Traces of Hecke operators in level 1 and Gaussian hypergeometric functions

TL;DR

This paper derives formulas connecting traces of p-th Hecke operators at level 1 with finite field hypergeometric functions, extending previous results to all odd primes p.

Contribution

It generalizes existing trace formulas to all odd primes p, linking Hecke operator traces with hypergeometric functions and elliptic curve Frobenius traces.

Findings

Formulas for Hecke operator traces in terms of hypergeometric functions.

Extension of previous work to all odd primes p.

Connection between elliptic curve Frobenius traces and hypergeometric functions.

Abstract

We provide formulas for traces of p-th Hecke operators in level 1 in terms of values of finite field 2F1-hypergeometric functions, extending previous work of the author to all odd primes p, instead of only those p=1 (mod 12). We first give a general level 1 trace formula in terms of the trace of Frobenius on a family of elliptic curves, and then we draw on recent work of Lennon to produce level 1 trace formulas in terms of hypergeometric functions for all primes p >3.