Duke Mathematical Journal

p-adic periods, p-adic L-functions, and the p-adic uniformization of Shimura curves

Massimo Bertolini and Henri Darmon

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Article information

Source
Duke Math. J., Volume 98, Number 2 (1999), 305-334.

Dates
First available in Project Euclid: 19 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1077228215

Digital Object Identifier
doi:10.1215/S0012-7094-99-09809-5

Mathematical Reviews number (MathSciNet)
MR1695201

Zentralblatt MATH identifier
1037.11045

Subjects
Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]
Secondary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

Citation

Bertolini, Massimo; Darmon, Henri. $p$ -adic periods, $p$ -adic $L$ -functions, and the $p$ -adic uniformization of Shimura curves. Duke Math. J. 98 (1999), no. 2, 305--334. doi:10.1215/S0012-7094-99-09809-5. https://projecteuclid.org/euclid.dmj/1077228215


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References

  • [B-SDGP] Katia Barré-Sirieix, Guy Diaz, François Gramain, and Georges Philibert, Une preuve de la conjecture de Mahler-Manin, Invent. Math. 124 (1996), no. 1-3, 1–9.
  • [BD1] M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves, Invent. Math. 126 (1996), no. 3, 413–456.
  • [BD2] Massimo Bertolini and Henri Darmon, Heegner points, $p$-adic $L$-functions, and the Cerednik-Drinfeld uniformization, Invent. Math. 131 (1998), no. 3, 453–491.
  • [Boi] X. Boichut, On the Mazur-Tate-Teitelbaum $p$-adic conjecture for elliptic curves with bad reduction at $p$, in preparation.
  • [BC] J.-F. Boutot and H. Carayol, Uniformisation $p$-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld, Astérisque (1991), no. 196-197, 7, 45–158 (1992), in Courbes modulaires et courbes de Shimura (Orsay, 1987/1988), Soc. Math. France, Marseille.
  • [C] I. V. Čerednik, Uniformization of algebraic curves by discrete arithmetic subgroups of $\rm PGL\sb2(k\sbw)$ with compact quotient spaces, Mat. Sb. (N.S.) 100(142) (1976), no. 1, 59–88, 165, (in Russian); English trans. in Math. USSR-Sb. 29 (1976), 55–78.
  • [Dag] H. Daghigh, Modular forms, quaternion algebras, and special values of $L$-functions, Ph.D. thesis, McGill University, 1997.
  • [Dr] V. G. Drinfeld, Coverings of $p$-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29–40.
  • [GvdP] Lothar Gerritzen and Marius van der Put, Schottky groups and Mumford curves, Lecture Notes in Mathematics, vol. 817, Springer, Berlin, 1980.
  • [GS] Ralph Greenberg and Glenn Stevens, $p$-adic $L$-functions and $p$-adic periods of modular forms, Invent. Math. 111 (1993), no. 2, 407–447.
  • [Gr] Benedict H. Gross, Heights and the special values of $L$-series, Number theory (Montreal, Que., 1985) eds. H. Kisilevsky and J. Labute, CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187.
  • [GZ] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320.
  • [JL] H. Jacquet and R. P. Langlands, Automorphic forms on $\rm GL(2)$, Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin, 1970.
  • [KKT] K. Kato, M. Kurihara, and Tsuji, forthcoming work.
  • [K1] Christoph Klingenberg, On $p$-adic $L$-functions of Mumford curves, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) eds. B. Mazur and G. Stevens, Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 277–315.
  • [M] Y. I. Manin, $p$-adic automorphic functions, J. Soviet Math. 5 (1976), 279–333.
  • [MTT] 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.
  • [RT] Kenneth A. Ribet and Shuzo Takahashi, Parametrizations of elliptic curves by Shimura curves and by classical modular curves, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11110–11114.
  • [Sch] P. Schneider, Rigid-analytic $L$-transforms, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 216–230.
  • [Se] Jean-Pierre Serre, Arbres, amalgames, $\rm SL\sb2$, Astérisque, vol. 46, Société Mathématique de France, Paris, 1977.
  • [Sh] Takuro Shintani, On construction of holomorphic cusp forms of half integral weight, Nagoya Math. J. 58 (1975), 83–126.
  • [T] Jeremy T. Teitelbaum, Values of $p$-adic $L$-functions and a $p$-adic Poisson kernel, Invent. Math. 101 (1990), no. 2, 395–410.
  • [Vi] Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980.