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Fast evaluation of Riemann theta functions in any dimension

Noam D. Elkies, Jean Kieffer

TL;DR

This work addresses the challenge of numerically evaluating Riemann theta functions in arbitrary genus with certified error control. It introduces a uniform quasi-linear algorithm for reduced inputs, backed by a robust FLINT implementation that tracks all arithmetic errors and supports derivatives, with complexity bounds of the form $2^{O(g\log^2 g)} \mathsf{M}(N)\log N$. The authors also demonstrate a powerful application to the explicit inverse Galois problem by computing theta-constants on genus-6 abelian varieties to produce degree-65 polynomials with conjectural Galois group $\mathrm{SL}_2(\mathbb{F}_{64})$, illustrating the practical reach of their method. Collectively, this work advances both the theory and computation of theta functions, enabling high-precision, high-genus evaluations and enabling new number-theoretic explorations across modular forms, abelian varieties, and Galois theory.

Abstract

We describe an algorithm to numerically evaluate Riemann theta functions in any dimension in quasi-linear time in terms of the required precision, uniformly on reduced input. This algorithm is implemented in the FLINT number theory library and vastly outperforms existing software. As an application, we evaluate the theta constants attached to certain special abelian varieties of dimension 6 to construct explicit polynomials of degree 65 over $\mathbb{Q}$ with conjectural Galois group $\mathrm{SL}_2(\mathbb{F}_{64})$.

Fast evaluation of Riemann theta functions in any dimension

TL;DR

This work addresses the challenge of numerically evaluating Riemann theta functions in arbitrary genus with certified error control. It introduces a uniform quasi-linear algorithm for reduced inputs, backed by a robust FLINT implementation that tracks all arithmetic errors and supports derivatives, with complexity bounds of the form . The authors also demonstrate a powerful application to the explicit inverse Galois problem by computing theta-constants on genus-6 abelian varieties to produce degree-65 polynomials with conjectural Galois group , illustrating the practical reach of their method. Collectively, this work advances both the theory and computation of theta functions, enabling high-precision, high-genus evaluations and enabling new number-theoretic explorations across modular forms, abelian varieties, and Galois theory.

Abstract

We describe an algorithm to numerically evaluate Riemann theta functions in any dimension in quasi-linear time in terms of the required precision, uniformly on reduced input. This algorithm is implemented in the FLINT number theory library and vastly outperforms existing software. As an application, we evaluate the theta constants attached to certain special abelian varieties of dimension 6 to construct explicit polynomials of degree 65 over with conjectural Galois group .

Paper Structure

This paper contains 37 sections, 47 theorems, 199 equations, 8 figures, 15 algorithms.

Table of Contents

  1. Introduction
  2. Riemann theta functions
  3. State of the art on evaluation algorithms
  4. Our contributions
  5. Overview of the fast algorithm
  6. Structure of the paper
  7. Notation
  8. Argument reduction and transformation formulas
  9. The reduction algorithm
  10. Reduced matrices
  11. Reducing the first argument
  12. Argument reduction and interval arithmetic
  13. The transformation formula
  14. Decomposition in elementary matrices
  15. Summation algorithms
  16. ...and 22 more sections

Key Result

Theorem 1.4

Given $g\in \mathbb{Z}_{\geq 1}$, $N\in \mathbb{Z}_{\geq 2}$, and a reduced point $(z,\tau)\in \mathbb{C}^g\times \mathcal{H}_g$, one can compute $\mathop{\mathrm{\hbox{$\widetilde{\space}$}\theta}}\nolimits_{a,b}(z,\tau)$ for all characteristics $a,b\in \{0,1\}^g$ up to an error of at most $2^{-N}$

Figures (8)

  • Figure 1: Comparing \ref{['thm:summation-bound-old', 'thm:summation-bound-new']} with $\mathop{\mathrm{Im}}\nolimits(\tau) = I_g$.
  • Figure 2: Possible workflow for three input vectors $0,z_1,z_2$ with $h=2$.
  • Figure 3: The tree $T(t,3)$.
  • Figure 4: Time in seconds to evaluate $\theta_{0,0}(0,\tau)$ by summation.
  • Figure 5: Time in seconds to evaluate $\theta_{0,0}(0,\tau)$ with FLINT, $1\leq g\leq 4$.
  • ...and 3 more figures

Theorems & Definitions (108)

  • Theorem 1.4
  • Definition 2.5
  • Proposition 2.7
  • proof
  • Definition 2.8
  • Proposition 2.9
  • Remark 2.11
  • Definition 2.12
  • Lemma 2.13
  • proof
  • ...and 98 more