On the first derivatives of the cyclotomic Katz p-adic L-functions for CM fields

Abstract: Buyukboduk and Sakamoto in 2019 proposed a precise conjectural formula relating the leading coefficient at the trivial zero s=0 of the cyclotomic Katz p-adic L-functions associated with ray class characters of a CM field K to suitable L-invariants/regulators of K. They were able to prove this formula in most cases when K is an imaginary quadratic field thanks to the existence of the Euler system of elliptic units/Rubin-Stark elements. In this talk, we will present a formula relating the first derivative of the cyclotomic Katz p-adic L-functions for general CM fields attached to ring class characters to the product of the L-invariant and the value of the improved Katz p-adic L-function at s=0. In particular, when the trivial zero occurs at s=0, we prove that the Katz p-adic L-function has a simple zero at s=0 if certain L-invariant is non-vanishing. Our method uses the congruence of Hilbert CM forms and does reply on the existence of the conjectural Rubin-Stark elements. This is a joint work with Adel Betina.

algebraic geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Columbia Automorphic Forms and Arithmetic Seminar