Sign choices in the AGM for genus two theta constants

Jean Kieffer (1, 2)

1 LFANT - Lithe and fast algorithmic number theory
2 IMB - Institut de Mathématiques de Bordeaux

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.

Mots clés: Theta functions; Algorithms; Borchardt sequences

Domaines: Théorie des nombres [math.NT]

Jean Kieffer. Sign choices in the AGM for genus two theta constants. 2020. ⟨hal-02967220v1⟩