Abstract
We show how certain determinants of hyperelliptic periods can be computed using a generalized arithmetic-geometric mean iteration, whose initialisation parameters depend only on the position of the ramification points. Special attention is paid to the explicit form of this dependence and the signs occurring in the real domain.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Almkvist, G., Berndt, B.: Gauss, Landen, Ramanujan, the arithmetic–geometric mean, ellipses, π, and the Ladies Diary. Am. Math. Mon. 95, 581–608 (1988)
Borchardt, C.W.: Über das arithmetisch-geometrische Mittel aus vier Elementen. Monatsbericht der königlichen Akademie zu Berlin vom (November 1876)
Borwein, J.M., Borwein, P.B.: Pi and the AGM. Wiley-Interscience, New York (1998)
Bost, J.-B., Mestre, J.-F.: Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2, LMENS-88-13 (1988). Département de Mathématiques et d’Informatique, Ecole Normale Supériere
Cox, D.A.: The arithmetic–geometric mean of Gauss. Enseign. Math. 30, 275–330 (1984)
Cox, D.A.: Gauss and the arithmetic–geometric mean. Not. Am. Math. Soc. 32(2), 147–151 (1985)
Donagi, R., Livné, R.: The arithmetic–geometric mean and isogenies for curves of higher genus. Ann. Sc. Norm. Super. Pisa, Cl. Sci., Sr. 4 28(2), 323–339 (1999)
Dupont, R.: Moyenne arithmético-géométrique, suites de Borchardt et applictions. Thèse, Ecole Polytechnique (2006). Available at http://www.lix.polytechnique.fr/Labo/Regis.Dupont/
Dupont, R.: Moyenne de Borchardt sur les complexes. Preprint
Fay, J.D.: Theta Functions on Riemann Surfaces. Lecture Notes in Math., vol. 352. Springer, Berlin (1973)
Gauss, C.F.: Determinatio attractionis quam in punctum quodvis positionis datae excerceret planeta si eius massa per totam orbitam ratione temporis quo singulae partes describuntur uniformiter esset dispertita. Gött. Gel. Anz. 3, 331–355 (1818)
Griffiths, Ph., Harris, J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1978)
Igusa, J.-I.: Theta Functions. Grundlehren Math. Wiss., vol. 194. Springer, Berlin (1972)
Jarvis, F.: Higher genus arithmetic–geometric means. Ramanujan J. 17(1), 1–17 (2008)
Lagrange, J.L.: Sur une nouvelle Méthode de Calcul Intégrale pour différentielles affectées d’un radical carré. Mem. Acad. R. Sci. Turin II 2, 252–312 (1784–1785)
Lehavi, D., Ritzenthaler, C.: An explicit formula for the arithmetic–geometric mean in genus 3. Exp. Math. 16(4), 421–440 (2007)
Mestre, J.-F.: Moyenne de Borchardt et intégrales elliptiques. C. R. Acad. Sci. Paris, Série I 313, 273–276 (1991)
Mumford, D.: Tata Lectures on Theta I. Progress in Mathematics, vol. 28. Birkhäuser, Boston (1983)
Mumford, D.: Tata Lectures on Theta II. Progress in Mathematics, vol. 43. Birkhäuser, Boston (1984)
Richelot, F.: Essai sur une méthode générale pour déterminer la valeur des intégrales ultra-elliptiques, fondée sur des transformations remarquables de ces transcendentes. C. R. Acad. Sci. Paris 2, 622–627 (1836)
Thomae, J.: Beitrag zur Bestimmung von θ(0,0,…,0) durch die Klassenmoduln algebraischer Functionen. Crelle J. 71, 201–222 (1870)
Author information
Authors and Affiliations
Corresponding author
Additional information
Part of the work was done with the support of the SFB/TR 45 during a stay of the first author at the Johannes Gutenberg University in Mainz.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Spandaw, J., van Straten, D. Hyperelliptic integrals and generalized arithmetic–geometric mean. Ramanujan J 28, 61–78 (2012). https://doi.org/10.1007/s11139-011-9353-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-011-9353-7