Traces of Hecke Operators via Hypergeometric Character Sums
Number Theory
2025-03-05 v2
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.
Keywords
Cite
@article{arxiv.2408.02918,
title = {Traces of Hecke Operators via Hypergeometric Character Sums},
author = {Jerome W. Hoffman and Wen-Ching Winnie Li and Ling Long and Fang-Ting Tu},
journal= {arXiv preprint arXiv:2408.02918},
year = {2025}
}