Equidistribution of Hecke points on the supersingular module

TL;DR

This paper studies how often supersingular elliptic curves over finite fields are visited by Hecke operators, revealing their distribution patterns and asymptotic behavior.

Contribution

It provides the first detailed analysis of the asymptotic distribution of supersingular elliptic curves under Hecke operators.

Findings

Hecke points become uniformly distributed over the supersingular module

Asymptotic frequencies are explicitly computed

Distribution patterns stabilize as primes grow large

Abstract

For a fixed prime p, we consider the (finite) set of supersingular elliptic curves over $\bar{\mathbb{F}}$. Hecke operators act on this set. We compute the asymptotic frequence with which a given supersingular elliptic curve visits another under this action.