Vol. 11, No. 8, 2017

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

Ashay A. Burungale and Haruzo Hida

Vol. 11 (2017), No. 8, 1921–1951

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 KF 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.

Hilbert modular Shimura variety, Hecke stable subvariety, Iwasawa $\mu$-invariant, Katz $p$-adic L-function
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 19F27
Received: 26 September 2016
Revised: 21 November 2016
Accepted: 6 February 2017
Published: 15 October 2017
Ashay A. Burungale
Département de Mathématiques
Institute Galilée
Université Paris 13
93430 Villetaneuse
Haruzo Hida
Department of Mathematics
Los Angeles, CA 90095-1555
United States