Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

TL;DR

This paper establishes an upper bound for the trace of Hecke operators on cusp forms, with applications to elliptic curves over finite fields, advancing beyond classical hypotheses.

Contribution

It introduces a new Petersson formula for newforms with general nebentype characters, enabling improved trace bounds and applications to finite field elliptic curves.

Findings

Derived an upper bound for Hecke operator traces.

Developed a Petersson formula for newforms with nebentype characters.

Applied bounds to the analytic theory of elliptic curves over finite fields.

Abstract

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.