## Algebra & Number Theory

### $\mathfrak{p}$-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 $p$ 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 $G ̂m ⊗ Op$ ($p$-rigidity). Let $K∕F$ be an imaginary quadratic CM extension such that each prime of $F$ above $p$ splits in $K$ and $λ$ a Hecke character of $K$. Partly based on $p$-rigidity, we prove that the $μ$-invariant of the anticyclotomic Katz $p$-adic L-function of $λ$ equals the $μ$-invariant of the full anticyclotomic Katz $p$-adic L-function of $λ$. An analogue holds for a class of Rankin–Selberg $p$-adic L-functions. When $λ$ is self-dual with the root number $− 1$, we prove that the $μ$-invariant of the cyclotomic derivatives of the Katz $p$-adic L-function of $λ$ equals the $μ$-invariant of the cyclotomic derivatives of the Katz $p$-adic L-function of $λ$. Based on previous works of the authors and Hsieh, we consequently obtain a formula for the $μ$-invariant of these $p$-adic L-functions and derivatives. We also prove a $p$-version of a conjecture of Gillard, namely the vanishing of the $μ$-invariant of the Katz $p$-adic L-function of $λ$.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 8 (2017), 1921-1951.

Dates
Revised: 21 November 2016
Accepted: 6 February 2017
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842828

Digital Object Identifier
doi:10.2140/ant.2017.11.1921

Mathematical Reviews number (MathSciNet)
MR3720935

Zentralblatt MATH identifier
06806365

#### Citation

Burungale, Ashay; Hida, Haruzo. $\mathfrak{p}$-rigidity and Iwasawa $\mu$-invariants. Algebra Number Theory 11 (2017), no. 8, 1921--1951. doi:10.2140/ant.2017.11.1921. https://projecteuclid.org/euclid.ant/1510842828

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