Sign choices in the AGM for genus two theta constants

Kieffer, Jean

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2022), pp. 37-58.

Les algorithmes existants pour le calcul de thêta-constantes en genre $2$ en temps quasilinéaire utilisent des suites de Borchardt, un analogue de la moyenne arithmético-géométrique pour quatre nombres complexes. Dans cet article, nous montrons que ces suites de Borchardt sont constituées uniquement de bons choix de signes, comme c’est le cas en genre $1$. Ce résultat permet de lever les indéterminations de signes lors du calcul de thêta-constantes en genre $2$ sans recours à l’intégration numérique.

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 only given by good choices of square roots, as in the genus $1$ case. This removes the sign indeterminacies when computing genus $2$ theta constants without relying on numerical integration.

Classification : 11Y35, 11Y16, 11F41, 11F27
Mots clés : Theta functions, Genus $2$, Algorithms, Borchardt mean

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