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Vincent Pilloni

Publications and source records attributed to Vincent Pilloni.

10 recordsLinked to original sources
arXiv2025

Modularity theorems for abelian surfaces

We prove the modularity of a positive proportion of abelian surfaces over $\mathbf{Q}$. More precisely, we prove the modularity of abelian surfaces which are ordinary at $3$ and are $3$-distinguished, subject to some assumptions on the $3$-torsion representation (a "big image" hypothesis, and a technical hypothesis on the action of a decomposition group at $2$). We employ a 2-3 switch and a new classicality theorem (in the style of Lue Pan) for ordinary $p$-adic Siegel modular forms.

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arXiv2023

p-adic interpolation of Gauss--Manin connections on nearly overconvergent modular forms and p-adic L-functions

In this paper, we give a new geometric definition of nearly overconvergent modular forms and $p$-adically interpolate the Gauss-Manin connection on this space. This can be seen as an ``overconvergent'' version of the unipotent circle action on the space of $p$-adic modular forms, as constructed by Gouv\^{e}a and Howe. This improves on results of Andreatta--Iovita and has applications to the construction of Rankin--Selberg and triple product $p$-adic $L$-functions.

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arXiv2022

Higher Hida theory and p-adic L-functions for GSp(4)

We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2).

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arXiv2021

Abelian Surfaces over totally real fields are Potentially Modular

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

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arXiv2021

Higher Coleman Theory

We develop local cohomology techniques to study the finite slope part of the coherent cohomology of Shimura varieties. The local cohomology groups we consider are a generalization of overconvergent modular forms, and they are defined by using a stratification on the Shimura variety obtained from the Bruhat stratification on a flag variety via the Hodge-Tate period map. We construct a spectral sequence from the local cohomologies to the classical cohomology and use it to obtain classicality and vanishing results. We also develop a theory of p-adic families and construct eigenvarieties. As an application, we prove some new properties of Galois representations arising from certain non-regular algebraic cuspidal automorphic representations.

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arXiv2019

Hecke operators and the coherent cohomology of Shimura varieties

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain $p$-adic estimates of Satake parameters of certain non-regular self dual automorphic representations of $\mathrm{GL}_n$.

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arXiv2012

P-Adic families of Siegel modular cuspforms

Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.

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arXiv2010

Partial canonical subgroups

The reduction of Siegel varieties modulo a prime number p is stratified by the multiplicative rank of the p-divisible group of the universal abelian variety. For r\geq 0 the maximal multiplicative subgroup of the restriction of the p-torsion group of the universal abelian variety to the r-th stratum lifts canonically to the tube of this stratum and defines a partial canonical subgroup of rank r. We prove that this subgroup extends in a finite flat way on some strict neighborhood of the tube. On the ordinary stratum and on its neighborhood, we recover the usual canonical subgroup considered by Abbes and Mokrane, and Andreatta and Gasbarri. ----- La reduction des varietes de Siegel modulo un nombre premier p est stratifiee par le rang multiplicatif du groupe p-divisible de la variete abelienne universelle. Pour r\geq 0, le sous-groupe multiplicatif maximal de la restriction du groupe de p-torsion de la variete abelienne universelle a la r-ieme strate se releve canoniquement sur le tube de cette strate et definit un sous-groupe canonique partiel de rang r. Nous montrons qu'il existe un voisinage strict du tube sur lequel ce sous-groupe s'etend de maniere finie et plate. Sur la strate ordinaire et au voisinage de celle-ci, on retrouve le sous-groupe canonique usuel etudie par Abbes et Mokrane d'une part, Andreatta et Gasbarri d'autre part.

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