The $${\mathcal {L}}$$ L -invariant, the dual $${\mathcal {L}}$$
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The L-invariant, the dual L-invariant, and families Jonathan Pottharst1 Dedicated to Glenn Stevens on the occasion of his 60th birthday.
Received: 17 July 2014 / Accepted: 20 August 2014 © Fondation Carl-Herz and Springer International Publishing Switzerland 2016
Abstract Given a rank two trianguline family of (ϕ, )-modules having a noncrystalline semistable member, we compute the Fontaine–Mazur L-invariant of that member in terms of the logarithmic derivative, with respect to the Sen weight, of the value at p of the trianguline parameter. This generalizes prior work, in the case of Galois representations, due to Greenberg–Stevens and Colmez. Keywords p-adic Galois representations · Semistable representations · L-invariants · Eigenvarieties Résumé Étant donnée une famille trianguline de rang deux de modules (φ, ) dont un membre est semi-stable non cristallin, nous calculons l’invariant-L de Fontaine-Mazur de ce membre en fonction de la dérivée logarithmique, par rapport au poids de Sen, de la valeur du paramètre triangulin pour p. Il s’agit d’une généralisation d’un travail antérieur dans le cas de représentations de Galois, dû à Greenberg-Stevens et Colmez. Mathematics Subject Classification
11F80
1 Introduction In the remarkable paper [4], Greenberg and Stevens proved a formula, conjectured by Mazur, Tate, and Teitelbaum in [6], for the derivative at s = 1 of the p-adic L-function of an elliptic curve E/Q when p is a prime of split multiplicative reduction. The novel quantity in this formula was the so-called L-invariant, namely L(E) = log p (q E )/ ord p (q E ) where an an . The proof of the q E ∈ pZp generates the kernel of the Tate uniformization Gm,Q → EQ p p Greenberg–Stevens Theorem had two main steps. On the one hand, a two-variable p-adic L-function was constructed, allowing the sought derivative to be computed in terms of the
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Jonathan Pottharst [email protected] San Francisco, USA
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derivative of the Hecke U p -operator with respect to the weight. On the other hand, a local Galois cohomology computation was used to relate this derivative of U p to the L-invariant, as in the formula [4, (0.15)]. This short paper extends [4, (0.15)], by extending the technique of its proof, to (ϕ, )modules over the Robba ring. The main result, which is proved in Sect. 3.3, is as follows. Theorem. Let X be an analytic space over a finite extension E of Qp , let δ, η : Qp × → O× X be continuous characters, and let the (ϕ, )-module D over R X be an extension of R X (δ) by R X (η). Assume that P ∈ X is such that the specialization D0 of D at P is, up to twist, noncrystalline semistable. Then the differential form d log(ηδ −1 ( p)) − L(D0 ) · d wt(ηδ −1 ) ∈ X/E vanishes at P. We hope the extension might motivate the generalization of other techniques of [4] to eigenvarieties whose triangulations possess a two-step graded piece that is noncrystalline semistable up to twist, and not necessarily étale. Although we had originally hoped to study the Hodge–Tate property in a uniform manner by formul
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