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

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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.

Résumé

La conjecture de Gross en rang 1 pour les fonctions L p-adiques de Deligne–Ribet a été résolue par Darmon-Dasgupta-Pollack et Ventullo au moyen de congruences d’Eisenstein parmi les formes modulaires de Hilbert. Le but de cet article est de prouver un analogue de la conjecture de Gross pour les fonctions L p-adiques de Katz des corps quadratiques imaginaires, via les congruences entre formes CM et formes non-CM. Le nouvel ingrédient est l’application de la méthode de Rankin-Selberg p-adique pour construire une famille de Hida non-CM qui est congruente à une famille de Hida de formes CM pour la spécialisation \(1+\varepsilon \).

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Acknowledgements

This work has its root in the authors’ participation in the program Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers during 2015–2017. The authors would like to thank Shinichi Kobayashi and Nobuo Tsuzuki for their support and hospitality during the period of this program. The authors are grateful to the referees for the suggestions and comments on the improvement of the manuscript.

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Correspondence to Ming-Lun Hsieh.

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To Bernadette Perrin-Riou, on her 65th birthday.

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Hsieh was partially supported by a MOST grant MOST 108-2628-M-001-009-MY4 and 110-2628-M-001-004-. Chida was supported by JSPS KAKENHI Grant Number JP18K03202.

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Chida, M., Hsieh, ML. The derivative formula of p-adic L-functions for imaginary quadratic fields at trivial zeros. Ann. Math. Québec 47, 1–30 (2023). https://doi.org/10.1007/s40316-022-00198-6

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