Asymptotique des nombres de Betti des vari\'et\'es arithm\'etiques

TL;DR

This paper investigates the growth of Betti numbers in towers of arithmetic varieties, specifically Siegel and orthogonal group varieties, using Waldspurger's theorem to establish bounds aligned with conjectures by Sarnak and Xue.

Contribution

It provides new bounds on Betti number growth for certain arithmetic varieties, connecting automorphic forms and geometric topology.

Findings

Established lower and upper bounds for Betti numbers growth.

Applied Waldspurger's theorem to geometric problems.

Supported conjectures by Sarnak and Xue.

Abstract

We study the question of the growth of Betti numbers of certain arithmetic varieties in tower of congruence coverings. In fact, our results are about Siegel varieties and varieties associated to orthogonal groups. We explain how a theorem of Waldspurger can be used to obtain lower and upper bound. Our results are in the direction of conjectures made by Sarnak and Xue.