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Summation of Poincaré Theta Series in the Schottky Model

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Abstract

New algorithms for approximate summation of Poincaré theta series in the Schottky model of real hyperelliptic curves are proposed. As a result, for the same output accuracy estimate, the amount of computations is reduced by several times in the case of slow convergence and by tens of percent in the usual situations. For the sum of the Poincaré series over the subtree on descendants of a given node, a new estimate in terms of the series member at this node is obtained.

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ACKNOWLEDGMENTS

I am grateful to my supervisor A.B. Bogatyrev for suggesting the problem and participating in valuable discussions. I am also grateful to S.A. Goreinov, O.A. Grigor’ev, and M.S. Smirnov for valuable discussions.

Funding

The research concerning the new estimate for the sum of the Poincaré series was supported by the Moscow Center for Fundamental and Applied Mathematics, Department at the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (agreement no. 075-15-2019-1624). The rest of this study was supported by the Russian Science Foundation, grant no. 21-11-00325.

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  1. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, 119333, Moscow, Russia

    S. Yu. Lyamaev

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  1. S. Yu. Lyamaev
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Correspondence to S. Yu. Lyamaev.

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Translated by I. Ruzanova

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Lyamaev, S.Y. Summation of Poincaré Theta Series in the Schottky Model. Comput. Math. and Math. Phys. 62, 1059–1073 (2022). https://doi.org/10.1134/S0965542522070053

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  • DOI: https://doi.org/10.1134/S0965542522070053

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