Duke Mathematical Journal

The $p$-adic sigma function

B. Mazur and J. Tate

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Article information

Source
Duke Math. J. Volume 62, Number 3 (1991), 663-688.

Dates
First available: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077296511

Mathematical Reviews number (MathSciNet)
MR1104813

Zentralblatt MATH identifier
0735.14020

Digital Object Identifier
doi:10.1215/S0012-7094-91-06229-0

Subjects
Primary: 11G07: Elliptic curves over local fields [See also 14G20, 14H52]
Secondary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)

Citation

Mazur, B.; Tate, J. The p -adic sigma function. Duke Mathematical Journal 62 (1991), no. 3, 663--688. doi:10.1215/S0012-7094-91-06229-0. http://projecteuclid.org/euclid.dmj/1077296511.


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References

  • [Ba] I. Barsotti, Considerazioni sulle funzioni theta, Symposia Mathematica, Vol. III (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 247–277.
  • [Be] D. Bernardi, Hauteur $p$-adique sur les courbes elliptiques, Seminar on Number Theory, Paris 1979–80, Progr. Math., vol. 12, Birkhäuser Boston, Mass., 1981, pp. 1–14.
  • [B-G-S] D. Bernardi, C. Goldstein, and N. Stephens, Notes $p$-adiques sur les courbes elliptiques, J. Reine Angew. Math. 351 (1984), 129–170.
  • [Bou] N. Bourbaki, Éléments de mathématique. Fascicule XXVII. Algèbre commutative. Chapitre 1: Modules plats. Chapitre 2: Localisation, Actualités Scientifiques et Industrielles, No. 1290, Herman, Paris, 1961.
  • [Br] L. Breen, Fonctions thêta et théorème du cube, Lecture Notes in Mathematics, vol. 980, Springer-Verlag, Berlin, 1983.
  • [Cr] V. Cristante, Theta functions and Barsotti-Tate groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 181–215.
  • [K] N. M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1973, 69–190. Lecture Notes in Mathematics, Vol. 350.
  • [L] S. Lang, Elliptic functions, 2nd ed. ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987.
  • [M-T 1] B. Mazur and J. Tate, Canonical height pairings via biextensions, Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195–237.
  • [M-T 2] B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1–48.
  • [McC] J. McCabe, $P$-adic theta functions, Ph.D. thesis, Harvard, 1968.
  • [Me] W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, vol. 264, Springer-Verlag, Berlin, 1972.
  • [Mo]1 H. Morikawa, Theta functions and abelian varieties over valuation fields of rank one. I, Nagoya Math. J. 20 (1962), 1–27.
  • [Mo]2 H. Morikawa, On theta functions and abelian varieties over valuation fields of rank one. II. Theta functions and abelian functions of characteristic $p(>0)$. , Nagoya Math. J. 21 (1962), 231–250.
  • [Ne1] A. Néron, Hauteurs et fonctions thêta, Rend. Sem. Mat. Fis. Milano 46 (1976), 111–135 (1978).
  • [Ne2] A. Néron, Fonctions thêta $p$-adiques, Sympos. Math., Vol. XXIV (Sympos., INDAM, Rome, 1979), Academic Press, London, 1981, pp. 315–345.
  • [Ne3] A. Néron, Fonctions thêta $p$-adiques et hauteurs $p$-adiques, Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), Progr. Math., vol. 22, Birkhäuser Boston, Boston, MA, 1982, pp. 149–174.
  • [No] P. Norman, $p$-adic theta functions, Amer. J. Math. 107 (1985), no. 3, 617–661.
  • [P-R1] B. Perrin-Riou, Descente infinie et hauteur $p$-adique sur les courbes elliptiques à multiplication complexe, Invent. Math. 70 (1982/83), no. 3, 369–398.
  • [P-R2] B. Perrin-Riou, Sur les hauteurs $p$-adiques, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 6, 291–294.
  • [T] J. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206.