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Sign choices in the AGM for genus two theta constants
[Choix de signes dans l’AGM pour les thêta constantes en genre deux]
Jean Kieffer1
1 Harvard University, Mathematics Department, Cambridge, MA, 02138, United States
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.

Reçu le :
Publié le :
DOI : 10.5802/pmb.45
Classification : 11Y35, 11Y16, 11F41, 11F27
Mots-clés : Theta functions, Genus $2$, Algorithms, Borchardt mean

Jean Kieffer 1

1 Harvard University, Mathematics Department, Cambridge, MA, 02138, United States
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Jean Kieffer. Sign choices in the AGM for genus two theta constants. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2022), pp. 37-58. doi : 10.5802/pmb.45. https://pmb.centre-mersenne.org/articles/10.5802/pmb.45/

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