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arXiv · 1409.8584

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

Isao Ishikawa
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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 $\mathbf{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of ${\rm GL}(2)$-type associated to a Hilbert modular form of weight 2.

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Isao Ishikawa. 2014-09-30. Integrals on $p$-adic upper half planes and Hida families over totally real fields. https://arxiv.org/abs/1409.8584

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