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The radius of convergence of the p-adic sigma function

Abstract

The purpose of this article is to investigate the radius of convergence of the p-adic sigma function of elliptic curves, especially when p is a prime of supersingular reduction. As an application, we prove certain p-divisibility of critical values of Hecke L-functions of imaginary quadratic fields at inert primes.

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Acknowledgments

The authors would like to deeply thank the referee for carefully reading an earlier version of this manuscript, and for the thoughtful and insightful comments which helped to greatly improve the mathematical precision and overall presentation of the paper. The major part of this work was done while the first author was visiting the École Normale Supérieure at Paris, and the second author the Université Paris 6. The second author would like to thank Pierre Colmez for discussion concerning this article.

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Correspondence to Kenichi Bannai.

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This work was supported in part by KAKENHI (21674001, 25707001, 26247004, 15H03610, 16K13742), and the JSPS Core-to-core program “Foundation of a Global Research Cooperative Center in Mathematics focused on Number Theory and Geometry”.

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Bannai, K., Kobayashi, S. & Yasuda, S. The radius of convergence of the p-adic sigma function. Math. Z. 286, 751–781 (2017). https://doi.org/10.1007/s00209-016-1783-x

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Keywords

  • Supersingular Reduction
  • Supersingular Case
  • Colmez
  • Lubin-Tate Group
  • Ordinary Reduction