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arXiv · 2205.14711

The derivative formula of $p$-adic $L$-functions for imaginary quadratic fields at trivial zeros

Masataka Chida·Ming-Lun Hsieh
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Abstract

The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue of the Gross conjecture for the Katz $p$-adic $L$-functions attached to imaginary quadratic fields via the congruences between CM forms and non-CM forms. The new ingredient is to apply the $p$-adic Rankin-Selberg method to construct a non-CM Hida family which is congruent to a Hida family of CM forms at the $1+\varepsilon$ specialization.

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Masataka Chida, Ming-Lun Hsieh. 2022-05-29. The derivative formula of $p$-adic $L$-functions for imaginary quadratic fields at trivial zeros. https://doi.org/10.1007/s40316-022-00198-6

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