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Distribution of toric periods of modular forms on definite quaternion algebras

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Abstract

Let D be a definite quaternion algebra over \(\mathbb {Q}\) and \(\mathcal {O}\) an Eichler order in D of square-free level. We study distribution of the toric periods of algebraic modular forms of level \(\mathcal {O}\). We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on \(\mathcal {O}\), we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld’s conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic L-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients \(\{a(n)\}_n\) of weight \(\frac{3}{2}\) modular forms, where n ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.

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Data availability

The data that support the findings of this study are available in http://wakatsuki.w3.kanazawa-u.ac.jp/Figures.html.

Notes

  1. Another way to define \(\coprod _{[I]\in {{\,\mathrm{Cl}\,}}(\mathcal {O})}{{\,\mathrm{Emb}\,}}(\mathfrak {o}, \mathcal {O}(I))_{/\sim }\) is to consider the \(D^\times \)-conjugate action on the set \(\coprod _I {{\,\mathrm{Emb}\,}}(\mathfrak {o}, \mathcal {O}(I))_{/\sim }\), where I runs over all right fractional \(\mathcal {O}\)-ideals.

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Acknowledgements

The authors thank Tamotsu Ikeda, Toshiki Matsusaka, Masataka Chida and Kimball Martin for helpful discussions and valuable comments. The authors also thank Siegfried Böcherer, Rainer Schulze-Pillot and Winfried Kohnen for answering many questions. M.S. was partially supported by Grant-in-Aid for JSPS Fellows No.20J00434. S.W. was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No.18K03235 and (B) No.21H00972. S.Y. was partially supported by JSPS Grant-in-Aid for Scientific Research (C) No.20K03537.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakumamachi, Kanazawa, Ishikawa, 920-1192, Japan

    Miyu Suzuki & Satoshi Wakatsuki

  2. Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, Tokyo, 192-0397, Japan

    Shun’ichi Yokoyama

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  1. Miyu Suzuki
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Suzuki, M., Wakatsuki, S. & Yokoyama, S. Distribution of toric periods of modular forms on definite quaternion algebras. Res. number theory 8, 90 (2022). https://doi.org/10.1007/s40993-022-00389-8

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