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

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. purpose this paper is to prove an analogue Katz attached imaginary quadratic fields via congruences between CM forms non-CM new ingredient apply Rankin-Selberg method construct a Hida family which congruent at $1+\varepsilon$ specialization.