p-rigidity and Iwasawa μ-invariants

Burungale, Ashay; Hida, Haruzo

doi:10.2140/ant.2017.11.1921

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

Ashay A. Burungale and Haruzo Hida

Vol. 11 (2017), No. 8, 1921–1951

DOI: 10.2140/ant.2017.11.1921

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 ${\hat{G}}_{m} \otimes O_{p}$ ( $p$ -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 $λ$ .

Keywords

Hilbert modular Shimura variety, Hecke stable subvariety, Iwasawa $\mu$-invariant, Katz $p$-adic L-function

Mathematical Subject Classification 2010

Primary: 11G18

Secondary: 19F27

Milestones

Received: 26 September 2016

Revised: 21 November 2016

Accepted: 6 February 2017

Published: 15 October 2017

Authors

Ashay A. Burungale
Département de Mathématiques LAGA Institute Galilée Université Paris 13 93430 Villetaneuse France

Haruzo Hida
Department of Mathematics UCLA Los Angeles, CA 90095-1555 United States

Vol. 11, No. 8, 2017