Computation of Hecke eigenvalues (mod p) via quaternions

2022/11/09 by Yiannis Fam, Fam, Yiannis
Mathematics · #11R52 (Primary) 11F30 (Secondary) #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2211.04708

openalex publication_date 2022/11/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a 1987 letter, Serre proves that the systems of Hecke eigenvalues arising from mod p modular forms (of fixed level Γ(N) coprime to p, and any weight k) are the same as those arising from functions Ω(N) → \mathbb F_p, where Ω(N) is some double quotient of D^× (\mathbb A_f) and D is the unique quaternion algebra over \mathbb Q ramified at \p,∞\. We present an algorithm which then computes these Hecke eigenvalues on the quaternion side in a combinatorial manner.

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