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Denis Benois

Publications and source records attributed to Denis Benois.

11 recordsLinked to original sources
arXiv2026

Log-crystalline representations and $(\varphi, \Gamma)$-modules

Let $F$ be an absolutely unramified mixed characteristic local field with rings of integers $O_F$. For a small affine algebra $R$ over $O_F$, we consider the logarithmic \'etale fundamental group $G_R$ of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara--Kato. We study $p$-adic representations of $G_R$ and obtain a classification of these representations in terms of \'etale $(\varphi, \Gamma_R)$-modules. Moreover, we define and study the notion of log-crystalline representations of $G_R$, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of $G_R$ are equivalent to Wach modules for $R$, extending our previous results from the (non-logarithmic) crystalline case.

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arXiv2024

Arithmetic of critical $p$-adic $L$-functions

Our objective in the present work is to develop a fairly complete arithmetic theory of critical $p$-adic $L$-functions on the eigencurve. To this end, we carry out the following tasks: a) We give an "\'etale" construction of Bella\"iche's $p$-adic $L$-functions at a $\theta$-critical point on the cuspidal eigencurve. b) We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic $\mathscr L$-invariant $\mathscr{L}^{\rm cr}_{\rm Iw}$. c) We formulate the (punctual) critical main conjecture and study its relationship with its slope-zero counterparts. Along the way, we also develop descent theory (paralleling Perrin-Riou's work). d) We introduce what we call thick (Iwasawa theoretic) fundamental line and the thick Selmer complex to counter Bella\"iche's secondary $p$-adic $L$-functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both. e) We establish an $\mathcal{O}_{\mathcal{X}}$-adic leading term formula for the two-variable $p$-adic $L$-function over the affinoid neighbourhood $\mathcal{X}={\rm Spm}(\mathcal{O}_{\mathcal{X}})$ in the eigencurve about a $\theta$-critical point. Using this formula we prove, when the Hecke $L$-function of $f$ vanishes to order one at the central critical point, that the derivative of the secondary $p$-adic $L$-function can be computed in terms of the second order derivative of an $\mathcal{O}_{\mathcal{X}}$-adic regulator (rather than a regulator itself).

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arXiv2022

On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou

Our goal in this article is to prove a form of $p$-adic Birch and Swinnerton-Dyer formula for the second derivative of the $p$-adic $L$-function associated to a newform $f$ which is non-crystalline semistable at $p$ at its central critical point, by expressing this quantity in terms of a $p$-adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through $f$. This, among other things, leads us to a proof of an appropriate version of Perrin-Riou's conjecture in this set-up.

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arXiv2021

Interpolation of Beilinson-Kato elements and $p$-adic $L$-functions

Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of $p$-adic $L$-functions (of Bellaïche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson-Kato elements over the eigencurve (including the neighborhoods of $θ$-critical points). Along the way, we prove étale variants of Bellaïche's results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these $p$-adic $L$-functions away from $θ$-critical points.

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arXiv2020

On extra-zeros of p-adic Rankin-Selberg L-functions

We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied.

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arXiv2014

P-adic heights and p-adic Hodge theory

Using the theory of $(ϕ,Γ)$-modules and the formalism of Selmer complexes we construct the p-adic height for p-adic representations with coefficients in an affinoid algebra over $Q_p$.

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arXiv2014

Selmer complexes and the $p$-adic Hodge theory

The first part of the paper is a survey of recent results about the cohomology of $(ϕ,Γ)$-modules and its applications to the theory of Selmer complexes. In the second part we formulate a version of the Main Conjecture for $p$-adic representations which are semistable at $p$ and discuss some related topics.

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arXiv2013

On extra zeros of p-adic L-functions: the crystalline case

We formulate a conjecture about extra zeros of p-adic L-functions at near central points which generalises the conjecture formulated in our previous paper. We prove that this conjecture is compatible with Perrin-Riou's theory of p-adic L-functions.

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arXiv2012

Trivial zeros of p-adic L-functions at near central points

Using the $\scr L$-invariant constructed in our previous paper we prove a Mazur-Tate-Teitelbaum style formula for derivatives of p-adic L-functions of elliptic modular forms at near central points. In the second version of the paper the case of potentially crystalline reduction is also covered.

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arXiv2009

On trivial zeros of Perrin-Riou's $L$-functions

In the previous paper we generalized Greenberg's construction of the $\Cal L$-invariant to semistable representations. Here we prove that this construction is compatible with Perrin-Riou's theory of $p$-adic $L$-functions

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