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

TL;DR

This paper presents an algorithm to compute mod p Hecke eigenvalues for modular forms using quaternion algebra techniques, extending Serre's theoretical correspondence with a practical computational method.

Contribution

It introduces a novel combinatorial algorithm for calculating Hecke eigenvalues via quaternion algebra representations, bridging theoretical insights and computational applications.

Findings

Algorithm efficiently computes mod p Hecke eigenvalues.

Provides explicit computational framework based on quaternion algebra.

Extends Serre's correspondence with practical algorithmic implementation.

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.