Osaka Journal of Mathematics

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

Isao Ishikawa

Full-text: Open access

Abstract

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.

Article information

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

Dates
First available in Project Euclid: 4 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1475601834

Mathematical Reviews number (MathSciNet)
MR3554859

Zentralblatt MATH identifier
06654666

Subjects
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]

Citation

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. https://projecteuclid.org/euclid.ojm/1475601834


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