Duke Mathematical Journal

Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties

David Helm

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Abstract

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

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

Dates
First available in Project Euclid: 16 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1289916771

Digital Object Identifier
doi:10.1215/00127094-2010-061

Mathematical Reviews number (MathSciNet)
MR2738581

Zentralblatt MATH identifier
1205.11070

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

Citation

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


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