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A continuity of cycle integrals of modular functions

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Abstract

In this paper, we study a continuity of the “values” of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean topology.

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Acknowledgements

I would like to show my greatest appreciation to Professor Takuya Yamauchi for introducing me to this topic and giving much advice. I am deeply grateful to Dr. Toshiki Matsusaka for many discussions and valuable comments. I would like to express my gratitude to Paloma Bengoechea for her encouragement. Finally, we thank the referee for helpful suggestions and comments which substantially improved the presentation of our paper.

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  1. Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku, Sendai, 980-8578, Japan

    Yuya Murakami

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  1. Yuya Murakami
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Correspondence to Yuya Murakami.

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Murakami, Y. A continuity of cycle integrals of modular functions. Ramanujan J 55, 1177–1187 (2021). https://doi.org/10.1007/s11139-020-00307-5

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  • DOI: https://doi.org/10.1007/s11139-020-00307-5

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