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

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 $1+\epsilon$ specialization.

Final version. The reference numbers differ from the journal version. To appear in Annales Mathematiques du Quebec (Special birthday issue for Bernadette Perrin-Riou)

References in corpus (2)

• On extra-zeros of p-adic Rankin-Selberg L-functions
• CM congruence and trivial zeros of the Katz $p$-adic $L$-functions for CM fields