Darmon, Dasgupta and Pollack in 2011 applied the Eisenstein congruence for Hilbert modular forms to prove the rank one Gross conjecture for Deligne-Ribet p-adic L-functions under some technical assumptions. These assumptions were later lifted by Ventullo.
In this talk, we will apply their ideas in the setting of CM congruence to study the exceptional zero conjecture for the Katz p-adic L-functions associated with ring class characters of imaginary quadratic fields.
The main result is a precise first derivate formula of the Katz p-adic L-functions at the exceptional zero, which proves a formula proposed in a recent work of Betina and Dimitrov.
This is a joint work with Masataka Chida.
Note the unusual time!