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Derivatives of Eisenstein series of weight 2 and intersections of modular correspondences

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Abstract

We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight \(\frac{g}{2}\) and genus g. When \(g=4\), the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices.

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Acknowledgements

Cho is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA2001-04. Yamauchi is partially supported by JSPS Grant-in-Aid for Scientific Research (B) 19H01778. Yamana is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 18K03210. Yamana also thanks Max Planck Institut für Mathematik for an excellent working environment. We would like to thank Stephen Kudla for very stimulating discussions.

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Authors and Affiliations

  1. Department of Mathematics, POSTECH, 77, Cheongam-ro, Nam-gu, Pohang-si, 37673, Gyeongsangbuk-do, Korea

    Sungmun Cho

  2. Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Shunsuke Yamana

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

    Takuya Yamauchi

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  1. Sungmun Cho
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  2. Shunsuke Yamana
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  3. Takuya Yamauchi
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Correspondence to Shunsuke Yamana.

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Communicated by Jens Funke.

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Cho, S., Yamana, S. & Yamauchi, T. Derivatives of Eisenstein series of weight 2 and intersections of modular correspondences. Abh. Math. Semin. Univ. Hambg. 92, 27–52 (2022). https://doi.org/10.1007/s12188-022-00256-4

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  • DOI: https://doi.org/10.1007/s12188-022-00256-4

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