Fast evaluation of Riemann theta functions in any dimension

TL;DR

This paper introduces a fast, quasi-linear time algorithm for numerically evaluating Riemann theta functions in any dimension, significantly improving performance over existing methods and enabling advanced applications in algebraic geometry.

Contribution

The authors develop and implement a novel algorithm for efficient Riemann theta function evaluation applicable in any dimension, with demonstrated practical and theoretical advantages.

Findings

Algorithm achieves quasi-linear time complexity.

Implementation in FLINT library outperforms existing software.

Application to abelian varieties yields explicit polynomials with conjectural Galois groups.

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 $\mathrm{SL}_2(\mathbb{F}_{64})$.