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Computing theta functions in quasi-linear time in genus two and above

Part of: Abelian varieties and schemes Curves Deformations of analytic structures

Published online by Cambridge University Press:  26 August 2016

Hugo Labrande  and
Emmanuel Thomé
Hugo Labrande
Affiliation:
Université de Lorraine, LORIA (UMR CNRS 7503), INRIA Nancy, 615 rue du jardin botanique, 54602 Villers-lès-Nancy Cedex, France University of Calgary, Department of Computer Science, 2500 University Dr NW, Calgary, Alberta, Canada T2N 1N4 email hugo.labrande@inria.fr
Emmanuel Thomé
Affiliation:
Université de Lorraine, LORIA (UMR CNRS 7503), INRIA Nancy, 615 rue du jardin botanique, 54602 Villers-lès-Nancy Cedex, France email emmanuel.thome@inria.fr

Abstract

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We outline an algorithm to compute $\unicode[STIX]{x1D703}(z,\unicode[STIX]{x1D70F})$ in genus two in quasi-linear time, borrowing ideas from the algorithm for theta constants and the one for $\unicode[STIX]{x1D703}(z,\unicode[STIX]{x1D70F})$ in genus one. Our implementation shows a large speed-up for precisions as low as a few thousand decimal digits. We also lay out a strategy to generalize this algorithm to genus  $g$ .

MSC classification

Primary: 14K25: Theta functions
Secondary: 14H42: Theta functions; Schottky problem 32G81: Applications to physics

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