Algebra & Number Theory

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

Ashay Burungale and Haruzo Hida

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 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

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

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

Permanent link to this document
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

Subjects
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]

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

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


Export citation

References

  • M. Brakočević, “Anticyclotomic $p$-adic $L$-function of central critical Rankin–Selberg $L$-value”, Int. Math. Res. Not. 2011:21 (2011), 4967–5018.
  • A. A. Burungale, “On the $\mu$-invariant of the cyclotomic derivative of a Katz $p$-adic $L$-function”, J. Inst. Math. Jussieu 14:1 (2015), 131–148.
  • A. A. Burungale, “An $l\ne p$-interpolation of genuine $p$-adic L-functions”, Res. Math. Sci. 3 (2016), Paper No. 16, 26.
  • A. A. Burungale, “Non-triviality of generalised Heegner cycles over anticyclotomic towers: a survey”, pp. 279–306 in $p$-adic aspects of modular forms, edited by B. Balasubramanyam et al., World Sci., Hackensack, NJ, 2016.
  • A. A. Burungale, “On the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves”, J. Inst. Math. Jussieu 16:1 (2017), 189–222.
  • A. Burungale, “$\mathfrak{p}$-rigidity and $\mathfrak{p}$-independence of quaternionic modular forms modulo $p$”, preprint.
  • A. Burungale and D. Disegni, “On the non-vanishing of p-adic heights for CM abelian varieties, and the arithmetic of Katz p-adic L-functions”, preprint.
  • A. A. Burungale and H. Hida, “André–Oort conjecture and nonvanishing of central $L$-values over Hilbert class fields”, Forum Math. Sigma 4 (2016), e20, 26.
  • A. Burungale and M.-L. Hsieh, “The vanishing of $\mu$-invariant of $p$-adic Hecke $L$-functions for CM fields”, Int. Math. Res. Not. 2013:5 (2013), 1014–1027.
  • C.-L. Chai, “Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli”, Invent. Math. 121:3 (1995), 439–479.
  • C.-L. Chai, “Families of ordinary abelian varieties: canonical coordinates, $p$-adic monodromy,Tate-linear subvarieties and Hecke orbits”, preprint, 2003, hook http://www.math.upenn.edu/~chai/papers_pdf/fam_ord_av.pdf \posturlhook.
  • C.-L. Chai, “Hecke orbits as Shimura varieties in positive characteristic”, pp. 295–312 in International Congress of Mathematicians, II, Eur. Math. Soc., Zürich, 2006.
  • E. Eischen, M. Harris, J. Li, and C. Skinner, “$p$-adic L-functions for unitary groups”, preprint, 2016.
  • R. Gillard, “Remarques sur l'invariant mu d'Iwasawa dans le cas CM”, J. Théor. Nombres Bordeaux $(2)$ 3:1 (1991), 13–26.
  • M. Harris, J.-S. Li, and C. M. Skinner, “$p$-adic $L$-functions for unitary Shimura varieties, I: Construction of the Eisenstein measure”, Doc. Math. Extra Vol. (2006), 393–464.
  • H. Hida, $p$-adic automorphic forms on Shimura varieties, Springer, 2004.
  • H. Hida, “Anticyclotomic main conjectures”, Doc. Math. Extra Vol. (2006), 465–532.
  • H. Hida, “Irreducibility of the Igusa tower”, Acta Math. Sin. $($Engl. Ser.$)$ 25:1 (2009), 1–20.
  • H. Hida, “Quadratic exercises in Iwasawa theory”, Int. Math. Res. Not. 2009:5 (2009), 912–952.
  • H. Hida, “The Iwasawa $\mu$-invariant of $p$-adic Hecke $L$-functions”, Ann. of Math. $(2)$ 172:1 (2010), 41–137.
  • H. Hida, “Vanishing of the $\mu$-invariant of $p$-adic Hecke $L$-functions”, Compos. Math. 147:4 (2011), 1151–1178.
  • H. Hida, Elliptic curves and arithmetic invariants, Springer, 2013.
  • H. Hida, “Local indecomposability of Tate modules of non-CM abelian varieties with real multiplication”, J. Amer. Math. Soc. 26:3 (2013), 853–877.
  • H. Hida and J. Tilouine, “Anti-cyclotomic Katz $p$-adic $L$-functions and congruence modules”, Ann. Sci. École Norm. Sup. $(4)$ 26:2 (1993), 189–259.
  • M.-L. Hsieh, “On the $\mu$-invariant of anticyclotomic $p$-adic $L$-functions for CM fields”, J. Reine Angew. Math. 688 (2014), 67–100.
  • M.-L. Hsieh, “Special values of anticyclotomic Rankin–Selberg $L$-functions”, Doc. Math. 19 (2014), 709–767.
  • N. M. Katz, “$p$-adic $L$-functions for CM fields”, Invent. Math. 49:3 (1978), 199–297.
  • N. Katz, “Serre–Tate local moduli”, pp. 138–202 in Algebraic surfaces (Orsay, France, 1976–78), Lecture Notes in Math. 868, Springer, 1981.
  • G. Shimura, “On analytic families of polarized abelian varieties and automorphic functions”, Ann. of Math. $(2)$ 78 (1963), 149–192.
  • G. Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series 46, Princeton University Press, 1998.