QM abelian varieties, hypergeometric character sums and modular forms

TL;DR

This paper presents explicit formulas involving hypergeometric character sums for traces of Hecke operators on cusp forms associated with specific Fuchsian groups derived from quaternion algebras, advancing understanding of their arithmetic properties.

Contribution

It provides new explicit formulas connecting hypergeometric character sums with Hecke operator traces on modular forms for certain quaternion-derived Fuchsian groups.

Findings

Explicit formulas for Hecke operator traces involving hypergeometric sums

Application to groups from quaternion algebra of discriminant 6

Enhanced understanding of modular forms associated with quaternionic Fuchsian groups

Abstract

This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(\Gamma)$ of weight $k$ for certain arithmetically defined Fuchsian subgroups $\Gamma\subset \mathrm{SL}_2 (\mathbf{R})$. In particular we consider the groups attached to the quaternion division algebra $B_6$ over $\mathbf{Q}$ of discriminant 6.