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Evaluating modular equations for abelian surfaces

Jean Kieffer

TL;DR

This work develops provably correct numerical methods for evaluating Siegel and Hilbert modular equations at given invariants of abelian surfaces, avoiding full polynomial precomputation despite the enormous size of these polynomials. Central to the approach is a certified genus-2 framework: compute period matrices via the AGM method, reduce to the Siegel/Hilbert fundamental domains, evaluate modular forms at the corresponding CM-like points, and reconstruct evaluations of the modular equations with certified accuracy. The authors establish new complexity and precision-loss bounds for key numerical steps (reduction, AGM, big period matrices, RM structures), and provide quasi-linear-time evaluation results for generic inputs over finite fields and number fields. They also describe practical implementations and delineate how these techniques enable Elkies-type isogeny computations for abelian surfaces, with potential extensions to genus 1. Overall, the paper offers a comprehensive, certified pipeline for evaluating high-dimensional modular equations that scales feasibly with output size and field complexity, opening doors to efficient isogeny-based point counting and related arithmetic-geometric computations.

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.

Evaluating modular equations for abelian surfaces

TL;DR

This work develops provably correct numerical methods for evaluating Siegel and Hilbert modular equations at given invariants of abelian surfaces, avoiding full polynomial precomputation despite the enormous size of these polynomials. Central to the approach is a certified genus-2 framework: compute period matrices via the AGM method, reduce to the Siegel/Hilbert fundamental domains, evaluate modular forms at the corresponding CM-like points, and reconstruct evaluations of the modular equations with certified accuracy. The authors establish new complexity and precision-loss bounds for key numerical steps (reduction, AGM, big period matrices, RM structures), and provide quasi-linear-time evaluation results for generic inputs over finite fields and number fields. They also describe practical implementations and delineate how these techniques enable Elkies-type isogeny computations for abelian surfaces, with potential extensions to genus 1. Overall, the paper offers a comprehensive, certified pipeline for evaluating high-dimensional modular equations that scales feasibly with output size and field complexity, opening doors to efficient isogeny-based point counting and related arithmetic-geometric computations.

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.

Paper Structure

This paper contains 36 sections, 47 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Modular equations
  3. Main results
  4. Overview of the algorithms
  5. Key ingredients of the proofs
  6. Organization of the paper
  7. Notation
  8. Acknowledgements
  9. Modular equations for abelian surfaces
  10. Invariants on the Siegel moduli space
  11. Siegel modular equations
  12. Denominators of Siegel modular equations
  13. Invariants on Hilbert moduli spaces
  14. Hilbert modular equations
  15. Denominators of Hilbert modular equations
  16. ...and 21 more sections

Key Result

Theorem 1.1

Given prime numbers $p$ and $\ell$ and a generic tuple $(j_1,j_2,j_3)\in\mathbb{F}_p^3$, one can evaluate $\Psi_{\ell,k}(j_1,j_2,j_3,X)$ and $\partial_{J_n} \Psi_{\ell,k}(j_1,j_2,j_3,X)$ for $1\leq k,n\leq 3$ as elements of $\mathbb{F}_p[X]$ in time $\hbox{[}.9]{\widetilde{O}}(\ell^6\log p)$.

Theorems & Definitions (94)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • proof
  • Proposition 2.11
  • ...and 84 more