Equidistribution of Hecke Orbits on the Picard group of definite Shimura curves

TL;DR

This paper proves that Hecke orbits of supersingular Drinfeld modules of rank 2 become uniformly distributed on the Picard group of certain Shimura curves over function fields, using automorphic forms techniques.

Contribution

It introduces an automorphic method to establish equidistribution of Hecke orbits on the Picard group of Shimura curves from definite quaternion algebras over function fields.

Findings

Hecke orbits of supersingular Drinfeld modules are equidistributed.

Bounds for coefficients of cuspidal automorphic forms are key to the proof.

The approach applies to Shimura curves over function fields.

Abstract

We prove an equidistribution result about Hecke orbits on the Picard group of Shimura curves coming from definite quaternion algebras over function fields. In particular, we show the equidistribution of Hecke orbits of supersingular Drinfeld modules of rank 2. Our approach is via the automorphic method, using bounds for coefficients of cuspidal automorphic forms of Drinfeld type as the main tool.