Skip to main content
SpringerLink
Log in

The radius of convergence of the p-adic sigma function

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Colmez, P.: Intégration sur les variétés \(p\)-adiques. Astérisque 248 (1998). http://smf4.emath.fr/Publications/Asterisque/1998/248/html/smf_ast_248.html. ISSN:0303-1179

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

    MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama, 223-8522, Japan

    Kenichi Bannai

  2. KiPAS, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama, 223-8522, Japan

    Kenichi Bannai

  3. Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-Ku, Fukuoka, 819-0395, Japan

    Shinichi Kobayashi

  4. Department of Mathematics, Osaka University, Toyonaka, Osaka, 560-0043, Japan

    Seidai Yasuda

Authors
  1. Kenichi Bannai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shinichi Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Seidai Yasuda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kenichi Bannai.

Additional information

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”.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-016-1783-x

Keywords

Search

Navigation