Algebra & Number Theory

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

Ashay Burungale and Haruzo Hida

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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.

Article information

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

Received: 26 September 2016
Revised: 21 November 2016
Accepted: 6 February 2017
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 19F27: Étale cohomology, higher regulators, zeta and L-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10]

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


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.

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