Computing newforms using supersingular isogeny graphs

TL;DR

This paper introduces algorithms leveraging supersingular isogeny graphs and Wiedemann's algorithm to compute and verify weight 2 cusp forms of prime levels up to 2,000,000, significantly advancing computational methods in modular forms.

Contribution

It presents novel algorithms based on supersingular isogeny graphs for computing q-expansions and verifying the uniqueness of cusp forms at high prime levels.

Findings

Successfully computed q-expansions for all weight 2 cusp forms up to level 2,000,000.

Verified the uniqueness of cusp forms of dimension 7 or more in specified levels.

Demonstrated the effectiveness of Mestre's graph method combined with Wiedemann's algorithm.

Abstract

We describe an algorithm that we used to compute the q-expansions of all weight 2 cusp forms of prime level at most 2,000,000 and dimension at most 6. We also present an algorithm that we used to verify that there was only one cusp form of dimension 7 or more per Atkin-Lehner eigenspace for prime levels between 10,000 and 1,000,000. Our algorithm is based on Mestre's M\'ethode des Graphes, and involves supersingular isogeny graphs and Wiedemann's algorithm for finding the minimal polynomial of sparse matrices over finite fields.