Construction de repr\'esentations galoisiennes de torsion, d'apr\`es Peter Scholze

TL;DR

This paper discusses Scholze's groundbreaking work on constructing torsion Galois representations linked to Hecke algebra characters in the cohomology of locally symmetric spaces for GL_n, advancing number theory and algebraic geometry.

Contribution

It presents Scholze's novel methods for constructing torsion Galois representations associated with Hecke algebra characters in the cohomology of locally symmetric varieties for GL_n.

Findings

Construction of torsion Galois representations for GL_n.

Connection between Hecke algebra characters and Galois representations.

Advancement in understanding the cohomology of locally symmetric spaces.

Abstract

Le but de cet expos\'e est de pr\'esenter les r\'esultats de Scholze sur la construction des repr\'esentations galoisiennes de torsion associ\'ees aux caract\`eres de l'alg\`ebre de Hecke apparaissant dans la cohomologie de torsion des espaces localement sym\'etriques associ\'es au groupe $\mathrm{GL}_n$. La phrase pr\'ec\'edente est expliqu\'ee plus en d\'etail dans la section 1. Toutes les erreurs et inexactitudes dans ce texte sont bien entendu dues \`a l'auteur et non \`a 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 $\mathrm{GL}_n$.