Publications Mathématiques de Besançon

Algèbre et Théorie des Nombres
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/

[1] Carl W. Borchardt Theorie des arithmetisch-geometrisches Mittels aus vier Elementen, Gesammelte Werke, Reimer, 1888, pp. 373-431

[2] Jean-Benoît Bost; Jean-François Mestre Moyenne arithmético-géométrique et périodes de courbes de genre 1 et 2, Gaz. Math., Soc. Math. Fr., Volume 38 (1988), pp. 36-64 | Zbl

[3] David A. Cox The arithmetic-geometric mean of Gauss, Enseign. Math., Volume 30 (1984), pp. 275-330 | MR | Zbl

[4] Régis Dupont Moyenne arithmético-géométrique, suites de Borchardt et applications, Ph. D. Thesis, École polytechnique (2006)

[5] Régis Dupont Fast evaluation of modular functions using Newton iterations and the AGM, Math. Comput., Volume 80 (2011) no. 275, pp. 1823-1847 | DOI | MR | Zbl

[6] Andreas Enge The complexity of class polynomial computation via floating point approximations, Math. Comput., Volume 78 (2009) no. 266, pp. 1089-1107 | DOI | MR | Zbl

[7] Andreas Enge Computing modular polynomials in quasi-linear time, Math. Comput., Volume 78 (2009) no. 267, pp. 1809-1824 | DOI | MR | Zbl

[8] Andreas Enge; Emmanuel Thomé Computing class polynomials for abelian surfaces, Exp. Math., Volume 23 (2014), pp. 129-145 | DOI | MR | Zbl

[9] Eberhard Freitag; Ricardo Salvati Manni On the variety associated to the ring of theta constants in genus 3, Am. J. Math., Volume 141 (2019) no. 3, pp. 705-732 | DOI | MR | Zbl

[10] Carl F. Gauss Werke, Dietrich, 1868

[11] Philipp Habegger; Fabien Pazuki Bad reduction of genus 2 curves with CM Jacobian varieties, Compos. Math., Volume 153 (2017) no. 12, pp. 2534-2576 | DOI | MR | Zbl

[12] M. Hohenwarter; M. Borcherds; G. Ancsin; B. Bencze; M. Blossier; J. Éliás; K. Frank; L. Gál; A. Hofstätter; F. Jordan; B. Karacsony; Z. Konečný; Z. Kovács; W. Küllinger; E. Lettner; S. Lizefelner; B. Parisse; C. Solyom-Gecse; M. Tomaschko GeoGebra 6.0.588.0, 2020

[13] Jun-Ichi Igusa On the graded ring of theta-constants, Am. J. Math., Volume 86 (1964) no. 1, pp. 219-246 | DOI | MR | Zbl

[14] Jun-Ichi Igusa On the graded ring of theta-constants (II), Am. J. Math., Volume 88 (1966) no. 1, pp. 221-236 | DOI | MR | Zbl

[15] Jun-Ichi Igusa Theta functions, Springer, 1972 | DOI | MR

[16] Frazer Jarvis Higher genus arithmetic-geometric means, Ramanujan J., Volume 17 (2008) no. 1, pp. 1-17 | DOI | MR | Zbl

[17] Helmut Klingen Introductory lectures on Siegel modular forms, Cambridge Studies in Advanced Mathematics, 20, Cambridge University Press, 1990 | DOI

[18] Hugo Labrande Computing Jacobi’s θ in quasi-linear time, Math. Comput., Volume 87 (2018), pp. 1479-1508 | DOI | MR | Zbl

[19] Hugo Labrande; Emmanuel Thomé Computing theta functions in quasi-linear time in genus 2 and above, LMS J. Comput. Math., Volume 19A (2016), pp. 163-177 special issue: Algorithmic Number Theory Symposium (ANTS XII) | DOI | MR | Zbl

[20] Enea Milio A quasi-linear time algorithm for computing modular polynomials in dimension 2, LMS J. Comput. Math., Volume 18 (2015), pp. 603-632 | DOI | MR | Zbl

[21] Enea Milio; Damien Robert Modular polynomials on Hilbert surfaces, J. Number Theory, Volume 216 (2020), pp. 403-459 | DOI | MR | Zbl

[22] Pascal Molin; Christian Neurohr Computing period matrices and the Abel–Jacobi map of superelliptic curves, Math. Comput., Volume 88 (2019) no. 316, pp. 847-888 | DOI | MR | Zbl

[23] David Mumford Tata lectures on theta. I, Progress in Mathematics, 28, Birkhäuser, 1983 | DOI | MR

[24] Marco Streng Complex multiplication of abelian surfaces, Ph. D. Thesis, Universiteit Leiden (2010)

[25] Marco Streng Computing Igusa class polynomials, Math. Comput., Volume 83 (2014), pp. 275-309 | DOI | MR | Zbl

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