr] and [G]. If V = M_{λ} we set L(M, D) = L(M_{λ}, D). A natural formulation of the trivial zero conjecture states as follows:

Conjecture. L_{p}(M, D, s) has a zero of order e at s = 0 and

(0.1)  lim_{s→0} \frac{L_{p}(M, D, s)}{s^{e}} = E^{+}(M, D) L(M^{∗}(1), D^{∗}) \frac{L(M, 0)}{Ω_{∞}(M)},

The ℓ-Invariant 7

where E^{+}(M, D) is the subproduct of E(M, D) obtained by ”excluding zero factors” and D^{∗} = Hom(D_{st}(V)/D, D_{st}(L(1))) is the dual regular module (see [Ben2] for more details). We refer to this statement as Greenberg’s conjecture because if M_{λ} is ordinary at p it coincides with the conjecture formulated in [G], p.166. Remark that if M_{λ} is crystalline at p, Greenberg’s conjecture is compatible with Perrin-Riou’s theory of p-adic L-functions [Ben3].

0.3. Consider the motive M_{f} attached to a normalized newform f = ∑^{∞} ; n=1a_{n}q^{n}

of weight 2k on Γ_{0}(Np) with (N, p) = 1. The complex L-function of M_{f} is

L(f, s) = ∑^{∞} ; n=1a_{n}n^{−s}. The twisted motive M_{f}(k) is critical. The eigenvalues of φ

acting on D_{st}(M_{f,λ}(k)) are α = p^{−k}a_{p} and β = p^{1−k}a_{p} with v_{p}(a_{p}) = k−1. The unique regular submodule of D_{st}(M_{f,k}(k)) is D = E_{λ}d where φ(d) = α d and L_{p}(M_{f}(k), D, s) = L_{p}(f, s + k) where L_{p}(f, s) is the classical p-adic L-function associated to a_{p} via the theory of modular symbols [Mn], [AV]. If a_{p} = p^{k−1}, the function L_{p}(f, s) vanishes at s = k. In this case several constructions of the L-invariant based on different ideas were proposed (see [Co1], [Tm], [Mr], [O], [Br]). Thanks to the work of many people it is known that they are all equal and we refer to [Cz3] and [BDI] for further information. As M_{f}(k) is self-dual (i.e. M_{f}(k) ≃ M_{f}^{∗}(1−k)) one has L(M_{f}^{∗}(1−k), D^{∗}) = L(M_{f}(k), D) (see also section 0.4 below). Moreover it is not difficult to prove that L(M_{f}(k), D) coincides with the L-invariant of Fontaine-Mazur L_{FM}(f) [Mr] ([Ben2], Proposition 2.3.7) and (0.1) takes the form of the Mazur-Tate-Teitelbaum conjecture

L′_{p}(f, k) = L(f) \frac{L(f, k)}{Ω_{∞}(f)}

where we write L(f) for an unspecified L-invariant and Ω_{∞}(f) for the Shimura period of f [MTT]. This conjecture was first proved by Greenberg and Stevens in the weight two case [GS1] [GS2]. In the unpublished note [St], Stevens generalized this approach to the higher weights. Other proofs were found by Kato, Kurihara and Tsuji (unpublished but see [Cz2]), Orton [O], Emerton [E] and by Bertolini, Darmon and Iovita [BDI]. The approach of Greenberg and Stevens is based on the study of families of modular forms and their padic L-functions. Namely, Hida (in the ordinary case) and Coleman [Co1] (in

general) constructed an analytic family f_{x} = ∑^{∞} ; n=1a_{n}(x)q^{n} of p-adic modular

forms for x ∈ ℂ_{p} passing through f with f = f_{2k}. Next, Panchishkin [Pa] and independently Stevens (unpublished) constructed a two-variable p-adic Lfunction L-function L_{p}(x, s) satisfying the following properties:

• L_{p}(2k, s) = L_{p}(f, s).

• L_{p}(x, x − s) = −⟨N⟩^{s−x}L_{p}(x, s).

• L_{p}(x, k) = (1 − p^{k−1}a_{p}(x)^{−1}) L^{∗}(x) where L_{p}^{∗}(x) is a p-adic analytic function such that L_{p}^{∗}(2k) = L(f, k)/Ω_{∞}(f). From these properties it follows easily that

L′_{p}(f, k) = −2 d log a_{p}(2k) \frac{L(f, k)}{Ω_{∞}(f)},

where d log a_{p}(x) = a_{p}(x)^{−1} \frac{da_{p}(x)}{dx}. Thus the Mazur-Tate-Teitelbaum conjecture is equivalent to the assertion that

(0.2)  L(f) = −2 d log a_{p}(2k).

This formula was first proved for weight two by Greenberg and Stevens. In the higher weight case several proofs of (0.2) have been proposed:

1. By Stevens [St], working with Coleman’s L-invariant L_{C}(f) defined in [Co1].

2. By Colmez [Cz5], working with the Fontaine-Mazur’s L-invariant L_{FM}(f) defined in [Mr].

3. By Colmez [Cz6], working with Breuil’s L-invariant L_{Br}(f) defined in [Br].

4. By Bertolini, Darmon and Iovita [BDI], working with Teitelbaum’s Linvariant L_{T}(f) [Tm] and Orton’s L-invariant L_{O}(f) [O].

0.4. In this paper, working with the L-invariant defined in [Ben2] we generalize (0.2) to some infinitesimal deformations of pseudo geometric representations. Our result is purely algebraic and is a direct generalization of Theorem 2.3.4 of [GS2] using the cohomology of (φ, Γ)-modules instead Galois cohomology. Let V be a pseudo-geometric representation with coefficients in L/ℚ_{p} which satisfies C1-4). Fix a regular submodule D. In view of (0.1) it is convenient to set

ℓ(V, D) = L(V^{∗}(1), D^{∗}).

Suppose that e = 1. Conjecturally this means that the p-adic L-function has a simple trivial zero. Then either D^{φ=p^{−1}} or (D^{∗})^{φ=p^{−1}} has dimension 1 over L. To fix ideas, assume that dim_{L} D^{φ=p^{−1}} = 1. Otherwise, as one expects a functional equation relating L_{p}(M, D, s) and L_{p}(M^{∗}(1), D^{∗}, −s) one can consider V^{∗}(1) and D^{∗} instead V and D. We distinguish two cases. In each case one can express ℓ(V, D) directly in terms of V and D.

• The crystalline case: D^{φ=p^{−1}} ∩ N (D_{st}(V)^{φ=1}) = {0}. Let D_{rig}^{†}(V) be the (φ, Γ)-module over the Robba ring R_{L} associated to V [Ber1], [Cz1]. Set D_{−1} = (1 − p^{−1}φ^{−1})D and D_{0} = D. The two step filtration D_{−1} ⊂ D_{0} ⊂ D_{st}(V) induces a filtration

F_{−1}D_{rig}^{†}(V) ⊂ F_{0}D_{rig}^{†}(V) ⊂ D_{rig}^{†}(V)

such that gr_{0}D_{rig}^{†}(V) ≃ R_{L}(δ) is the (φ, Γ)-module of rank 1 associated to a character δ : ℚ_{p}^{∗} ⟶ L^{∗} of the form δ(x) = |x|x^{m} with m ⩾ 1. The cohomology of (φ, Γ)-modules of rank 1 is studied in details in [Cz4]. Let η : ℚ_{p}^{∗} ⟶ L^{∗} be a continuous character. Colmez proved that H^{1}(R_{L}(η)) is a one dimensional L-vector space except for η(x) = |x|x^{m} with m ⩾ 1 and η(x) = x^{−n} with n ⩽ 0. In the exceptional cases H^{1}(R_{L}(η)) has dimension 2 and can be canonically decomposed into direct sum of one dimensional subspaces

(0.3) H^{1}(R_{L}(η)) ≃ H_{f}^{1}(R_{L}(η)) ⊕ H_{c}^{1}(R_{L}(η)), η(x) = |x|x^{m} or η(x) = x^{−n}

([Ben2], Theorem 1.5.7). The condition C1) implies that

(0.4)  H^{1}(V) ≃ ⨁_{l∈S} \frac{H^{1}(ℚ_{l}, V)}{H_{f}^{1}(ℚ_{l}, V)}

for a finite set of primes S. This isomorphism defines a one dimensional subspace H^{1}(D, V) of H^{1}(V) together with an injective localisation map κ_{D} : H^{1}(D, V) ⟶ H^{1}(R_{L}(δ)). Then ℓ(V, D) is the slope of Im(κ_{D}) with respect to the decomposition of H^{1}(R_{L}(δ)) into direct sum (0.3). Let

0 ⟶ V ⟶ V_{x} ⟶ L ⟶ 0

be an extension in the category of global Galois representations such that cl(x) ∈ H^{1}(D, V) is non zero. We equip D_{rig}^{†}(V_{x}) with a canonical filtration

{0} ⊂ F_{−1}D_{rig}^{†}(V_{x}) ⊂ F_{0}D_{rig}^{†}(V_{x}) ⊂ F_{1}D_{rig}^{†}(V_{x}) ⊂ D_{rig}^{†}(V_{x})

such that F_{i}D_{rig}^{†}(V_{x}) = F_{i}D_{rig}^{†}(V) for i = −1, 0 and gr_{1}D_{rig}^{†}(V_{x}) ≃ R_{L}. Let V_{A,x} be an infinitesimal deformation of V_{x} over A = L[T]/(T^{2}) endowed with a filtration F_{i}D_{rig}^{†}(V_{A,x}) such that F_{i}D_{rig}^{†}(V) = F_{i}D_{rig}^{†}(V_{A,x}) ⊗_{A} L. Write

gr_{0}D_{rig}^{†}(V_{A,x}) ≃ R_{A}(δ_{A,x}), gr_{1}D_{rig}^{†}(V_{A,x}) ≃ R_{A}(ψ_{A,x})

with δ_{A,x}, ψ_{A,x} : ℚ_{p}^{∗} ⟶ A^{∗}.

Theorem 1. Assume that \frac{d(δ_{A,x}ψ_{A,x}^{−1})(u)}{dT}|_{T =0} ≠ 0 for u ≡ 1 (mod p^{2}). Then

ℓ(V, D) = − log(u) \frac{d log(δ_{A,x}ψ_{A,x}^{−1})(p)}{d log(δ_{A,x}ψ_{A,x}^{−1})(u)}|_{T =0}

(note that the right hand side does not depend on the choice of u).

• The semistable case: D^{φ=p^{−1}} ⊂ N (D_{st}(V)^{φ=1}) . Set D_{−1} = (1 − p^{−1}φ^{−1})D, D_{0} = D and D_{1} = N^{−1}(D^{φ=p^{−1}}) ∩ D_{st}(V)^{φ=1}. The filtration

D_{−1} ⊂ D_{0} ⊂ D_{1} ⊂ D_{st}(V)

induces a filtration

F_{−1}D_{rig}^{†}(V) ⊂ F_{0}D_{rig}^{†}(V) ⊂ F_{1}D_{rig}^{†}(V) ⊂ D_{rig}^{†}(V)

Then gr_{0}D_{rig}^{†}(V) ≃ R_{L}(δ) and gr_{1}D_{rig}^{†}(V) ≃ R_{L}(ψ) where the characters δ and ψ are such that δ(x) = |x|x^{m} and ψ(x) = x^{−n} for some m ⩾ 1 and n ⩾ 0. Set M = F_{1}D_{rig}^{†}(V)/F_{−1}D_{rig}^{†}(V) and consider the map κ_{D} : H^{1}(M) ⟶ H^{1}(R_{L}(ψ)) induced by the projection M ⟶ R_{L}(ψ). The image of κ_{D} is a one dimensional L-subspace of H^{1}(R_{L}(ψ)) and ℓ(V, D) is the slope of Im(κ_{D}) with respect to (0.3). Assume that V_{A} is an infinitesimal deformation of V equipped with a filtration F_{i}D_{rig}^{†}(V_{A}) such that F_{i}D_{rig}^{†}(V) = F_{i}D_{rig}^{†}(V_{A}) ⊗_{A} L. Write gr_{0}D_{rig}^{†}(V_{A}) ≃ R_{A}(δ_{A}) and gr_{1}D_{rig}^{†}(V_{A}) ≃ R_{A}(ψ_{A}).

Theorem 2. Assume that

(0.5)  \frac{d(δ_{A}ψ_{A}^{−1})(u)}{dT}|_{T =0} ≠ 0 for u ≡ 1 (mod p^{2}).

Then

ℓ(V, D) = − log(u) \frac{d log(δ_{A}ψ_{A}^{−1}) (p)}{d log(δ_{A}ψ_{A}^{−1})(u)}|_{T =0}.

Remark that in the semistable case ℓ(V, D) = L(V, D).

For classical modular forms the existence of deformations having the above properties follows from the theory of Coleman-Mazur [CM] together with deep results of Saito and Kisin [Sa], [Ki]. Applying Theorem 2 to the representation M_{f,λ}(k) we obtain a new proof of (0.2) with the Fontaine-Mazur L-invariant. Remark that the local parameter T corresponds to the weight of a p-adic modular form and (0.5) holds automatically. In the general case the existence of deformations satisfying the above conditions should follow from properties of eigenvarieties of reductive groups [BC].

The formulations of Theorems 1 and 2 look very similar and the proof is essentially the same in the both cases. The main difference is that in the crystalline case the ℓ-invariant is global and contains information about the localisation map H^{1}(V) ⟶ H^{1}(ℚ_{p}, V). In the proof of Theorem 1 we consider V_{x} as a representation of the local Galois group but the construction of V_{x} depends on the isomorphism (0.4). In the semistable case the definition of ℓ(V, D) is purely local and the hypothesis C1-2) can be omitted. However C1-2) are essential for the formulation of Greenberg conjecture because (0.1) is meaningless if L(M, 0) = 0. One can compare our results with Hida’s paper [Hi] where the case of ordinary representations over totally real ground field is studued.

Here goes the organization of this paper. The §1 contains some background material. In section 1.1 we review the theory of (φ, Γ)-modules and in section 1.2 recall the definition of the ℓ-invariant following [Ben2]. The crystalline and semistable cases of trivial zeros are treated in §2 and §3 respectively. I would like to thank Pierre Parent for several very valuable discussions which helped me with the formulation of Theorem 1 and the referee for pointing out several inaccuracies in the first version of this paper.

It is a great pleasure to dedicate this paper to Andrei Alexandrovich Suslin on the occasion of his 60th birthday.

§1. The ℓ-invariant

1.1. (φ, Γ)-modules. ([Fo1], [Ber1], [Cz1]) 1.1.1. Let p be a prime number. Fix an algebraic closure \overline{ℚ}_{p} of ℚ_{p} and set G_{ℚ_{p}} = Gal(\overline{ℚ}_{p}/ℚ_{p}). We denote by ℂ_{p} the p-adic completion of \overline{ℚ}_{p} and write | · | for the absolute value on ℂ_{p} normalized by |p| = 1/p. For any 0 ⩽ r < 1 set

B(r, 1) = {z ∈ ℂ_{p} | p^{−1/r} ⩽ |z| < 1}.

Let χ : G_{ℚ_{p}} ⟶ ℤ_{p}^{∗} denote the cyclotomic character. Set H_{ℚ_{p}} = ker(χ) and Γ = G_{ℚ_{p}}/H_{ℚ_{p}}. The character χ will be often considered as an isomorphism χ : Γ →^{∼} ℤ_{p}^{∗}. Let L be a finite extension of ℚ_{p}. For any 0 ⩽ r < 1 we denote by B_{rig,L}^{†,r} the ring of p-adic functions f(π) = ∑_{k∈ℤ}a_{k}π^{k} (a_{k} ∈ L) which are

holomorphic on the annulus B(r, 1). The Robba ring over L is defined as R_{L} = ⋃_{r}B_{rig,L}^{†,r}. Recall that R_{L} is equipped with commuting, L-linear, continuous

actions of Γ and a frobenius φ which are defined by

\begin{matrix}γ(f(π)) = f((1 + π)^{χ(γ)} − 1), γ ∈ Γ, \\ φ(f(π)) = f((1 + π)^{p} − 1). \end{matrix}

Set t = log(1 + π) = ∑^{∞} ; n=1(−1)^{n−1} \frac{π^{n}}{n} . Remark that γ(t) = χ(γ) t and φ(t) = p t.

A finitely generated free R_{L}-module D is said to be a (φ, Γ)-module if it is equipped with commuting semilinear actions of Γ and φ and such that R_{L}φ(D) = D. The last condition means simply that φ(e_{1}), …, φ(e_{d}) is a basis of D if e_{1}, …, e_{d} is. Let δ : ℚ_{p}^{∗} ⟶ L^{∗} be a continuous character. We will write R_{L}(δ) for the (φ, Γ)-module R_{L}e_{δ} of rank 1 defined by

φ(e_{δ}) = δ(p) e_{δ}, γ(e_{δ}) = δ(χ(γ)) e_{δ}, γ ∈ Γ.

For any D we let D(χ) denote the φ-module D endowed with the action of Γ twisted by the cyclotomic character χ. Fix a topological generator γ ∈ Γ. For any (φ, Γ)-module D we denote by C_{φ,γ}(D) the complex

0 ⟶ D ⟶^{f} D ⊕ D ⟶^{g} D ⟶ 0

with f(x) = ((φ − 1)x, (γ − 1)x) and g(y, z) = (γ − 1)y − (φ − 1)z ([H1], [Cz4]). We shall write H^{∗}(D) for the cohomology of C_{φ,γ}(D). The main properties of these groups are the following 1) Long cohomology sequence. A short exact sequence of (φ, Γ)-modules

0 ⟶ D′ ⟶ D ⟶ D′′ ⟶ 0

gives rise to an exact sequence

0 ⟶ H^{0}(D′) ⟶ H^{0}(D) ⟶ H^{0}(D) ⟶^{Δ^{0}} H^{1}(D′) ⟶ ⋯ ⟶ H^{2}(D′′) ⟶ 0.

2) Euler-Poincaré characteristic. H^{i}(D) are finite dimensional L-vector spaces and

χ(D) = ∑^{2} ; i=0(−1)^{i} dim_{L} H^{i}(D) = −rg(D).

(see [H1] and [Li]).

3) Computation of the Brauer group. The map

cl(x) ↦ − (1 − \frac{1}{p})^{−1} (log χ(γ))^{−1}res(xdt)

is well defined and induces an isomorphism inv : H^{2}(R_{L}(χ)) →^{∼} L (see [H2]

[Ben1] and [Li]).

4) The cup-products. Let D and M be two (φ, Γ)-modules. For all i and j such that i + j ⩽ 2 define a bilinear map

∪ : H^{i}(D) × H^{j}(M) ⟶ H^{i+j}(D ⊗ M)

by

cl(x) ∪ cl(y) = cl(x ⊗ y) if i = j = 0,
cl(x) ∪ cl(y_{1}, y_{2}) = cl(x ⊗ y_{1}, x ⊗ y_{2}) if i = 0, j = 1,
cl(x_{1}, x_{2}) ∪ cl(y_{1}, y_{2}) = cl(x_{2} ⊗ γ(y_{1}) − x_{1} ⊗ φ(y_{2})) if i = 1, j = 1,
cl(x) ∪ cl(y) = cl(x ⊗ y) if i = 0, j = 2.

These maps commute with connecting homomorphisms in the usual sense.

5) Duality. Let D^{∗} = Hom_{R_{L}}(D, R_{L}). For i = 0, 1, 2 the cup product

(1.1)  H^{i}(D) × H^{2−i}(D^{∗}(χ)) ⟶^{∪} H^{2}(R_{L}(χ)) ≃ L

is a perfect pairing ([H2], [Li]).

1.1.2. Recall that a filtered (φ, N)-module with coefficients in L is a finite dimensional L-vector space M equipped with an exhausitive decreasing filtration Fil^{i}M, a linear bijective map φ : M ⟶ M and a nilpotent operator N : M ⟶ M such that φN = p φN. Filtered (φ, N)-modules form a ⊗- category which we denote by MF^{φ,N}. A filtered (φ, N)-module M is said to be a Dieudonné module if N = 0 on M. Filtered Dieudonné modules form a full subcategory MF^{φ} of MF^{φ,N}. It is not difficult to see that the series log(φ(π)/π^{p}) and log(γ(π)/π) (γ ∈ Γ) converge in R_{L}. Let log π be a transcendental element over the field of fractions of R_{L} equipped with actions of φ and Γ given by

φ(log π) = p log π + log (\frac{φ(π)}{π^{p}}) , γ(log π) = log π + log (\frac{γ(π)}{π}) .

Thus the ring R_{L,log} = R_{L}[log π] is equipped with natural actions of φ and

Γ and the monodromy operator N = − (1 − \frac{1}{p})^{−1} \frac{d}{d log π} . For any (φ, Γ)-

module D set

D_{st}(D) = (D ⊗_{R_{L}} R_{L,log}[1/t])^{Γ}

with t = log(1+π). Then D_{st}(D) is a finite dimensional L-vector space equipped with natural actions of φ and N such that Nφ = p φN. Moreover, it is equipped with a canonical exhaustive decreasing filtration Fil^{i}D_{st}(D) which is induced by the embeddings ι_{n} : B_{rig,L}^{†,r} ↪ L_{∞}[[t]], n ≫ 0 constructed in [Ber1] (see [Ber2] for more details). Set

D_{cris}(D) = D_{st}(D)^{N=0} = (D[1/t])^{Γ}.

Then

dim_{L} D_{cris}(D) ⩽ dim_{L} D_{st}(D) ⩽ rg(D)

and one says that D is semistable (resp. crystalline) if dim_{L} D_{cris}(D) = rg(D) (resp. if dim_{L} D_{st}(D) = rg(D)). If D is semistable, the jumps of the filtration Fil^{i}D_{st}(D) are called the Hodge-Tate weights of D and the tangent space of D is defined as t_{D}(L) = D_{st}(D)/Fil^{0}D_{st}(D). We let denote by M_{pst}^{φ,Γ} and M_{cris}^{φ,Γ} the categories of semistable and crystalline representations respectively. In [Ber2] Berger proved that the functors

( 1.2)  D_{st} : M_{pst}^{φ,Γ} ⟶ MF^{φ,N},  D_{cris} : M_{cris}^{φ,Γ} ⟶ MF^{φ}

are equivalences of ⊗-categories.

1.1.3. As usually, H^{1}(D) can be interpreted in terms of extensions. Namely, to any cocycle α = (a, b) ∈ Z^{1}(C_{φ,γ}(D)) one associates the extension

0 ⟶ D ⟶ D_{α} ⟶ R_{L} ⟶ 0

such that D_{α} = D ⊕ R_{L}e with φ(e) = e + a and γ(e) = e + b. This defines a canonical isomorphism

H^{1}(D) ≃ Ext^{1}(R_{L}, D).

We say that cl(α) ∈ H^{1}(D) is crystalline if dim_{L} D_{cris}(D_{α}) = dim_{L} D_{cris}(D)+1 and define

H_{f}^{1}(D) = {cl(α) ∈ H^{1}(D) | cl(α) is crystalline }.

It is easy to see that H_{f}^{1}(D) is a subspace of H^{1}(D). If D is semistable (even potentially semistable), one has

H^{0}(D) = Fil^{0}D_{st}(D)^{φ=1,N=0},

(1.3) dim_{L} H_{f}^{1}(D) = dim_{L} t_{D}(L) + dim_{L} H^{0}(D)

(see [Ben2], Proposition 1.4.4 and Corollary 1.4.5). Moreover, H_{f}^{1}(D) and H_{f}^{1}(D^{∗}(χ)) are orthogonal complements to each other under duality (1.1) ([Ben2], Corollary 1.4.10).

1.1.4. Let D be semistable (φ, Γ)-module of rank d. Assume that D_{st}(D)^{φ=1} = D_{st}(D) and that the all Hodge-Tate weights of D are ⩾ 0. Since Nφ = pφN this implies that N = 0 on D_{st}(D) and D is crystalline. The results of this section are proved in [Ben2] (see Proposition 1.5.9 and section 1.5.10). The canonical map D^{Γ} ⟶ D_{cris}(D) is an isomorphism and therefore H^{0}(D) ≃ D_{cris}(D) = D^{Γ} has dimension d over L. The Euler-Poincaré characteristic formula gives

dim_{L} H^{1}(D) = d + dim_{L} H^{0}(D) + dim_{L} H^{0}(D^{∗}(χ)) = 2d.

On the other hand dim_{L} H_{f}^{1}(D) = d by (1.3). The group H^{1}(D) has the following explicit description. The map

i_{D} : D_{cris}(D) ⊕ D_{cris}(D) ⟶ H^{1}(D),

i_{D}(x, y) = cl(−x, log χ(γ) y)

is an isomorphism. (Remark that the sign −1 and log χ(γ) are normalizing factors.) We let denote i_{D,f} and i_{D,c} the restrictions of i_{D} on the first and second summand respectively. Then Im(i_{D,f}) = H_{f}^{1}(D) and we set H_{c}^{1}(D) = Im(i_{D,c}). Thus we have a canonical decomposition

H^{1}(D) ≃ H_{f}^{1}(D) ⊕ H_{c}^{1}(D)

([Ben2], Proposition 1.5.9). Now consider the dual module D^{∗}(χ). It is crystalline, D_{cris}(D^{∗}(χ))^{φ=p^{−1}} = D_{cris}(D^{∗}(χ)) and the all Hodge-Tate weights of D^{∗}(χ) are ⩽ 0. Let

[ , ]_{D} : D_{cris}(D^{∗}(χ)) × D_{cris}(D) ⟶ L

denote the canonical pairing. Define

i_{D^{∗}(χ)} : D_{cris}(D^{∗}(χ)) ⊕ D_{cris}(D^{∗}(χ)) ⟶ H^{1}(D^{∗}(χ))

by

i_{D^{∗}(χ)}(α, β) ∪ i_{D}(x, y) = [β, x]_{D} − [α, y]_{D}.

As before, let i_{D∗(χ), f} and i_{D^{∗}(χ), c} denote the restrictions of i_{D} on the first and second summand respectively. From H_{f}^{1}(D^{∗}(χ)) = H_{f}^{1}(D)^{⊥} it follows that Im(i_{D^{∗}(χ), f}) = H_{f}^{1}(D^{∗}(χ)) and we set H_{c}^{1}(D^{∗}(χ)) = Im(i_{D^{∗}(χ), c}).

Write ∂ for the differential operator (1 + π)\frac{d}{dπ} .

Proposition 1.1.5. Let R_{L}(|x|x^{m}) be the (φ, Γ)-module R_{L}e_{δ} associated to the character δ(x) = |x|x^{m} (m ⩾ 1). Then

i) D_{cris}(R_{L}(|x|x^{m})) is the one-dimensional L-vector space generated by t^{−m}e_{δ}. Moreover D_{cris}(R_{L}(|x|x^{m})) = D_{cris}(R_{L}(|x|x^{m}))^{φ=p^{−1}} and the unique Hodge-Tate weight of R_{L}(|x|x^{m}) is −m.

ii) H^{0}(R_{L}(|x|x^{m})) = 0 and H^{1}(R_{L}(|x|x^{m})) is the two-dimensional

L-vector space generated by α_{m}^{∗} = − (1 − \frac{1}{p}) cl(α_{m}) and β_{m}^{∗} =

(1 − \frac{1}{p}) log χ(γ) cl(β_{m}) where

α_{m} = \frac{(−1)^{m−1}}{(m − 1)!} ∂^{m−1} (\frac{1}{π} + \frac{1}{2}, a) e_{δ}

with a ∈ R_{L}^{+} = R_{L} ∩ L[[π]] such that (1 − φ) a = (1 − χ(γ)γ) (\frac{1}{π} + \frac{1}{2}) and

β_{m} = \frac{(−1)^{m−1}}{(m − 1)!} ∂^{m−1} (b, \frac{1}{π} ) e_{δ}

with b ∈ R_{L} such that (1−φ) (\frac{1}{π} ) = (1−χ(γ)γ) b. Moreover i_{m,f}(1) = α_{m}^{∗} and

i_{m,c}(1) = β_{m}^{∗} where i_{m} denotes the map i defined in 1.1.4 for R_{L}(|x|x^{m}). In particular, H_{f}^{1}(R_{L}(|x|x^{m})) is generated by α_{m}^{∗} and H_{c}^{1}(R_{L}(|x|x^{m})) is generated by β_{m}^{∗}.

iii) Let x = cl(u, v) ∈ H^{1}(R_{L}(|x|x^{m})). Then

x = a cl(α_{m}) + b cl(β_{m})

with a = res(ut^{m−1}dt) and b = res(vt^{m−1}dt).

iv) The map

Res_{m} : R_{L}(|x|x^{m}) ⟶ L,

Res_{m}(α) = − (1 − \frac{1}{p})^{−1} (log χ(γ))^{−1}res (αt^{m−1}dt)

induces an isomorphism inv_{m} : H^{2}(R_{L}(|x|x^{m})) ≃ L. Moreover

inv_{m} (ω_{m}) = 1 where ω_{m} = (−1)^{m} (1 − \frac{1}{p}) \frac{log χ(γ)}{(m − 1)!} cl (∂^{m−1}(1/π))

Proof. The assertions i) and ii) are proved in [Cz4], sections 2.3-2.5 and [Ben2], Theorem 1.5.7 and (16). The assertions iii) and iv) are proved in [Ben2], Proposition 1.5.4 iii) Corollary 1.5.5.

1.1.6. In [Fo1], Fontaine worked out a general approach to the classification of p-adic representations in terms of (φ, Γ)-modules. Thanks to the work of Cherbonnier-Colmez [CC] and Kedlaya [Ke] this approach allows to construct an equivalence

D_{rig}^{†} : Rep_{L}(G_{ℚ_{p}}) ⟶ M_{ét}^{φ,Γ}

between the category of L-adic representations of G_{ℚ_{p}} and the category M_{ét}^{φ,Γ} of étale (φ, Γ)-modules in the sense of [Ke]. If V is a L-adic representation of G_{ℚ_{p}}, define

D_{st}(V) = D_{st}(D_{rig}^{†}(V)), D_{cris}(V) = D_{cris}(D_{rig}^{†}(V)).

Then D_{st} and D_{cris} are canonically isomorphic to classical Fontaine’s functors [Fo2], [Fo3] defined using the rings B_{st} and B_{cris} ([Ber1], Theorem 0.2). The continuous Galois cohomology H^{∗}(ℚ_{p}, V) = H_{cont}^{∗}(G_{ℚ_{p}}, V) is functorially isomorphic to H^{∗}(D_{rig}^{†}(V)) ([H1], [Li]). and under this isomorphism

H_{f}^{1}(D_{rig}^{†}(V)) ≃ H_{f}^{1}(ℚ_{p}, V)

where H_{f}^{1}(ℚ_{p}, V) = ker(H^{1}(ℚ_{p}, V) ⟶ H^{1}(ℚ_{p}, V ⊗ B_{cris})) is H_{f}^{1} of Bloch and Kato [BK].

1.2. The ℓ-invariant. 1.2.1. The results of this section are proved in [Ben2], 2.1-2.2. Denote by \overline{ℚ}^{(S)}/ℚ the maximal Galois extension of ℚ unramified outside S ∪ \{∞\} and set G_{S} = Gal(\overline{ℚ}^{(S)}/ℚ). If V is a L-adic representation of G_{S} we write H^{∗}(V) for the continuous cohomology of G_{S} with coefficients in V. If V is potentially semistable at p, set

H_{f}^{1}(ℚ_{l}, V) = \{ \begin{matrix}ker(H^{1}(ℚ_{l}, V) ⟶ H^{1}(ℚ_{l}^{nr}, V) & if l ≠ p, \\ H_{f}^{1}(D_{rig}^{†}(V)) & if l = p. \end{matrix}

The Selmer group of Bloch and Kato is defined by

H_{f}^{1}(V) = ker (H^{1}(V) ⟶ ⨁_{l∈S} \frac{H^{1}(ℚ_{l}, V)}{H_{f}^{1}(ℚ_{l}, V)}) .

Assume that V satisfies the condition C1-4) of 0.2. The Poitou-Tate exact sequence together with C1) gives an isomorphism

(1.4)  H^{1}(V) ≃ ⨁_{l∈S} \frac{H^{1}(ℚ_{l}, V)}{H_{f}^{1}(ℚ_{l}, V)}.

Recall that a (φ, N)-submodule D of D_{st}(V) is said to be regular if the canonical projection D ⟶ t_{V} (L) is an isomorphism. To any regular D we associate a filtration on D_{st}(V)

{0} ⊂ D_{−1} ⊂ D_{0} ⊂ D_{1} ⊂ D_{st}(V)

setting

D_{i} = \{ \begin{matrix}(1 − p^{−1}φ^{−1}) D + N(D^{φ=1}) & if i = −1, \\ D & if i = 0, \\ D + D_{st}(V)^{φ=1} ∩ N^{−1}(D^{φ=p^{−1}}) & if i = 1. \end{matrix}

By (1.2) this filtration induces a filtration on D_{rig}^{†}(V) by saturated (φ, Γ)- submodules

{0} ⊂ F_{−1}D_{rig}^{†}(V) ⊂ F_{0}D_{rig}^{†}(V) ⊂ F_{1}D_{rig}^{†}(V) ⊂ D_{rig}^{†}(V).

Set W = F_{1}D_{rig}^{†}(V)/F_{−1}D_{rig}^{†}(V). In [Ben2], Proposition 2.1.7 we proved that

(1.5)  W ≃ W_{0} ⊕ W_{1} ⊕ M,

where W_{0} and W_{1} are direct summands of gr_{0} (D_{rig}^{†}(V)) and gr_{1} (D_{rig}^{†}(V)) of

ranks dim_{L} H^{0}(W^{∗}(χ)) and dim_{L} H^{0}(W) respectively. Moreover M seats in a non split exact sequence

0 ⟶ M_{0} ⟶^{f} M ⟶^{g} M_{1} ⟶ 0

with rg(M_{0}) = rg(M_{1}), gr_{0} (D_{rig}^{†}(V)) = M_{0} ⊕ W_{0} and gr_{1} (D_{rig}^{†}(V)) = M_{1} ⊕ W_{1}. Set

e = rg(W_{0}) + rg(W_{1}) + rg(M_{0}).

Generalizing [G] we expect that the p-adic L-function L_{p}(V, D, s) has a zero of order e at s = 0. If W_{0} = 0, the main construction of [Ben2] associates to V and D an element L(V, D) ∈ L which can be viewed as a generalization of Greenberg’s L-invariant to semistable representations. Now assume that W_{1} = 0. Let D^{∗} = Hom(D_{st}(V)/D, D_{st}(ℚ_{p}(1))) be the dual regular space. As the decompositions (1.5) for the pairs (V, D) and (V^{∗}(1), D^{∗}) are dual to each other, one can define

ℓ(V, D) = L(V^{∗}(1), D^{∗}).

In this paper we do not review the construction of the L-invariant but give a direct description of ℓ(V, D) in terms of V and D in two important particular cases.

1.2.2. The crystalline case: W = W_{0} (see [Ben2], 2.2.6-2.2.7 and 2.3.3). In this case W is crystalline, W_{1} = M = 0 and F_{0}D_{rig}^{†}(V) = F_{1}D_{rig}^{†}(V). From the decomposition (1.5) it is not difficult to obtain the following description of H_{f}^{1}(ℚ_{p}, V) in the spirit of Greenberg’s local conditions:

(1.6)  H_{f}^{1}(ℚ_{p}, V) = ker (H^{1}(F_{0}D_{rig}^{†}(V)) ⟶ \frac{H^{1}(W)}{H_{f}^{1}(W)}) .

Let H^{1}(D, V) denote the inverse image of H^{1}(F_{0}D_{rig}^{†}(V))/H_{f}^{1}(ℚ_{p}, V) under the isomorphism (1.4). Thus one has a commutative diagram

(1.7)  \xymatrix{H^{1}(D, V) \ar[r] \ar[dr] & H^{1}(F_{0}D_{rig}^{†}(V)) \ar[d] \\  & H^{1}(D_{rig}^{†}(V))}

where the vertical map is injective ([Ben2], section 2.2.1). From (1.6) it follows that the composition map

κ_{D} : H^{1}(D, V) ⟶ H^{1}(F_{0}D_{rig}^{†}(V)) ⟶ H^{1}(W)

is injective. By construction, D_{cris}(W) = D/D_{−1} = D^{φ=p^{−1}}. As D is regular, the Hodge-Tate weights of W are ⩽ 0. Thus one has a decomposition

i_{W} : D_{cris} (W) ⊕ D_{cris} (W) ≃ H_{f}^{1} (W) ⊕ H_{c}^{1} (W) ≃ H^{1} (W) .

Denote by p_{D,f} and p_{D,c} the projection of H^{1} (W) on the first and the second direct summand respectively. We have a diagram

\xymatrix{ & D_{cris}(W) \\ H^{1}(D, V) \ar[ur]^{ρ_{D,f}} \ar[r]^{κ_{D}} \ar[dr]_{ρ_{D,c}} & H^{1}(W) \ar[u]_{p_{D,f}} \ar[d]^{p_{D,c}} \\  & D_{cris}(W)}

where ρ_{D,c} is an isomorphism. Then

ℓ(V, D) = det_{L} (ρ_{D,f} ∘ ρ_{D,c}^{−1} | D_{cris}(W)) .

1.2.3. The semistable case: W = M (see [Ben2], 2.2.3-2.2.4 and 2.3.3). In this case W is semistable , W_{0} = W_{1} = 0 and

(1.8)  H_{f}^{1}(ℚ_{p}, V) = ker (H^{1}(F_{1}D_{rig}^{†}(V)) ⟶ H^{1}(M_{1})) .

Let H^{1}(D, V) be the inverse image of H^{1}(F_{1}D_{rig}^{†}(V))/H_{f}^{1}(ℚ_{p}, V) under the isomorphism (1.4). Consider the exact sequence

\xymatrix{H^{1}(M_{0}) \ar[r]^{h_{1}(f)} & H^{1}(M) \ar[r]^{h_{1}(g)} & H^{1}(M_{1}) \ar[r]^{Δ^{1}} & H^{2}(M_{0}) \ar[r] & 0 . \\  & H^{1}(D, V) \ar[u]^{κ_{D}} \ar[ur]^{κ̄_{D}}}

By (1.8), the map κ̄_{D} is injective and it is not difficult to prove that the image of H^{1}(D, V) in H^{1}(M_{1}) coincides with Im(h_{1}(g)) ([Ben2], section 2.2.3). Thus in the semistable case the position of H^{1}(D, V) in H^{1}(M_{1}) is completely determined by the the restriction of V on the decomposition group at p. By construction, D_{st}(M_{1}) = D_{1}/D where (D_{1}/D)^{φ=1} = D_{1}/D and the Hodge-Tate weights of M_{1} are ⩾ 0. Again, one has an isomorphism

i_{M_{1}} : D_{cris} (M_{1}) ⊕ D_{cris} (M_{1}) ≃ H_{f}^{1} (M_{1}) ⊕ H_{c}^{1} (M_{1}) ≃ H^{1} (M_{1})

which allows to construct a diagram

\xymatrix{ & D_{st}(M_{1}) \\ Im(h_{1}(g)) \ar@{-->}[ur]^{ρ_{D,f}} \ar[r]^{κ_{D}} \ar[dr]_{ρ_{D,c}} & H^{1}(M_{1}) \ar[u]_{p_{D,f}} \ar[d]^{p_{D,c}} \\  & D_{st}(M_{1}).}

Then

(1.9)  ℓ(V, D) = L(V, D) = det_{L} (ρ_{D,f} ∘ ρ_{D,c}^{−1} | D_{st}(M_{1})) .

From (1.5) it is clear that if e = 1 then either W = W_{0} with rg(W_{0}) = 1 or W = M with rg(M_{0}) = rg(M_{1}) = 1. We consider these cases separately in the rest of the paper.

§2. The crystalline case

2.1. Let A = L[T]/(T^{2}) and let V_{A} be a free finitely generated A-module equipped with a A-linear action of G_{S}. One says that V_{A} is an infinitesimal deformation of a p-adic representation V if V ≃ V_{A} ⊗_{A} L. Write R_{A} = A⊗_{L} R_{L} and extend the actions of φ and Γ to R_{A} by linearity. A (φ, Γ)-module over R_{A} is a free finitely generated R_{A}-module D_{A} equipped with commuting semilinear actions of φ and Γ and such that R_{A}φ(D_{A}) = D_{A}. We say that D_{A} is an infinitesimal deformation of a (φ, Γ)-module D over R_{L} if D = D_{A} ⊗_{A} L.

2.2. Let V be a p-adic representation of G_{S} which satisfies the conditions C1-4) and such that W = W_{0}. Moreover we assume that rg(W) = 1. Thus W is a crystalline (φ, Γ)-module of rank 1 with D_{cris}(W) = D_{cris}(W)^{φ=p^{−1}} and such that Fil^{0}D_{cris}(W) = 0. This implies that

(2.1)  W ≃ R_{L}(δ) with δ(x) = |x|x^{m}, m ⩾ 1.

(see for example [Ben2], Proposition 1.5.8). Note that the Hodge-Tate weight of W is −m. The L-vector space H^{1}(D, V) is one dimensional. Fix a basis cl(x) ∈ H^{1}(D, V). We can associate to cl(x) a non trivial extension

0 ⟶ V ⟶ V_{x} ⟶ L ⟶ 0.

This gives an exact sequence of (φ, Γ)-modules

0 ⟶ D_{rig}^{†}(V) ⟶ D_{rig}^{†}(V_{x}) ⟶ R_{L} ⟶ 0.

From (1.7) it follows that there exists an extension in the category of (φ, Γ)- modules

0 ⟶ F_{0}D_{rig}^{†}(V) ⟶ D_{x} ⟶ R_{L} ⟶ 0

which is inserted in a commutative diagram

\xymatrix{0 \ar[r] & F_{0}D_{rig}^{†}(V) \ar[r] \ar[d] & D_{x} \ar[r] \ar[d] & R_{L} \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & D_{rig}^{†}(V) \ar[r] & D_{rig}^{†}(V_{x}) \ar[r] & R_{L} \ar[r] & 0.}

Define a filtration

{0} ⊂ F_{−1}D_{rig}^{†}(V_{x}) ⊂ F_{0}D_{rig}^{†}(V_{x}) ⊂ F_{1}D_{rig}^{†}(V_{x}) ⊂ D_{rig}^{†}(V_{x})

by F_{i}D_{rig}^{†}(V_{x}) = F_{i}D_{rig}^{†}(V) for i = −1, 0 and F_{1}D_{rig}^{†}(V_{x}) = D_{x}. Set

W_{x} = F_{1}D_{rig}^{†}(V_{x})/F_{−1}D_{rig}^{†}(V_{x}).

Thus one has a diagram

\xymatrix{0 \ar[r] & F_{0}D_{rig}^{†}(V) \ar[r] \ar[d] & D_{x} \ar[r] \ar[d] & R_{L} \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & W \ar[r] & W_{x} \ar[r] & R_{L} \ar[r] & 0.}

2.3. Let V_{A,x} be an infinitesimal deformation of V_{x}. Assume that D_{rig}^{†}(V_{A,x}) is equipped with a filtration by saturated (φ, Γ)-modules over R_{A}:

{0} ⊂ F_{−1}D_{rig}^{†}(V_{A,x}) ⊂ F_{0}D_{rig}^{†}(V_{A,x}) ⊂ F_{1}D_{rig}^{†}(V_{A,x}) ⊂ D_{rig}^{†}(V_{A,x})

such that F_{i}D_{rig}^{†}(V_{A,x})⊗_{A}L ≃ F_{i}D_{rig}^{†}(V_{x}) for all i. The quotients gr_{0}D_{rig}^{†}(V_{A,x}) and gr_{1}D_{rig}^{†}(V_{A,x}) are (φ, Γ)-modules of rank 1 over R_{A} and by [BC], Proposition 2.3.1 there exists unique characters δ_{A,x}, ψ_{A,x} : ℚ_{p}^{∗} ⟶ A^{∗} such that gr_{0}D_{rig}^{†}(V_{A,x}) ≃ R_{A}(δ_{A,x}) and gr_{1}D_{rig}^{†}(V_{A,x}) ≃ R_{A}(ψ_{A,x}). It is clear that δ_{A,x} (mod T) = δ and ψ_{A,x} (mod T) = 1. One has a diagram

\xymatrix{0 \ar[r] & F_{0}D_{rig}^{†}(V_{A}) \ar[r] \ar[d] & F_{1}D_{rig}^{†}(V_{A,x}) \ar[r] \ar[d] & R_{A}(ψ_{A}) \ar[r] \ar[d]^{=} & 0 \\ 0 \ar[r] & W_{A} \ar[r] & W_{A,x} \ar[r] & R_{A}(ψ_{A}) \ar[r] & 0}

with W_{A} = gr_{0}D_{rig}^{†}(V_{A,x}) and W_{A,x} = F_{1}D_{rig}^{†}(V_{A,x})/F_{−1}D_{rig}^{†}(V_{A,x}). Assume that

\frac{d(δ_{A,x}ψ_{A,x}^{−1})(u)}{dT}|_{T =0} ≠ 0, u ≡ 1 (mod p^{2})

(as the multiplicative group 1 + p^{2}ℤ_{p} is procyclic it is enough to assume that this holds for u = 1 + p^{2}.)

Theorem 1. Let V_{A,x} be an infinitesimal deformation of V_{x} which satisfies the above conditions. Then

ℓ(V, D) = − log χ(γ) \frac{d log(δ_{A,x}ψ_{A,x}^{−1})(p)}{d log(δ_{A,x}ψ_{A,x}^{−1})(χ(γ))}|_{T =0}.

This theorem will be proved in section 2.5. We start with an auxiliary result which plays a key role in the proof. Set δ(x) = |x|x^{m} (m ⩾ 1) and fix a character δ_{A} : ℚ_{p}^{∗} ⟶ A^{∗} such that δ_{A} (mod T) = δ. Consider the exact sequence

0 ⟶ R_{L}(δ) ⟶ R_{A}(δ_{A}) ⟶ R_{L}(δ) ⟶ 0

and denote by B_{δ}^{i} the connecting maps H^{i}(R_{L}(δ)) ⟶ H^{i+1}(R_{L}(δ)).

Proposition 2.4. One has

inv_{m}(B_{δ}^{1}(α_{m}^{∗})) = (log χ(γ))^{−1}d logδ_{A}(χ(γ))|_{T =0},
inv_{m}(B_{δ}^{1}(β_{m}^{∗})) = d logδ_{A}(p)|_{T =0}.

Proof. a) Recall that

α_{m}^{∗} = − (1 − \frac{1}{p}) \frac{(−1)^{m−1}}{(m − 1)!} cl (∂^{m−1} (\frac{1}{π} + \frac{1}{2}, a) e_{δ}) .

Let e_{A,δ} be a generator of R_{A}(δ_{A}) such that e_{δ} = e_{A,δ} (mod T). Directly from the definition of the connecting map

B_{δ}^{1}(α_{m}^{∗}) = − (1 − \frac{1}{p}) \frac{(−1)^{m−1}}{(m − 1)!} cl (\frac{1}{T} ((γ − 1) (∂^{m−1} (\frac{1}{π} + \frac{1}{2}) e_{A,δ}) −

− (φ − 1) (∂^{m−1}(a)e_{A,δ}))) .

Write

(γ − 1) (∂^{m−1} (\frac{1}{π} + \frac{1}{2}) e_{A,δ}) − (φ − 1) (∂^{m−1}(a)e_{A,δ}) =

= (χ(γ)^{−m}δ_{A}(χ(γ)) − 1) ∂^{m−1} (\frac{1}{π} + \frac{1}{2}) e_{A,δ} + z

where

z = (γ − χ(γ)^{−m}) ∂^{m−1} (\frac{1}{π} + \frac{1}{2}) δ_{A}(χ(γ))e_{A,δ} − (δ_{A}(p)φ − 1) ∂^{m−1}(a)e_{A,δ}.

Since δ_{A}(χ(γ)) ≡ χ(γ)^{m} (mod T), from the definition of a it follows that z ≡ 0 (mod T). On the other hand, as a ∈ R_{L}^{+} and

(γ − χ(γ)^{−m}) ∂^{m−1} (\frac{1}{π} + \frac{1}{2}) ∈ R_{L}^{+}

we obtain that z/T ∈ R_{L}^{+}e_{δ}. Thus the class of z/T in H^{2}(R_{L}(δ)) is zero. On the other hand, writing δ_{A} in the form

δ_{A}(u) = u^{m} + T \frac{dδ_{A}(u)}{dT}|_{T =0}

one finds that

\frac{χ(γ)^{−m}δ_{A}(χ(γ)) − 1}{T} = d log δ_{A}(χ(γ))|_{T =0}

Documenta Mathematica · Extra Volume Suslin (2010) 5–31

and the first formula follows from Proposition 1.1.5 iv).

b) By the definition of B_{δ}^{1}

B_{δ}^{1}(β_{m}^{∗}) = (1 − \frac{1}{p}) \frac{(−1)^{m−1} log χ(γ)}{(m − 1)!} cl (\frac{1}{T} ((γ − 1) (∂^{m−1}(b)e_{A,δ}) −

− (φ − 1) (∂^{m−1}(1/π)e_{A,δ}))) .

As

δ_{A}(p) (φ − δ(p)^{−1}) (∂^{m−1}(1/π)) = \frac{δ_{A}(p)}{δ(p)} (δ(χ(γ)) γ − 1) ∂^{m−1}(b)

we can write

(γ − 1) (∂^{m−1}(b)e_{A,δ}) − (φ − 1) (∂^{m−1}(1/π)e_{A,δ}) =

= −(δ(p)^{−1}δ_{A}(p) − 1) ∂^{m−1}(1/π) + w

where

w = (δ_{A}(χ(γ)) γ − 1) (∂^{m−1}b) e_{A,δ} + \frac{δ_{A}(p)}{δ(p)} (δ(χ(γ)) γ − 1) (∂^{m−1}b)e_{A,δ}.

Remark that

\frac{δ(p)^{−1}δ_{A}(p) − 1}{T} = −d log δ_{A}(p)|_{T =0}

On the other hand

res (∂^{m−1}(b) t^{m−1}dt) = 0

(see [Ben2], proof of Corollary 1.5.6). As res ((χ(γ)^{m}γ − 1) ∂^{m−1}(b) t^{m−1}dt) = 0, this implies that res (γ(∂^{m−1}b) t^{m−1}dt) = 0 and we obtain that Res_{m}(w) = 0. Thus

inv_{m}(B_{δ}^{1}(β_{m}^{∗})) = −dlog δ_{A}(p)|_{T =0}Res_{m}(ω_{m}) = dlog δ_{A}(p)|_{T =0}

and the Proposition is proved.

2.5. We pass to the proof of Theorem 1. By Proposition 1.1.5, H^{1}(W) is a two dimensional L-vector space generated by α_{m}^{∗} and β_{m}^{∗}. One has a commutative diagram with exact rows

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & W \ar[r] \ar[d] & W_{x} \ar[r] \ar[d] & R_{L} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W_{A} \ar[r] \ar[d] & W_{A,x} \ar[r] \ar[d] & R_{A}(ψ_{A,x}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W \ar[r] \ar[d] & W_{x} \ar[r] \ar[d] & R_{L} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

Twisting the middle row by ψ_{A,x}^{−1} and taking into account that ψ_{A,x} ≡ 1 (mod T) we obtain

(2.2)  \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & W \ar[r] \ar[d] & W_{x} \ar[r] \ar[d] & R_{L} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W_{A}(ψ_{A,x}^{−1}) \ar[r] \ar[d] & W_{A,x}(ψ_{A,x}^{−1}) \ar[r] \ar[d] & R_{A} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W \ar[r] \ar[d] & W_{x} \ar[r] \ar[d] & R_{L} \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

The connecting map Δ^{0} : H^{0}(R_{L}) ⟶ H^{1}(W) sends 1 to y = κ_{D}(cl(x)) and we can write

y = a α_{m}^{∗} + b β_{m}^{∗}

with a, b ∈ L. Directly from the definition of the ℓ-invariant one has

(2.3)  ℓ(V, D) = b^{−1}a.

The diagram (2.2) gives rise to a commutative diagram

\xymatrix{H^{0}(R_{L}) \ar[d]^{_{B_{0}}} \ar[r]^{Δ^{0}} & H^{1}(W) \ar[d]^{_{B{1 \atop W}}} \\ H^{1}(R_{L}) \ar[r]^{Δ^{1}} & H^{2}(W).}

Since the rightmost vertical row of (2.2) splits, the connecting map B^{0} is zero and

aB_{W}^{1} (α_{m}^{∗}) + bB_{W}^{1} (β_{m}^{∗}) = B_{W}^{1} (y) = 0.

As W_{A}(ψ_{A,x}^{−1}) ≃ R_{A}(δ_{A,x}ψ_{A,x}^{−1}), Proposition 2.4 gives

inv_{m}(B_{W}^{1} (α_{m}^{∗})) = (log(χ(γ))^{−1}d log(δ_{A,x}ψ_{A,x}^{−1})(χ(γ))|_{T =0},
inv_{m}(B_{W}^{1} (β_{m}^{∗})) = d log(δ_{A,x}ψ_{A,x}^{−1})(p)|_{T =0}.

Together with (2.3) this gives the Theorem.

§3. The semistable case

3.1. In this section we assume that V is a p-adic representation which satisfies the conditions C1-4) and such that W = M. Thus one has an exact sequence

(3.1)  0 ⟶ M_{0} ⟶^{f} W ⟶^{g} M_{1} ⟶ 0

where M_{0} and M_{1} are such that e = rg(M_{0}) = rg(M_{1}). We will assume that e = 1. Then

M_{0} = R_{L}e_{δ} ≃ R_{L}(δ), δ(x) = |x|x^{m}, m ⩾ 1,
M_{1} = R_{L}e_{ψ} ≃ R_{L}(ψ), ψ(x) = x^{−n}, n ⩾ 0

(see for example [Ben2], Lemma 1.5.2 and Proposition 1.5.8). Thus

{0} ⊂ F_{−1}D_{rig}^{†}(V) ⊂ F_{0}D_{rig}^{†}(V) ⊂ F_{1}D_{rig}^{†}(V) ⊂ D_{rig}^{†}(V)

with gr_{0}D_{rig}^{†}(V) ≃ R_{L}(δ) and gr_{1}D_{rig}^{†}(V) ≃ R_{L}(ψ). Assume that V_{A} is an infinitesimal deformation of V and that D_{rig}^{†}(V_{A}) is equipped with a filtration by saturated (φ, Γ)-modules over R_{A}

{0} ⊂ F_{−1}D_{rig}^{†}(V_{A}) ⊂ F_{0}D_{rig}^{†}(V_{A}) ⊂ F_{1}D_{rig}^{†}(V_{A}) ⊂ D_{rig}^{†}(V_{A})

such that

F_{i}D_{rig}^{†}(V_{A}) ⊗_{L} A ≃ F_{i}D_{rig}^{†}(V), −1 ⩽ i ⩽ 1.

Then

gr_{0}D_{rig}^{†}(V_{A}) ≃ R_{A}(δ_{A}), gr_{i}D_{rig}^{†}(V_{A}) ≃ R_{A}(ψ_{A}),

where δ_{A}, ψ_{A} : ℚ_{p}^{∗} ⟶ A^{∗} are such that δ_{A} (mod T) = δ and ψ_{A} (mod T) = ψ. As before, assume that

\frac{d(δ_{A}ψ_{A}^{−1})(u)}{dT}|_{T =0} ≠ 0, u ≡ 1 (mod p^{2}).

Theorem 2. Let V_{A} be an infinitesimal deformation of V which satisfies the above conditions. Then

(3.2)  ℓ(V, D) = − log χ(γ) \frac{d log(δ_{A}ψ_{A}^{−1})(p)}{d log(δ_{A}ψ_{A}^{−1})(χ(γ))}|_{T =0}

3.2. Proof of Theorem 2. The classes x_{n}^{∗} = −cl(t^{n}e_{ψ}, 0) and y_{n}^{∗} = log χ(γ) cl(0, t^{n}e_{ψ}) form a basis of H^{1}(M_{1}) and H_{f}^{1}(M_{1}) is generated by x_{n}^{∗} (see section 1.1.4). Consider the long cohomology sequence associated to (3.1):

⋯ ⟶ H^{1}(M_{0}) ⟶^{h_{1}(f)} H^{1}(W) ⟶^{h_{1}(g)} H^{1}(M_{1}) ⟶^{Δ^{1}} H^{2}(M_{0}) ⟶ · · · .

We can also consider the dual sequence 0 ⟶ M_{1}^{∗}(χ) ⟶ W^{∗}(χ) ⟶ M_{0}^{∗}(χ) ⟶ 0 and write

⋯ ⟶ H^{0}(M_{0}^{∗}(χ)) ⟶^{Δ_{∗}^{0}} H^{1}(M_{1}^{∗}(χ)) ⟶ H^{1}(W^{∗}(χ)) ⟶ H^{1}(M_{0}^{∗}(χ)) ⟶ · · · .

As M_{0}^{∗}(χ) = R_{L}e_{δ^{−1}χ} is isomorphic to R_{L}(x^{1−m}), the cohomology H^{0}(M_{0}^{∗}(χ)) is the one dimensional L-vector space generated by ξ = t^{m−1}e_{δ^{−1}χ}. Write

Δ_{∗}^{0}(ξ) = a α_{n+1}^{∗} + b β_{n+1}^{∗},

where α_{n+1}^{∗}, β_{n+1}^{∗} is the canonical basis of H^{1}(M_{1}^{∗}(χ)) ≃ R_{L}(|x|x^{n+1}). From the duality it follows that Im(Δ_{∗}^{0}) is orthogonal to ker(Δ^{1}) under the pairing

H^{1}(R_{L}(|x|x^{n+1})) × H^{1}(R_{L}(x^{−n})) ⟶^{∪} L

Since

α_{n+1}^{∗} ∪ x_{n}^{∗} = β_{n+1}^{∗} ∪ y_{n}^{∗} = 0, α_{n+1}^{∗} ∪ y_{n}^{∗} = −1, β_{n+1}^{∗} ∪ x_{n}^{∗} = 1

(see Proposition 1.1.5 ii), we obtain that Im(h_{1}(g)) = ker(Δ^{1}) is generated by ax_{n}^{∗} + by_{n}^{∗}. By the definition of the L-invariant

(3.3)  L(V, D) = b^{−1}a.

Set W_{A} = F_{1}D_{rig}^{†}(V)/F_{−1}D_{rig}^{†}(V). One has a commutative diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & R_{L}(ψ^{−1}χ) \ar[r] \ar[d] & W^{∗}(χ) \ar[r] \ar[d] & R_{L}(δ^{−1}χ) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & R_{A}(ψ_{A}^{−1}χ) \ar[r] \ar[d] & W_{A}^{∗}(χ) \ar[r] \ar[d] & R_{A}(δ_{A}^{−1}χ) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & R_{L}(ψ^{−1}χ) \ar[r] \ar[d] & W^{∗}(χ) \ar[r] \ar[d] & R_{L}(δ^{−1}χ) \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

Now the theorem can be proved either by twisting this diagram by δ_{A}χ^{−1} and applying the argument used in the proof of Theorem 2.3 or by the following direct computation. One has an anticommutative square

\xymatrix{H^{0}(R_{L}(δ^{−1}χ)) \ar[r]^{Δ_{∗}^{0}} \ar[d]^{_{B{0 \atop δ−1χ}}} & H^{1}(R_{L}(ψ^{−1}χ)) \ar[d]^{_{B{1 \atop ψ−1χ}}} \\ H^{1}(R_{L}(δ^{−1}χ)) \ar[r]^{Δ_{∗}^{1}} & H^{2}(R_{L}(ψ^{−1}χ)).}

Thus

(3.4)  B_{ψ^{−1}χ}^{1}Δ_{∗}^{0}(ξ) = −Δ_{∗}^{1}B_{δ^{−1}χ}^{0}(ξ).

From Proposition 2.3 it follows that

(3.5) inv_{n+1}(B_{ψ^{−1}χ}^{1}Δ_{∗}^{0}(ξ)) = a inv_{n+1}(B_{ψ^{−1}χ}^{1}(α_{n+1}^{∗})) + b (B_{ψ^{−1}χ}^{1}(β_{n+1}^{∗})) =

= −a log(χ(γ))^{−1}d log ψ_{A}(χ(γ))|_{T =0} − b d log ψ_{A}(p)|_{T =0}.

Fix a generator e_{A,δ^{−1}χ} of R_{A}(δ_{A}^{−1}χ). We can assume that e_{A,δ^{−1}χ} is a lifting of e_{δ^{−1}χ} and set ξ_{A} = t^{m−1}e_{A,δ^{−1}χ}. Directly by the definition of the connecting map

B_{δ^{−1}χ}^{0}(ξ) = \frac{1}{T} cl((φ − 1) ξ_{A}, (γ − 1) ξ_{A}) =

= \frac{1}{T} cl((p^{m−1}δ_{A}^{−1}(p) − 1) ξ_{A}, (χ(γ)^{m}δ_{A}^{−1}(χ(γ)) − 1) ξ_{A}) =

= −cl(d log δ_{A}(p) ξ, d log δ_{A}(χ(γ)) ξ)|_{T =0}.

Let ξ̂ be a lifting of ξ in W^{∗}(χ). Then

Δ_{∗}^{1}B_{δ^{−1}χ}^{0}(ξ) = −cl(d log δ_{A}(p) (γ − 1) ξ̂− d log δ_{A}(χ(γ)) (φ − 1) ξ̂)|_{T =0}.

On the other hand, Δ_{∗}^{0}(ξ) = cl((φ − 1) ξ̂, (γ − 1) ξ̂) and by Proposition 1.1.5 iii)

res((φ − 1) (ξ̂) t^{n}dt) = (1 − \frac{1}{p}) a,

res((γ − 1) (ξ̂) t^{n}dt) = log(χ(γ)) (1 − \frac{1}{p}) b.

Thus,

(3.6) inv_{n+1}(Δ_{∗}^{1}B_{δ^{−1}χ}^{0}(ξ)) =

= b d log δ_{A}(p)|_{T =0} + a log(χ(γ))^{−1}d log δ_{A}(χ(γ))|_{T =0}.

From (3.4), (3.5) and (3.6) we obtain that

a (log χ(γ))^{−1}dlog(δ_{A}ψ_{A}^{−1})(χ(γ))|_{T =0} = −b d log(δ_{A}ψ_{A}^{−1}) (p)|_{T =0}.

Together with (3.3) this prove the theorem.

3.4. Remark. It would be interesting to generalize Theorems 1 and 2 to the case e > 1. For this one should first understand what kind of filtrations on D_{rig}^{†}(V) appears naturally if V comes from automorphic forms [BC].

3.5. Modular forms. Let f be a normalized newform of weight x_{0} = 2k which is split multiplicative at p. Let V = M_{f,λ} be the λ-adic representation associated to f by Deligne [D]. The structure of D_{st}(V) is well known (see for example [Cz2]) Namely, D_{st}(V) = Ld_{1} + Ld_{2} with N(d_{2}) = d_{1}, N(d_{1}) = 0, φ(d_{2}) = p^{k}d_{2} and φ(d_{1}) = p^{k−1}d_{1}. Thus D_{st}(V (k)) = Ld_{1}^{(k)} + Ld_{2}^{(k)} with φ(d_{2}^{(k)}) = d_{2}^{(k)}, φ(d_{1}^{(k)}) = p^{−1}d_{1}^{(k)} and D = D_{cris}(V (k)) = Ld_{1}^{(k)} is the unique regular subspace of D_{st}(V (k)). It is clear that D_{−1} = 0, D_{1} = D_{st}(V (k)) and for the associated filtration on D_{rig}^{†}(V (k)) we have F_{0}D_{rig}^{†}(V (k)) = (D ⊗ R_{L}[1/t])∩D_{rig}^{†}(V (k)), F_{1}D_{rig}^{†}(V (k)) = D_{rig}^{†}(V (k)). In [Ben2], Proposition 2.2.6 it is proved that L(V (k), D) coincides with the L-invariant of Fontaine-Mazur L_{FM}(f). In [Co2], Coleman constructed an analytic family of overconvergent modular

forms f_{x} = ∑^{∞}_{n=1}a_{n}(x)q^{n} on an affinoid disk U containing 2k which satisfies the

following conditions

• For any x ∈ ℕ ∩ U the form f_{x} is classical.

• f_{x_{0}} = f. Moreover, one can interpolate the p-adic representations associated to classical forms f_{x} (x ∈ ℕ ∩ U) and construct a two dimensional representation V of G_{ℚ} over the Tate algebra O(U) of U such that

• For any integer x ∈ ℕ in U the Galois representation V_{x} obtained by specialization of V at x is isomorphic to the λ-adic representation associated to f_{x} [CM]. In particular, it is semistable with the Hodge-Tate weights (0, x − 1) [Fa]. By continuity this implies that for all x ∈ U the Hodge-Tate-Sen weights of V_{x} are (0, x − 1).

• ∧^{2}V_{x} ≃ L_{x} (χ^{1−2k} ⟨χ⟩^{2k−x}) where as usually ⟨χ⟩ denotes the projection of χ and L_{x} is the field of coefficients of V_{x}.

• (B_{cris}^{φ=a_{p}(x)} ˆ⊗V)^{G_{ℚp}} is locally free of rank 1 on U [Sa], [Ki].

Let O_{x_{0}} denote the local ring of U at x_{0} and let A = O_{x_{0}}/(T^{2}) where T = x−x_{0} is a local parameter at x_{0}. Then V_{A} = V⊗_{O(U)}O_{x_{0}} of V = V_{x_{0}} is an infinitesimal deformation of V = V_{x_{0}}. It is not difficult to see that

F_{0}D_{rig}^{†}(V_{A}) = R_{A} ⊗_{L} D_{cris}(D_{rig}^{†}(V_{A}))^{φ=a_{p}(x)}

is a saturated (φ, Γ)-submodule of D_{rig}^{†}(V_{A}) ( [BC], Lemma 2.5.2 iii)). We see immediately that F_{0}D_{rig}^{†}(V_{A}) ≃ R_{A}(δ_{A}) where δ_{A}(u) = 1 for u ∈ ℤ_{p}^{∗} and δ_{A}(p) = a_{p}(2k) + a′_{p}(2k)T (mod T^{2}) with a_{p}(2k) = p^{k−1}. Set F_{1}D_{rig}^{†}(V_{A}) = D_{rig}^{†}(V_{A}). As

⟨χ(γ)⟩ = exp((2k − x) log χ(γ)) = 1 − (log χ(γ))T (mod T^{2}) we obtain that

(ψ_{A}δ_{A}) (p) = 1, (ψ_{A}δ_{A}) (χ(γ)) = 1 − (log χ(γ))T (mod T^{2})

Thus ψ_{A}(χ(γ)) = 1 − log χ(γ) T (mod T^{2}) and d log ψ_{A}(χ(γ))|_{T =0} = − log χ(γ). Twisting V_{A} by χ^{k} we obtain an infinitesimal deformation V_{A}(k) of V (k). The formula (3.2) writes

L(V (k), D) = −2 d log a_{p}(2k).

In particular we obtain that L_{FM}(f) = −2 d log a_{p}(2k). The first direct proof of this result was done in [Cz5] using Galois cohomology computations inside the rings of p-adic periods. Remark that in [Cz6], Colmez used the theory of (φ, Γ)-modules to prove this formula with Breuil’s L-invariant. His approach is based on the local Langlands correspondence for two-dimensional trianguline representations.

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Institut de Mathématiques Université Bordeaux I 351 cours de la Libération 33405 Talence Cedex France denis.benois@math.u-bordeaux1.fr 32

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