# On the exceptional zero conjectures of Katz p-adic L-functions for CM fields

19.10.2021 13:15 - 14:45

This talk is based on a joint work with Ming-Lun Hsieh studying  the exceptional zeros conjecture of Katz p-adic L-functions for CM fields. We give under some hypothesis on the branch character a formula for the derivative at $$s=0$$ along the cyclotomic direction. The formula  involves a certain L-invariant  and can be viewed as the CM variant of the rank one Gross conjecture for Deligne-Ribet p-adic L-functions.
The main ingredient is to apply the p-adic Rankin-Selberg method to construct a non-CM Hida family  such that its Fourier coefficients away from p are congruent to a linear combination of CM families at the 1 + $$\varepsilon$$ infinitesimal specialization, and its coefficients at p are related to the derivative of the anti-cyclotomic Katz p-adic L-function, and which we compute using Galois deformation theory.