Search

Given a irreducible automorphic representation $\Pi$ of a similitude group $G/\mathbb Q$ giving rise to a KHT-Shimura variety, given a local congruence of the local component of $\Pi$ at a fixed place $p$, we justify the existence of a global automorphic representation $\Pi'$ with the same weight and the same level outside $p$ than $\Pi$, such that $\Pi$ and $\Pi'$ are weakly congruent. The arguments rest on the separation of the various contributions coming either from torsion or on the distinct families of automorphic representations, to the modulo $l$ reduction of the cohomology of Harris-Taylor perverse sheaves., Comment: 3 figures

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 $\overline{��_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2i��/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimura variety and $l$ the residual characteristic.

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., Comment: compared to arXiv:1610.10020, to construct torsion we simply use completed cohomology

We tackle three related problems. The first deals with freeness of localized cohomology groups of Harris-Taylor perverse sheaves, defined on the special fiber of some Kottwitz-Harris-Taylor Shimura variety. We then study the nilpotent monodromy operator acting both on the global cohomology of KHT Shimura variety and on the perverse sheaf of vanishing cycles. We then exhibe cases of level fixing phenomenon in the sense where the level at some fixed place of any rise in characteristic zero of an irreducible automorphic representation, is fixed equals to the one modulo l.

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., Nous donnons d'une part un procédé relativement souple de construction de classes de cohomologie de torsion dans la cohomologie d'une variété de Shimura de type Kottwitz-Harris-Taylor à coefficients dans un système local pas trop régulier. D'autre part nous montrons qu'associée à toute classe de cohomologie de torsion, il existe une infinité de représentations irréductibles automorphes cohomologiques, en caractéristique nulle, qui sont deux à deux non isomorphes et faiblement congruentes au sens.

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., arXiv admin note: text overlap with arXiv:1712.06824

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., Comment: written in french. arXiv admin note: text overlap with arXiv:1512.02008

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., 27 pages, written in french

International audience; When the level at $l$ of a Shimura variety of Kottwitz-Harris-Taylor is not maximal, its cohomology withcoefficients 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 oron the Galois representation $\overline \rho_{\mathfrak m}$ associated to it.Concerning the torsion, in a rather restricted case than \cite{scholze-cara}, we prove that the torsiondoesn't give new Satake parameters systems by showing that each torsion cohomology class can be raised in the freepart of the cohomology of a Igusa variety.; Lorsque le niveau en $l$ d'une variété de Shimura de Kottwitz-Harris-Taylor n'est pas maximal, sa cohomologie à coefficients dans un $\overline{\mathbb Z}_l$-système local n'est en général pas libre.Afin d'obtenir des énoncés d'annulation de la torsion, on localise en un idéal maximal $\mathfrak m$de l'algèbre de Hecke. Nous prouvons alors un énoncé d'annulation de la torsion de ces localisés, reposant soit sur $\mathfrak m$ directement, soit sur la représentation galoisienne $\overline \rho_{\mathfrak m}$qui lui est associée. En ce qui concerne la torsion, dans un cadrebien moins général que \cite{scholze-cara}, nous obtenons de même que la torsion ne fournitpas de nouveaux systèmes de paramètres de Satake, en prouvant que toute classe de torsion se relèvedans la partie libre de la cohomologie d'une variété d'Igusa.

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 $l$., This paper has been withdrawn by the author because its principal result is now enclosed in the paper arXiv:1503.03303

The principal aim of this paper is to construct torsion cohomology classes in the initial terms of a spectral sequence computing the cohomology of a Kottwitz-Harris-Taylor Shimura variety. Beside we produce some global congruences between automorphic representations., in French, accepted for publication in Annales de l'Institut Fourier 2015

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., Comment: in French, accepted for publication at Bulletin de la SMF (2014). arXiv admin note: text overlap with arXiv:0707.4396

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., 61 pages, in French

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., 28 pages, in French. arXiv admin note: text overlap with arXiv:0707.4396, arXiv:1309.1946

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., in French, accepted for publication at Bulletin de la SMF (2014). arXiv admin note: text overlap with arXiv:0707.4396

This article is the $\mathrm{Z}_l$-version of my paper "Monodromie du faisceau pervers des cycles \'evanescents de quelques vari\'et\'es de Shimura simples" in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $\mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $\mathrm{Z}_l$ version of my paper "Conjecture de monodromie-poids pour quelques vari\'t\'es de Shimura unitaires" in Compositio vol 146 part 2, pp. 367-403. The aim of this second article will be to study the torsion of the cohomology groups of these Shimura varieties., Comment: the new version is now arXiv:1309.1946

This article is the $\mathrm{Z}_l$-version of my paper "Monodromie du faisceau pervers des cycles ��vanescents de quelques vari��t��s de Shimura simples" in Invent. Math. 2009 vol 177 pp. 239-280, where we study the vanishing cycles of some unitary Shimura variety. The aim is to prove that the cohomology sheaves of this complexe are free so that, thanks to the main theorem of Berkovich on vanishing cycles, we can deduce that the $\mathrm{Z}_l$-cohomology of the model of Deligne-Carayol is free. There will be a second article which will be the $\mathrm{Z}_l$ version of my paper "Conjecture de monodromie-poids pour quelques varit��s de Shimura unitaires" in Compositio vol 146 part 2, pp. 367-403. The aim of this second article will be to study the torsion of the cohomology groups of these Shimura varieties., the new version is now arXiv:1309.1946

We describe the connected components of Igusa's varieties of second species defined par Harris and Taylor in their book. There are in bijection with the characters of the inversible group of the maximal order of the division algebra attached to them., Comment: 5 pages

In the first half of the paper, we translate in the geometric situation of Drinfeld varieties, the principal results of the Harris and Taylor's book. We give in particular the restriction to the open strata of the vanishing cycles sheaves in terms of some local systems named Harris-Taylor's local systems which we calculate the alternated sum of the cohomology group with compact supports. In the last half of the paper, we describe the monodromy filtration of the vanishing cycles perverse sheaf and the spectral sequence associated to it. Thanks to the Berkovich-Fargues' theorem, we obtain the description of the local monodromy filtration of the Deligne-Carayol model., Comment: 180 pages, 10 figures

In the first half of the paper, we translate in the geometric situation of Drinfeld varieties, the principal results of the Harris and Taylor's book. We give in particular the restriction to the open strata of the vanishing cycles sheaves in terms of some local systems named Harris-Taylor's local systems which we calculate the alternated sum of the cohomology group with compact supports. In the last half of the paper, we describe the monodromy filtration of the vanishing cycles perverse sheaf and the spectral sequence associated to it. Thanks to the Berkovich-Fargues' theorem, we obtain the description of the local monodromy filtration of the Deligne-Carayol model., 180 pages, 10 figures

In the geometric situation of the simple Shimura varieties of Kottwitz studied in Harris and Taylor's book, we describe the monodromy filtration of the vanishing cycles complex and the spectral sequence associated to it. We prove in particular that this filtration coincides with the weight one up to shift. Thanks to the Berkovich-Fargues' theorem, we deduce the description of the local monodromy filtration of the Deligne-Carayol model. In application, we obtain the description of the local components of cohomological automorphic representations of certains unitary groups, a global Jacquet-Langlands correspondence between two such groups and a proof of the weight-monodromy conjecture for the cohomology of these Shimura varieties., Comment: 83 pages, 13 figures, some minor changes