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Pascal Boyer

Publications and source records attributed to Pascal Boyer.

At least 19 recordsLinked to original sources
arXiv2025

Filtrations of the perverse sheaf of nearby cycles in the semi stable situation

In the strict semi stable reduction situation, we describe the various filtrations of the perverse sheaf of nearby cycles in terms of irreducible perverse sheaves together with the action of the monodromy operator. We then study the spectral sequences associated to these filtration computing the sheaf cohomology groups. Finally we propose an illustration of how it can be used to compute the cohomology groups. Considering the similarity with the results of my paper at Duke Math. Journal,, it could be a good introduction before reading loc. cit.

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arXiv2023

Ihara's Lemma for $\mathrm{GL}_d$: the limit case

Clozel, Harris and Taylor proposed conjectural generalizations of the classical Ihara's lemma for $\mathrm{GL}_2$, to higher dimensional similitude groups. We prove these conjectures in the so called limit case, which after base change is the essential one, under any hypothesis allowing level raising.

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arXiv2019

Level lowering: a Mazur principle in higher dimension

For a maximal ideal $\mathfrak m$ of some anemic Hecke algebra $\mathbb{T}^S_\xi$ of a similitude group of signature $(1,d-1)$, one can associate a Galois $\overline{\mathbb F}_l$-representation $\overline \rho_{\mathfrak m}$ as well as a Galois $\mathbb{T}_{\xi,\mathfrak m}^S$-representation $\rho_{\mathfrak m}$.For $l\geq d$, on can also define a monodromy operator $\overline N_{\mathfrak m}$ as well as $N_{\widetilde{\mathfrak m}}$ for every prime ideal $\widetilde{\mathfrak m} \subset \mathfrak m$, giving rise to partitions $\underline{\bar d_{\mathfrak m}}$ and $\underline d_{\widetilde{\mathfrak m}}$ of $d$. As with Mazur's principle for $GL_2$, analysing the difference between these partitions, we infer informations about the liftings of $\overline \rho_{\mathfrak m}$ in characteristic zero known as level lowering problem.

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arXiv2019

Galois irreducibility implies cohomology freeness for KHT Shimura varieties

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{\rho_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2i\pi/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimura variety and $l$ the residual characteristic.

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arXiv2019

Construction of torsion cohomology classes for KHT Shimura varieties

Let $Sh_K(G,\mu)$ be a Shimura variety of KHT type, as introduced in Harris-Taylor book, associated to some similitude group $G/\mathbb Q$ and a open compact subgroup $K$ of $G(\mathbb A)$. For any irreducible algebraic $\overline{\mathbb Q}_l$-representation $\xi$ of $G$, let $V_\xi$ be the $\mathbb Z_l$-local system on $Sh_K(G,\mu)$. From my paper about p-stabilization, we know that if we allow the local component $K_l$ of $K$ to be small enough, then there must exists some non trivial cohomology classes with coefficient in $V_\xi$. The aim of this paper is then to construct explicitly such torsion classes with the control of $K_l$. As an application we obtain the construction of some new automorphic congruences between tempered and non tempered automorphic representations of the same weight and same level at $l$.

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arXiv2018

Local Ihara's lemma and applications

Persistence of non-degeneracy is a phenomenon which appears in the theory of $\overline{\mathbb Q}_l$-representations of the linear group: every irreducible submodule of the restriction to the mirabolic sub-representation of a non-degenerate irreducible representation is non-degenerate. This is not true anymore in general, if we look at the modulo $l$ reduction of some stable lattice. As in the Clozel-Harris-Taylor generalization of global Ihara's lemma, we show that this property, called non-degeneracy persistence and related to the notion of essentially absolutely irreducible and generic representations in the work of Emerton-Helm, remains true for lattices given by the cohomology of Lubin-Tate spaces. As an global application, we give a new construction of automorphic congruences in the Ribet spirit.

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arXiv2018

$p$-stabilization in higher dimension

Using $l$-adic completed cohomology in the context of Shimura varieties of Kottwitz-Harris-Taylor type attached to some fixed similitude group $G$, we prove, allowing to increase the levet at $l$, some new automorphic congruences between any degenerate automorphic representation with a non degenerate one of the same weight.

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arXiv2017

Torsion classes in the cohomology of KHT Shimura varieties

A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to each such torsion class an infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent. Then, using completed cohomology, we construct torsion classes in regular weight and then deduce explicit examples of such automorphic congruences.

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arXiv2016

Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces

In my 2009 paper at Inventiones, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle $\Psi$ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of $\Psi$. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

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arXiv2016

Automorphic congruences and torsion in the cohomology of a simple unitary Shimura variety

We first give a relative flexible process to construct torsion cohomology classes for Shimura varieties of Kottwitz-Harris-Taylor type with coefficient in a non too regular local system. We then prove that associated to each torsion cohomology class, there exists a infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent.

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arXiv2015

On the $\bar{\mathbb F}_l$-cohomology of a simple unitary Shimura variety

We study the torsion cohomology classes of Shimura varieties of type Kottwitz-Harris-Taylor and we show that " up to an arbitrary place " one can raise them to an automorphic representation. In application, to any mod $l$ system of Hecke eigenvalues appearing in the $\bar{\mathbb F}_l$-cohomology of a Shimura's variety of Kottwitz-Harris-Taylor type, we associate a $\bar{\mathbb F}_l$-Galois representation which Frobenius eigenvalues are given by Hecke's. Compared to the highly more general construction of Scholze, we gain both the simplicity of the proof and the control at places ramified and at those dividing $l$.

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arXiv2015

Ihara's lemma and level rising in higher dimension

A key ingredient in the Taylor-Wiles proof of Fermat last theorem is the classical Ihara's lemma which is used to rise the modularity property between some congruent galoisian representations. In their work on Sato-Tate, Clozel-Harris-Taylor proposed a generalization of the Ihara's lemma in higher dimension for some similitude groups. The main aim of this paper is then to prove some new instances of this generalized Ihara's lemma by considering some particular non pseudo Eisenstein maximal ideals of unramified Hecke algebras. As a consequence, we prove a level rising statement.

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arXiv2015

Mazur's principle for U(1,d)

The Mazur principle give simple conditions for an irreducible unramified $\overline{\mathbb{F}_l}$-representation coming from a modular form of level $\Gamma_0(Np)$ to come for some modular form of level $\Gamma_0(N)$. The aim of this work is to give a generalization of this principle in higher dimension for some particular extended inner forms non quasi split of a unitary group studying the torsion cohomology classes of Shimura varieties of Kottwitz-Harris-Taylor type within its link with the local monodromy degeneracy.

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arXiv2015

Sur la torsion dans la cohomologie des vari\'et\'es de Shimura de Kottwitz-Harris-Taylor

When the level at $l$ of a Shimura variety of Kottwitz-Harris-Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb Z}_l$-local system isn't in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak m$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak m$ itself or on the Galois representation $\overline \rho_{\mathfrak m}$ associated to it. Concerning the torsion, in a rather restricted case than the work of Caraiani-Scholze, we prove that the torsion doesn't give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

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arXiv2013

Sur les extensions interm\'ediaires des syst\`emes locaux d'Harris-Taylor

In the geometric situation of some simple unitary Shimura varieties studied by Harris and Taylor, I have built two filtrations of the perverse sheaf of vanishing cycles. The graduate of the first are the $p$-intermediate extension of some local Harris-Taylor's local systems, while for the second, obtained by duality, they are the $p+$-intermediate extensions. In this work, we describe the difference between these $p$ and $p+$ intermediate extension. Precisely, we show, in the case where the local system is associated to an irreducible cuspidal representation whose reduction modulo $l$ is supercuspidal, that the two intermediate extensions are the same. Otherwise, if the reduction modulo $l$ is just cuspidal, we describe the $l$-torsion of their difference.

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arXiv2013

Filtrations de stratification de quelques vari\'et\'es de Shimura simples

We define and study new filtrations called of stratification of a perverse sheaf on a scheme; beside the cases of the weight or monodromy filtrations, these filtrations are available whatever are the ring of coefficients. We illustrate these constructions in the geometric situation of the simple unitary Shimura varieties of Harris and Taylor's book for the perverse sheaves of Harris-Taylor and the complex of vanishing cycles, introduced and studied in my 2009 paper at inventiones. In the situation studied in loc. cit., we show how to use these filtrations to simplify the principal step of this paper; the cases of finite field or ring of integer of a local field will be studied in the next published paper.

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arXiv2013

La cohomologie des espaces de Lubin-Tate est libre

The principal result of this work is the freeness in the $ \overline{\mathbb Z}_l$-cohomology of the Lubin-Tate tower. The strategy is of global nature and relies on studying the filtration of stratification of the perverse sheaf of vanishing cycles of some Shimura varieties of Kottwitz-Harris-Taylor types, whose graduates can be explicited as some intermediate extension of some local system constructed in the book of Harris andTaylor. The crucial point relies on the study of the difference between such extension for the two classical $t$-structures $p$ and $p+$. The main ingredients use the theory of derivative for representations of the mirabolic group.

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