Certified Newton schemes for the evaluation of low-genus theta functions
Abstract
Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N=60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2022
- DOI:
- 10.48550/arXiv.2203.02000
- arXiv:
- arXiv:2203.02000
- Bibcode:
- 2022arXiv220302000K
- Keywords:
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- Mathematics - Numerical Analysis;
- Mathematics - Number Theory;
- 14K25;
- 11Y35;
- 65D20 (Primary);
- 32A10 (Secondary)
- E-Print:
- This paper reproduces and improves on part of arXiv:2010.10094, to be updated separately