On the $\mu$-invariant of the cyclotomic derivative of Katz p-adic L-function

TL;DR

This paper investigates the $$-invariant of the cyclotomic derivative of Katz p-adic L-functions in cases where the anticyclotomic Katz p-adic L-function vanishes, with implications for CM modular forms and Iwasawa theory.

Contribution

It establishes the $$-invariant of the cyclotomic derivative in a specific vanishing scenario, linking it to non-vanishing results and Iwasawa main conjecture approaches.

Findings

Proves non-vanishing of the anticyclotomic regulator for certain CM forms.

Connects the $$-invariant to the behavior of Katz p-adic L-functions.

Supports recent advances in Eisenstein ideal and main conjecture studies.

Abstract

When the branch character has root number -1, the corresponding anticyclotomic Katz p-adic L-function identically vanishes. In this case, we study the $\mu$-invariant of the cyclotomic derivative of Katz p-adic L-function. As an application, this proves the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with the root number -1. The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.