Equidistribution in supersingular Hecke orbits

TL;DR

This paper proves an optimal equidistribution result for Hecke operators on supersingular Hecke orbits in Shimura varieties, linking convergence rates to known bounds from the Ramanujan conjecture for automorphic representations.

Contribution

It establishes an equidistribution theorem with optimal convergence rates for Hecke orbits on certain Shimura varieties, connecting these rates to automorphic bounds.

Findings

Proves equidistribution of Hecke orbits in supersingular strata.

Links convergence rates to Ramanujan bounds for automorphic representations.

Achieves optimal estimates on the rate of convergence.

Abstract

We prove an equidistribution result for Hecke operators acting on the basic stratum of certain Shimura varieties. We relate the rate of convergence to the bounds from the Ramanujan conjecture of certain cuspidal automorphic representations on Gl_n for which this conjecture is known, and therefore we obtain optimal estimates on the rate of convergence.