$\mathfrak{p}$-rigidity and Iwasawa $\mu$-invariants

Abstract

Let $F$ be a totally real field with ring of integers $O$ and $p$ be an odd prime unramified in $F$. Let $\mathfrak{p}$ be a prime above $p$. We prove that a mod $p$ Hilbert modular form associated to $F$ is determined by its restriction to the partial Serre–Tate deformation space ${\hat{\mathbb{G}}}_m\otimes O_{\mathfrak{p}}$ ($\mathfrak{p}$-rigidity). Let $K∕F$ be an imaginary quadratic CM extension such that each prime of $F$ above $p$ splits in $K$ and $\lambda$ a Hecke character of $K$. Partly based on $\mathfrak{p}$-rigidity, we prove that the $\mu$-invariant of the anticyclotomic Katz $\mathfrak{p}$-adic L-function of $\lambda$ equals the $\mu$-invariant of the full anticyclotomic Katz $p$-adic L-function of $\lambda$. An analogue holds for a class of Rankin–Selberg $p$-adic L-functions. When $\lambda$ is self-dual with the root number $-1$, we prove that the $\mu$-invariant of the cyclotomic derivatives of the Katz $\mathfrak{p}$-adic L-function of $\lambda$ equals the $\mu$-invariant of the cyclotomic derivatives of the Katz $p$-adic L-function of $\lambda$. Based on previous works of the authors and Hsieh, we consequently obtain a formula for the $\mu$-invariant of these $\mathfrak{p}$-adic L-functions and derivatives. We also prove a $\mathfrak{p}$-version of a conjecture of Gillard, namely the vanishing of the $\mu$-invariant of the Katz $\mathfrak{p}$-adic L-function of $\lambda$.