Evaluating modular equations for abelian surfaces

TL;DR

This paper develops efficient algorithms for evaluating modular equations of Siegel and Hilbert types for abelian surfaces, leveraging complex approximations and establishing new correctness and complexity results for key numerical algorithms.

Contribution

It introduces novel algorithms for evaluating modular equations of abelian surfaces over various fields, with proven correctness and complexity bounds, including for specific quadratic fields.

Findings

Algorithms are efficient and provably correct for key cases.

New complexity bounds for reduction, AGM, and period matrix computations.

Applicable to abelian surfaces over number fields and finite fields.

Abstract

We design efficient algorithms to evaluate modular equations of Siegel and Hilbert type for abelian surfaces over number fields or finite fields using complex approximations. Their output is provably correct when the associated graded ring of modular forms over Z is explicitly known; this includes the Siegel case, and the Hilbert case for the quadratic fields of discriminant 5 and 8. As part of the proofs, we establish new correctness and complexity results for certain key numerical algorithms on period matrices in genus 2, namely the reduction algorithm to the fundamental domain, the AGM method, and computing big period matrices and RM structures.