Computation of Hecke eigenvalues (mod $p$) via quaternions

Abstract

In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod $p$ modular forms (of fixed level $\Gamma (N)$ coprime to $p$, and any weight $k$) are the same as those arising from functions $\Omega (N)\to {\bar{\mathbb{F}}}_p$, where $\Omega (N)$ is some double quotient of $D^{\times }({\mathbb{A}}_f)$ and $D$ is the unique quaternion algebra over $\mathbb{Q}$ ramified at $\{p,\infty \}$. We present an algorithm which then computes these Hecke eigenvalues on the quaternion side in a combinatorial manner.

27 pages. Edited to correct an error enumerating double cosets