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The sigma function over a family of curves with a singular fiber

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Abstract

In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves Xs defined by the equation

$${y^3} = x\left( {x - s} \right)\left( {x - {b_1}} \right)\left( {x - {b_2}} \right)$$

in the affine (x, y) plane, for s ∈ Dε:= {s ∈ ℂ∥s∣ < ε}. We compare the sigma function over the punctured disc Dµ*:= Dε ∖ {0} with the extension over s = 0 that specializes to the sigma function of the normalization \({X_{\hat 0}}\) of the singular curve Xs=0 by investigating explicitly the behavior of a basis of the first algebraic de Rham cohomology group and its period integrals. We demonstrate, using modular properties, that sigma, unlike the theta function, has a limit. In particular, we obtain the limit of the theta characteristics and an explicit description of the theta divisor translated by the Riemann constant.

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Acknowledgments

The third-named author thanks Tadashi Ashikaga for helpful and crucial comments, which led to Appendix C. Further, he is also grateful to Chris Eilbeck, Victor Enolskii and Yoshihiro Ônishi for critical discussions and comments, and Takeo Ohsawa and Hajime Kaji for valuable comments. The third named author thanks the participants in Numadu-Shizuoka Kenkyukai and, specially, its organizer Yoshinori Machida. The second- and third-named authors were supported by the Grant-in-Aid for Scientific Research (C) of Japan Society for the Promotion of Science, Grant No. 18K04830 and Grant No. 16K05187 respectively. The first- and the third-named authors thank Victor Enolskii, Julia Bernatska and Tony Shaska for their hospitality at the conference at Kiev August, 2018. The authors thank the anonymous reviewer for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Polytechnic University of Catalonia, Barcelona, 08034, Spain

    Yuri Fedorov

  2. Department of Mathematics, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Atsugi, Kanagawa, 243-0292, Japan

    Jiyro Komeda

  3. Graduate School of Natural Science & Technology, Kanazawa University, Kakuma Kanazawa, 920-1192, Japan

    Shigeki Matsutani

  4. Department of Mathematics and Statistics, Boston University, Boston, MA, 02215-2411, USA

    Emma Previato

  5. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

    Kazuhiko Aomoto

Authors
  1. Yuri Fedorov
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  2. Jiyro Komeda
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  3. Shigeki Matsutani
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  4. Emma Previato
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  5. Kazuhiko Aomoto
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Correspondence to Shigeki Matsutani.

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Fedorov, Y., Komeda, J., Matsutani, S. et al. The sigma function over a family of curves with a singular fiber. Isr. J. Math. 250, 345–402 (2022). https://doi.org/10.1007/s11856-022-2340-4

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  • DOI: https://doi.org/10.1007/s11856-022-2340-4

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