On the -invariant of the cyclotomic derivative of a Katzp-adic -function

Abstract

AbstractWhen the branch character has root number$- 1$, the corresponding anticyclotomic Katz$p$-adic$L$-function vanishes identically. For this case, we determine the$\mu $-invariant of the cyclotomic derivative of the Katz$p$-adic$L$-function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number$- 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.