Searcharxiv⌕ Search

arXiv subjects

Sophie Morel

Publications and source records attributed to Sophie Morel.

14 recordsLinked to original sources
arXiv2024

Mixed $\ell$-adic complexes for schemes over number fields

If $X$ is a variety over a number field, Annette Huber has defined a category of "horizontal" (or "almost everywhere unramified") $\ell$-adic complexes and $\ell$-adic perverse sheaves on $X$. For such objects, the notion of weights makes sense (in the sense of Deligne), just as in the case of varieties over finite fields. However, contrary to what happens in that last case, mixed perverse sheaves (or mixed locally constant sheaves) on $X$ do not have a weight filtration in general, even when $X$ is a point. The goal of this paper is to show how to avoid this problem by working directly in the derived category of the abelian category of perverse sheaves that do admit a weight filtration. As an application, the methods of a previous paper of the author to calculate the intermediate extension of a pure perverse sheaf apply over any finitely generated field, and not just over a finite field.

math.AG↗
arXiv2023

Shimura varieties

These are the notes of a course on Shimura varieties that I gave at the 2022 IHES summer school on the Langlands program. Lecture 1 gives an introduction to Shimura varieties over the complex numbers (defined here as a special type of locally symmetric spaces) and to the general theory of canonical models; it also discusses in more detail the example of the Siegel modular varieties. Lecture 2 presents some families of Shimura varieties (PEL type, Hodge type, abelian type) and the results that are known about their canonical and integral models. Finally, lecture 3 discusses the cohomology of Shimura varieties, concentrating mostly on the compact non-endoscopic case.

math.NT↗
arXiv2023

The four operations on perverse motives

Let $k$ be a field of characteristic zero with a fixed embedding $σ:k\hookrightarrow \mathbb{C}$ into the field of complex numbers. Given a $k$-variety $X$, we use the triangulated category of étale motives with rational coefficients on $X$ to construct an abelian category $\mathscr{M}(X)$ of perverse mixed motives. We show that over $\mathrm{Spec}(k)$ the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories $\mathrm{D}^{\mathrm{b}}(\mathscr{M}(X))$ are equipped with the four operations of Grothendieck (for morphisms of quasi-projective $k$-varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo-Migliorini and provide an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.

math.AG↗
arXiv2023

Pizza and 2-structures

Let $\mathcal{H}$ be a Coxeter hyperplane arrangement in $n$-dimensional Euclidean space. Assume that the negative of the identity map belongs to the associated Coxeter group $W$. Furthermore assume that the arrangement is not of type $A_1^n$. Let $K$ be a measurable subset of the Euclidean space with finite volume which is stable by the Coxeter group $W$ and let $a$ be a point such that $K$ contains the convex hull of the orbit of the point $a$ under the group $W$. In a previous article the authors proved the generalized pizza theorem: that the alternating sum over the chambers $T$ of $\mathcal{H}$ of the volumes of the intersections $T\cap(K+a)$ is zero. In this paper we give a dissection proof of this result. In fact, we lift the identity to an abstract dissection group to obtain a similar identity that replaces the volume by any valuation that is invariant under affine isometries. This includes the cases of all intrinsic volumes. Apart from basic geometry, the main ingredient is a theorem of the authors where we relate the alternating sum of the values of certain valuations over the chambers of a Coxeter arrangement to similar alternating sums for simpler subarrangements called $2$-structures introduced by Herb to study discrete series characters of real reduced groups.

math.CO↗
arXiv2023

Comparison of different definitions of pseudocharacters

We prove that the definitions of a $d$-dimensional pseudocharacter (or pseudorepresentation) given by Chenevier and V. Lafforgue agree over any ring. We also compare the scheme of Lafforgue's $G$-pseudocharacters of a group with its $G$-character variety.

math.AG↗
arXiv2022

A generalization of combinatorial identities for stable discrete series constants

This article is concerned with the constants that appear in Harish-Chandra's character formula for stable discrete series of real reductive groups, although it does not require any knowledge about real reductive groups or discrete series. In Harish-Chandra's work the only information we have about these constants is that they are uniquely determined by an inductive property. Later Goresky-Kottwitz-MacPherson and Herb gave different formulas for these constants. In this article we generalize these formulas to the case of arbitrary finite Coxeter groups (in this setting, discrete series no longer make sense), and give a direct proof that the two formulas agree. We actually prove a slightly more general identity that also implies the combinatorial identity underlying the discrete series character identities of Morel. We deduce this identity from a general abstract theorem giving a way to calculate the alternating sum of the values of a valuation on the chambers of a Coxeter arrangement. We also introduce a ring structure on the set of valuations on polyhedral cones in Euclidean space with values in a fixed ring. This gives a theoretical framework for the valuation appearing in Appendix A of the Goresky-Kottwitz-MacPherson paper. In Appendix B we extend the notion of $2$-structures (due to Herb) to pseudo-root systems.

math.CO↗
arXiv2022

Sharing pizza in n dimensions

We introduce and prove the $n$-dimensional Pizza Theorem: Let $\mathcal{H}$ be a hyperplane arrangement in $\mathbb{R}^{n}$. If $K$ is a measurable set of finite volume, the {pizza quantity} of $K$ is the alternating sum of the volumes of the regions obtained by intersecting $K$ with the arrangement $\mathcal{H}$. We prove that if $\mathcal{H}$ is a Coxeter arrangement different from $A_{1}^{n}$ such that the group of isometries $W$ generated by the reflections in the hyperplanes of $\mathcal{H}$ contains the map $-\mathrm{id}$, and if $K$ is a translate of a convex body that is stable under $W$ and contains the origin, then the pizza quantity of $K$ is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of $\mathcal{H}$ that we call the {even restricted arrangement}. More generally, we prove that for a class of arrangements that we call {even} (this includes the Coxeter arrangements above) and for a {sufficiently symmetric} set $K$, the pizza quantity of $K+a$ is polynomial in $a$ for $a$ small enough, for example if $K$ is convex and $0\in K+a$. We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at $a$ having radius $R\geq\|a\|$ vanishes for a Coxeter arrangement $\mathcal{H}$ with $|\mathcal{H}|-n$ an even positive integer. We also prove the Pizza Theorem for the surface volume: When $\mathcal{H}$ is a Coxeter arrangement and $|\mathcal{H}| - n$ is a nonnegative even integer, for an $n$-dimensional ball the alternating sum of the $(n-1)$-dimensional surface volumes of the regions is equal to zero.

math.CO↗
arXiv2019

Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties

In the computation of the intersection cohomology of Shimura varieties, or of the $L^2$ cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about the origin of the identities, our paper is entirely combinatorial and does not require any knowledge of Shimura varieties or of representation theory.

math.CO↗
arXiv2018

On the cohomology of certain non-compact Shimura varieties (with an appendix by Robert Kottwitz)

The goal of this paper is to calculate the trace of the composition of a Hecke correspondence and a (high enough) power of the Frobenius at a good place on the intersection cohomology of the Satake-Baily-Borel compactification of certain Shimura varieties, to stabilize the result for Shimura varieties associated to unitary groups over $\mathbb{Q}$ and to give applications of this calculations using base change from these unitary groups to $GL_n$. ----- Le but de ce texte est de calculer la trace d'une correspondance de Hecke composee avec une puissance (assez grande) du Frobenius en une bonne place sur la cohomologie d'intersection de la compactification de Satake-Baily-Borel de certaines varietes de Shimura, de stabiliser le resultat obtenu pour les varietes de Shimura associees aux groupes unitaires sur $\mathbb{Q}$, et de donner des applications de ces calculs en utilisant le changement de base de ces groupes unitaires a $GL_n$.

math.AG↗
arXiv2018

Complexes d'intersection des compactifications de Baily-Borel : Le cas des variétés de Siegel

In this work, we calculate the trace of a Hecke correspondance composed with a power of the Frobenius endomorphism on the fibre of the intersection complexes of the Baily-Borel compactification of a Siegel modular variety. Our main tool is Pink's theorem about the restriction to the strata of the Baily-Borel compactification of the direct image of a local system on the Shimura variety. To use this theorem, we give a new construction of the intermediate extension of a pure perverse sheaf as a weight truncation of the full direct image. More generally, we are able to define analogs in positive characteristic of the weighted cohomology complexes introduced by Goresky, Harder and MacPherson.

math.NT↗
arXiv2018

Cohomologie d'intersection des variétés modulaires de Siegel, suite

In this work, we study the intersection cohomology of Siegel modular varieties. The goal is to express the trace of a Hecke operator composed with a power of the Frobenius endomorphism (at a good place) on this cohomology in terms of the geometric side of Arthur's invariant trace formula for well-chosen test functions. Our main tools are the results of Kottwitz about the contribution of the cohomology with compact support and about the stabilization of the trace formula, Arthur's $L^2$ trace formula and the fixed point formula of the author. We "stabilize" this last formula, ie express it as a sum of stable distributions on the general symplectic groups and its endoscopic groups, and obtain the formula conjectured by Kottwitz. Applications of the results of this article have already been given by Kottwitz, assuming Arthur's conjectures. Here, we give weaker unconditional applications in the cases of the groups $\mathrm{GSp}_4$ and $\mathrm{GSp}_6$.

math.RT↗
arXiv2018

Construction de représentations galoisiennes de torsion, d'après Peter Scholze

Le but de cet exposé est de présenter les résultats de Scholze sur la construction des représentations galoisiennes de torsion associées aux caractères de l'algèbre de Hecke apparaissant dans la cohomologie de torsion des espaces localement symétriques associés au groupe $\mathrm{GL}_n$. La phrase précédente est expliquée plus en détail dans la section 1. Toutes les erreurs et inexactitudes dans ce texte sont bien entendu dues à l'auteur et non à Scholze. -- The goal of this lecture is to present Scholze's results about the construction of torsion Galois representations associated to the characters of the Hecke algebra appearing in the cohomology of a locally symmetric variety for the group $\mathrm{GL}_n$.

math.NT↗