Duke Mathematical Journal

Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties

David Helm

Full-text: Access denied (no subscription detected)

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


Let G be a unitary group over a totally real field, and let X be a Shimura variety associated to G. For certain primes p of good reduction for X, we construct cycles Xτ0,i on the characteristic p fiber of X. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on X. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group G, which is isomorphic to G at all finite places but not isomorphic to G at archimedean places. More precisely, each cycle Xτ0,i has a natural desingularization X~τ0,i, which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety X associated to G. We exploit this relationship to construct an injection of the étale cohomology of X into that of X. This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of G to automorphic representations of G.

Article information

Duke Math. J., Volume 155, Number 3 (2010), 483-518.

First available in Project Euclid: 16 November 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]


Helm, David. Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties. Duke Math. J. 155 (2010), no. 3, 483--518. doi:10.1215/00127094-2010-061. https://projecteuclid.org/euclid.dmj/1289916771

Export citation


  • M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math 36 (1969), 23--58.
  • A. J. De Jong, The moduli spaces of principally polarized abelian varieties with $\Gamma_0(p)$-level structure, J. Algebraic Geom. 2 (1993), 667--688.
  • G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, with an appendix by David Mumford, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.
  • W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
  • A. Ghitza, Hecke eigenvalues of Siegel modular forms ( mod $p$) and algebraic modular forms, J. Number Theory 106 (2004), 345--384.
  • —, All Siegel Hecke eigensystems (mod $p$) are cuspidal, Math. Res. Lett. 13 (2006), 813--823.
  • A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Séminaire de Mathématiques Supérieures 45, Presses de l'Université de Montréal, Montréal, 1974.
  • M. Harris and J.-P. Labesse, Conditional base change for unitary groups, Asian J. Math. 8 (2004), 653--684.
  • M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, with an appendix by V. G. Berkovich, Ann. of Math. Stud. 151, Princeton Univ. Press, Princeton, 2001.
  • D. Helm, A geometric Jacquet-Langlands correspondence for $U(2)$ Shimura varieties.
  • R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373--444.
  • J.-P. Labesse, Changement de base CM et séries discrètes, preprint, 2009.
  • K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, Ph.D. dissertation, Harvard University, Cambridge, Massachusetts, 2008.
  • W. Messing, The Crystals Associated to Barsotti-Tate Groups, Lecture Notes in Math. 264, Springer, Berlin, 1972.
  • J. S. Milne, Étale cohomology, Princeton Math. Ser. 33, Princeton Univ. Press, Princeton, 1980.
  • T. Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63--135.
  • K. A. Ribet, ``Bimodules and abelian surfaces'' in Algebraic Number Theory, Adv. Stud. Pure Math. 17, Academic Press, Boston, 1989, 359--407.
  • —, On modular representations of $\gal(\overline{\QQ}/\QQ)$ arising from modular forms, Invent. Math. 100 (1990), 431--476.
  • J.-P. Serre, Two letters on quaternions and modular forms (mod $p$), with an appendix by R. Livné, Israel J. Math. 95 (1996), 281--299.