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Effective estimates for traces of singular moduli

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Abstract

Traces of singular moduli can be approximated by exponential sums of quadratic irrationals. Recently Andersen and Duke used theory of Maass forms to estimate generalized twisted traces with power-saving error bounds. We establish an asymptotic formula with effective error bounds for such traces. Our methods depend on an explicit bound for sums of Kloosterman sums on \(\Gamma _{0}(4)\).

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Data availability

The code generated as part of this work is available at [21], including the searching results. The dataset of Maass forms published by Seymour-Howell is available at [20].

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Acknowledgements

The authors thank the referee for a careful reading and with helpful comments and suggestions of an earlier version of this manuscript. The authors also thank Scott Ahlgren for a lot of delightful discussions and suggestions and thank Nick Andersen for insightful comments on our result. The first author was partially supported by the Alfred P. Sloan Foundation’s MPHD Program, awarded in 2017.

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  1. Oscar E. González, Qihang Sun have contributed equally to this work.

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  1. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    Oscar E. González & Qihang Sun

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  1. Oscar E. González
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Correspondence to Qihang Sun.

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González, O.E., Sun, Q. Effective estimates for traces of singular moduli. Res. number theory 10, 29 (2024). https://doi.org/10.1007/s40993-024-00517-6

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