Existing algorithms to compute genus 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 case. This removes the sign indeterminacies when computing genus theta constants without relying on numerical integration.
Les algorithmes existants pour le calcul de thêta-constantes en genre 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 . Ce résultat permet de lever les indéterminations de signes lors du calcul de thêta-constantes en genre sans recours à l’intégration numérique.
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Keywords: Theta functions, Genus $2$, Algorithms, Borchardt mean
Jean Kieffer 1
@article{PMB_2022____37_0, author = {Jean Kieffer}, title = {Sign choices in the {AGM} for genus two theta constants}, journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres}, pages = {37--58}, publisher = {Presses universitaires de Franche-Comt\'e}, year = {2022}, doi = {10.5802/pmb.45}, language = {en}, url = {https://pmb.centre-mersenne.org/articles/10.5802/pmb.45/} }
TY - JOUR AU - Jean Kieffer TI - Sign choices in the AGM for genus two theta constants JO - Publications mathématiques de Besançon. Algèbre et théorie des nombres PY - 2022 SP - 37 EP - 58 PB - Presses universitaires de Franche-Comté UR - https://pmb.centre-mersenne.org/articles/10.5802/pmb.45/ DO - 10.5802/pmb.45 LA - en ID - PMB_2022____37_0 ER -
%0 Journal Article %A Jean Kieffer %T Sign choices in the AGM for genus two theta constants %J Publications mathématiques de Besançon. Algèbre et théorie des nombres %D 2022 %P 37-58 %I Presses universitaires de Franche-Comté %U https://pmb.centre-mersenne.org/articles/10.5802/pmb.45/ %R 10.5802/pmb.45 %G en %F PMB_2022____37_0
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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