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Values of modular functions at real quadratics and conjectures of Kaneko

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Abstract

In 2008, Kaneko made several interesting observations about the values of the modular j invariant at real quadratic irrationalities. The values of modular functions at real quadratics are defined in terms of their cycle integrals along the associated geodesics. In this paper we prove some of the conjectures of Kaneko for a general modular function.

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Acknowledgements

We thank the referee for his/her many useful remarks which greatly improved the exposition of the paper.

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Authors and Affiliations

  1. Mathematics Department, ETH, 8092, Zürich, Switzerland

    P. Bengoechea & Ö. Imamoglu

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  1. P. Bengoechea
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  2. Ö. Imamoglu
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Correspondence to P. Bengoechea.

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Communicated by Kannan Soundararajan.

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Bengoechea’s research is supported by SNF Grant 173976. Imamoglu’s is research supported by SNF Grant 200021-185014.

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Bengoechea, P., Imamoglu, Ö. Values of modular functions at real quadratics and conjectures of Kaneko. Math. Ann. 377, 249–266 (2020). https://doi.org/10.1007/s00208-020-01979-6

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  • DOI: https://doi.org/10.1007/s00208-020-01979-6

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