Algebra & Number Theory

Families of nearly ordinary Eisenstein series on unitary groups

Xin Wan

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

We use the doubling method to construct p-adic L-functions and families of nearly ordinary Klingen Eisenstein series from nearly ordinary cusp forms on unitary groups of signature (r,s) and Hecke characters, and prove the constant terms of these Eisenstein series are divisible by the p-adic L-function, following earlier constructions of Eischen, Harris, Li, Skinner and Urban. We also make preliminary computations for the Fourier–Jacobi coefficients of the Eisenstein series. This provides a framework to do Iwasawa theory for cusp forms on unitary groups.

Article information

Source
Algebra Number Theory, Volume 9, Number 9 (2015), 1955-2054.

Dates
Received: 12 February 2014
Revised: 27 June 2015
Accepted: 18 August 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842420

Digital Object Identifier
doi:10.2140/ant.2015.9.1955

Mathematical Reviews number (MathSciNet)
MR3435811

Zentralblatt MATH identifier
1334.11083

Subjects
Primary: 11R23: Iwasawa theory

Keywords
Iwasawa theory ordinary Klingen Eisenstein series unitary groups $p$-adic $L$-function

Citation

Wan, Xin. Families of nearly ordinary Eisenstein series on unitary groups. Algebra Number Theory 9 (2015), no. 9, 1955--2054. doi:10.2140/ant.2015.9.1955. https://projecteuclid.org/euclid.ant/1510842420


Export citation

References

  • W. L. Baily, Jr. and A. Borel, “Compactification of arithmetic quotients of bounded symmetric domains”, Ann. of Math. $(2)$ 84 (1966), 442–528.
  • A. Borel, “Some finiteness properties of adele groups over number fields”, Inst. Hautes Études Sci. Publ. Math. 16 (1963), 5–30.
  • F. Bruhat and J. Tits, “Groupes réductifs sur un corps local”, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251.
  • W. Casselman, “The unramified principal series of $\mathfrak{p}$-adic groups, I: The spherical function”, Compositio Math. 40:3 (1980), 387–406.
  • P. Deligne and D. Mumford, “The irreducibility of the space of curves of given genus”, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.
  • A. Grothendieck, “Eléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I”, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 5–167.
  • E. Eischen and X. Wan, “$p$-adic $L$-functions of finite slope forms on unitary groups and Eisenstein series”, J. Math. Inst. Jussieu (online publication November 2014).
  • E. Eischen, M. Harris, J. Li, and C. Skinner, “$p$-adic $L$-functions for unitary Shimura varieties, II”, in preparation.
  • P. B. Garrett, “Pullbacks of Eisenstein series; applications”, pp. 114–137 in Automorphic forms of several variables (Katata, 1983), edited by I. Satake and Y. Morita, Progr. Math. 46, Birkhäuser, Boston, 1984.
  • P. B. Garrett, “Integral representations of Eisenstein series and $L$-functions”, pp. 241–264 in Number theory, trace formulas and discrete groups (Oslo, 1987), edited by K. E. Aubert et al., Academic Press, Boston, 1989.
  • S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit constructions of automorphic $L$-functions, Lecture Notes in Mathematics 1254, Springer, Berlin, 1987.
  • M. Harris, “Eisenstein series on Shimura varieties”, Ann. of Math. $(2)$ 119:1 (1984), 59–94.
  • H. Hida, “Non-vanishing modulo $p$ of Hecke $L$-values”, pp. 735–784 in Geometric aspects of Dwork theory, vol. I–II, edited by A. Adolphson et al., de Gruyter, Berlin, 2004.
  • H. Hida, $p$-adic automorphic forms on Shimura varieties, Springer, New York, 2004.
  • H. Hida, “Irreducibility of the Igusa tower over unitary Shimura varieties”, pp. 187–203 in On certain $L$-functions, edited by J. Arthur et al., Clay Math. Proc. 13, Amer. Math. Soc., 2011.
  • M.-L. Hsieh, “Ordinary $p$-adic Eisenstein series and $p$-adic $L$-functions for unitary groups”, Ann. Inst. Fourier $($Grenoble$)$ 61:3 (2011), 987–1059.
  • M.-L. Hsieh, “Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields”, J. Amer. Math. Soc. 27:3 (2014), 753–862.
  • T. Ikeda, “On the theory of Jacobi forms and Fourier–Jacobi coefficients of Eisenstein series”, J. Math. Kyoto Univ. 34:3 (1994), 615–636.
  • R. E. Kottwitz, “Points on some Shimura varieties over finite fields”, J. Amer. Math. Soc. 5:2 (1992), 373–444.
  • S. S. Kudla, “Splitting metaplectic covers of dual reductive pairs”, Israel J. Math. 87:1–3 (1994), 361–401.
  • K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, Ph.D. thesis, Harvard University, 2008, hook http://search.proquest.com/docview/304599957 \posturlhook.
  • K.-W. Lan, “Elevators for degenerations of PEL structures”, Math. Res. Lett. 18:5 (2011), 889–907.
  • K.-W. Lan, “Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties”, J. Reine Angew. Math. 664 (2012), 163–228.
  • K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs Series 36, Princeton University Press, NJ, 2013.
  • K.-W. Lan, “Compactifications of PEL-Type Shimura Varieties and Kuga Families with Ordinary Loci”, preprint, 2013, hook http://www.math.umn.edu/~kwlan/articles/cpt-ram-ord.pdf \posturlhook.
  • K.-W. Lan, “Compactifications of PEL-type Shimura varieties in ramified characteristics”, preprint, 2014, hook http://www.math.umn.edu/~kwlan/articles/cpt-ram-flat.pdf \posturlhook.
  • E. M. Lapid and S. Rallis, “On the local factors of representations of classical groups”, pp. 309–359 in Automorphic representations, $L$-functions and applications: progress and prospects, edited by J. W. Cogdell et al., Ohio State Univ. Math. Res. Inst. Publ. 11, de Gruyter, Berlin, 2005.
  • V. Platonov and A. Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics 139, Academic Press, Boston, 1994.
  • I. Reiner, Maximal orders, London Mathematical Society Monographs 5, Academic Press, London–New York, 1975.
  • W. Scharlau, Quadratic and Hermitian forms, Grundlehren der Math. Wissenschaften 270, Springer, Berlin, 1985.
  • M. Demazure and A. Grothendieck, Schémas en groupes, Tome I: Propriétés générales des schémas en groupes, Exposés I–VII (Séminaire de Géométrie Algébrique du Bois Marie 1962–1964), Lecture Notes in Math. 151, Springer, Berlin, 1970.
  • M. Demazure and A. Grothendieck, Schémas en groupes, Tome III: Structure des schémas en groupes réductifs, Exposés XIX–XXVI (Séminaire de Géométrie Algébrique du Bois Marie 1962–1964), Lecture Notes in Math. 153, Springer, Berlin, 1970.
  • G. Shimura, Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics 93, Amer. Math. Soc., 1997.
  • C. Skinner, “A converse of Gross, Zagier and Kolyvagin”, preprint, 2014. \codarefhtttp://msp.org/idx/arx/1405.7294arXiv 1405.7294
  • C. Skinner and E. Urban, “The Iwasawa main conjectures for $\rm GL\sb 2$”, Invent. Math. 195:1 (2014), 1–277.
  • T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics 9, Birkhäuser, Boston, 1998.
  • X. Wan, preprint, 2013, hook {http://www.math.columbia.edu/~xw2295/Hilbert \posturlhook.
  • X. Wan, “The Iwasawa main conjecture for Hilbert modular forms”, Forum Math. Sigma 3 (2015), e18.