Osaka Journal of Mathematics

Integrals on $p$-adic upper half planes and Hida families over totally real fields

Isao Ishikawa

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Bertolini--Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbb{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of $\mathrm{GL}(2)$-type associated to a Hilbert modular form of weight 2.

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Osaka J. Math., Volume 53, Number 4 (2016), 1089-1124.

First available in Project Euclid: 4 October 2016

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Zentralblatt MATH identifier

Primary: 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27] 11G10: Abelian varieties of dimension > 1 [See also 14Kxx] 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]


Ishikawa, Isao. Integrals on $p$-adic upper half planes and Hida families over totally real fields. Osaka J. Math. 53 (2016), no. 4, 1089--1124.

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  • A. Ash and G. Stevens: $p$-adic deformations of cohomology classes of subgroups of $\mathrm{GL}(n,\mathbf{Z})$, Collect. Math. 48 (1997), 1–30.
  • J.-F. Boutot and H. Carayol: Uniformisation $p$-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfel'd, Astérisque 196197 (1991), 45–158.
  • M. Bertolini and H. Darmon: Hida families and rational points on elliptic curves, Invent. Math. 168 (2007), 371–431.
  • M. Bertolini and H. Darmon: The rationality of Stark–Heegner points over genus fields of real quadratic fields, Ann. of Math. (2) 170 (2009), 343–370.
  • M Bertolini, H. Darmon, A. Iovita and M. Spiess: Teitelbaum's exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411–449.
  • J.F. Boutot and T. Zink: The $p$-adic uniformisation of Shimura curves, preprint 95-107, Sonderforschungsbereich 343, Universität Bielefeld (1995).
  • A. Dabrowski: $p$-adic $L$-functions of Hilbert modular forms, Ann. Inst. Fourier (Grenoble) 44 (1994), 1025–1041.
  • S. Dasgupta: Stark–Heegner points on modular Jacobians, Ann. Sci. École Norm. Sup. (4) 38 (2005), 427–469.
  • L. Gerritzen and M. van der Put: Schottky Groups and Mumford Curves, Lecture Notes in Mathematics 817, Springer, Berlin, 1980.
  • R. Greenberg and G. Stevens: $p$-adic $L$-functions and $p$-adic periods of modular forms, Invent. Math. 111 (1993), 407–447.
  • H. Hida: On $p$-adic Hecke algebras for $\mathrm{GL}_{2}$ over totally real fields, Ann. of Math. (2) 128 (1988), 295–384.
  • H. Hida: On nearly ordinary Hecke algebras for $\mathrm{GL}(2)$ over totally real fields; in Algebraic Number Theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, MA, 139–169, 1989.
  • V.A. Kolyvagin and D.Yu. Logachëv: Finiteness of SH over totally real fields, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), 851–876.
  • C.P. Mok: Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields, Comment. Math. Helv. 86 (2011), 867–945.
  • C.P. Mok: The exceptional zero conjecture for Hilbert modular forms, Compos. Math. 145 (2009), 1–55.
  • B. Mazur, J. Tate and J. Teitelbaum: On $p$-adic analogues of the conjectures of Birch and Swinnerton–Dyer, Invent. Math. 84 (1986), 1–48.
  • G. Shimura: Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten, Publishers, Tokyo; Princeton Univ. Press, Princeton, N.J., 1971.
  • M.-F. Vignéras: Arithmétique des Algèbres de Quaternions, Lecture Notes in Mathematics 800, Springer, Berlin, 1980.
  • S.-W. Zhang: Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27–147.
  • S.-W. Zhang: Gross–Zagier formula for $\mathrm{GL}_{2}$, Asian J. Math. 5 (2001), 183–290.