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The radius of convergence of the p-adic sigma function


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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  1. 1.

    Abramowitz, M., Stegun, I.A., (eds.): Weierstrass elliptic and related functions. Chap. 18 In: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. Dover, New York, pp. 627–671 (1972)

  2. 2.

    André, Y.: Period mappings and differential equations. From \(C\) to \(C_p\), Tôhoku-Hokkaidô lectures in arithmetic geometry, With appendices by Kato, F., Tsuzuki, N., MSJ Memoirs 12, Mathematical Society of Japan, Tokyo (2003)

  3. 3.

    Bannai, K., Kobayashi, S.: Algebraic theta functions and \(p\)-adic interpolation of Eisenstein–Kronecker numbers. Duke Math. J. 153(2), 229–295 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bernardi, D., Goldstein, C., Stephens, N.: Notes \(p\)-adiques sur les courbes elliptiques. J. Reine Angew. Math. 351, 129–170 (1984)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Boxall, J.: \(p\)-adic interpolation of logarithmic derivatives associated to certain Lubin–Tate formal groups. Ann. Inst. Fourier 36(3), 1–27 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Colmez, P.: Périodes \(p\)-adiques des variétés abéliennes. Math. Ann. 292, 629–644 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Colmez, P.: Périodes des variétés abéliennes à Multiplication Complexe. Ann. Math. 138, 625–683 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Colmez, P.: Intégration sur les variétés \(p\)-adiques. Astérisque 248 (1998). ISSN:0303-1179

  9. 9.

    de Shalit, E.: Iwasawa Theory of Elliptic Curves with Complex Multiplication. Academic Press, Boston (1987)

    MATH  Google Scholar 

  10. 10.

    Fontaine, J.-M.: Groupes \(p\)-divisibles sur les corps locaux, Astérisque, No. 47–48. Société Mathématique de France, Paris (1977)

  11. 11.

    Fontaine, J.-M.: Le corps des périodes \(p\)-adiques. With an appendix by Pierre Colmez. Périodes \(p\)-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223, pp. 59–111 (1994)

  12. 12.

    Fujiwara, Y.: On divisibilities of special values of real analytic Eisenstein series. J. Fac. Sci. Univ. Tokyo 35, 393–410 (1988)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Honda, T.: On the theory of commutative formal groups. J. Math. Soc. Jpn. 22, 213–246 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Katz, N.: \(p\)-adic interpolation of real analytic Eisenstein series. Ann. Math. 104, 459–571 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Katz, N.: Formal groups and \(p\)-adic intergration. Astérisque 41–42, 55–65 (1977)

    MATH  Google Scholar 

  16. 16.

    Katz, N.: Cristalline cohomology, Dieudonné modules, and Jacobi sums. In: Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979), Tata Inst. Fundamental. Res., Bombay, pp. 165–246 (1981)

  17. 17.

    Manin, Ju. I., Vishik, S.: \(p\)-adic Hecke series for imaginary quadratic fields. Math. Sbornik V 95 (137), No. 3 (11) (1974)

  18. 18.

    Mazur, B., Tate, J.: The \(p\)-adic sigma function. Duke. Math. 62(3), 663–688 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Ogus, A.: A \(p\)-adic analogue of the Chowla-Selberg formula, \(p\)-adic analysis (Trento, 1989). Lect. Notes Math. 1454, 319–341 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Schneider, P., Teitelbaum, J.: \(p\)-adic Fourier theory. Doc. Math. 6, 447–481 (2001)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Silverman, J.: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106. Springer, New York (1992)

    Google Scholar 

  22. 22.

    Tsuji, T.: \(p\)-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. Math. 137(2), 233–411 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Yasuda, S.: Explicit \(t\)-expansions for the elliptic curve \(y^{2}=4(x^{3}+Ax+B)\). Proc. Jpn. Acad. Ser. A Math. Sci. 89(9), 123–127 (2013)

    Article  MATH  Google Scholar 

  24. 24.

    Weil, A.: Elliptic Functions According to Eisenstein and Kronecker. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  25. 25.

    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

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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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Corresponding author

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).

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