Sign choices in the AGM for genus two theta constants

TL;DR

This paper improves algorithms for computing genus 2 theta constants by resolving sign ambiguities through square root choices, eliminating the need for numerical integration and enhancing computational efficiency.

Contribution

It demonstrates that Borchardt sequences in genus 2 can be uniquely determined by appropriate sign choices, extending genus 1 techniques to genus 2.

Findings

Sign choices resolve Borchardt sequence ambiguities

Algorithm achieves quasi-linear time complexity

Eliminates reliance on numerical integration

Abstract

Existing algorithms to compute genus 2 theta constants in quasi-linear time use Borchardt sequences, an analogue of the arithmetic-geometric mean for four complex numbers. In this paper, we show that these Borchardt sequences are given by good choices of square roots only, as in the genus 1 case. This removes the sign indeterminacies in the algorithm without relying on numerical integration.