Evaluating modular equations for abelian surfaces

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.

References in corpus (8)

• Fast algorithms for computing isogenies between elliptic curves
• Rigorous computation of the endomorphism ring of a Jacobian
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• A quasi-linear time algorithm for computing modular polynomials in dimension 2
• Sign choices in the AGM for genus two theta constants
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• Counting points on abelian surfaces over finite fields with Elkies's method