Traces of Hecke Operators via Hypergeometric Character Sums

TL;DR

This paper derives explicit formulas for Hecke operator traces on cusp forms using hypergeometric character sums, providing a uniform geometric approach applicable to modular and Shimura curves.

Contribution

It introduces a new uniform geometric method to compute Hecke traces and eigenvalues using hypergeometric character sums, extending to elliptic and Shimura curves.

Findings

Explicit formulas for Hecke traces in terms of hypergeometric sums

Method applies uniformly to modular and Shimura curves

Can be used to compute eigenvalues of Hecke operators

Abstract

In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well.