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1 December 2010 Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties
David Helm
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Duke Math. J. 155(3): 483-518 (1 December 2010). DOI: 10.1215/00127094-2010-061

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.

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David Helm. "Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties." Duke Math. J. 155 (3) 483 - 518, 1 December 2010. https://doi.org/10.1215/00127094-2010-061

Information

Published: 1 December 2010
First available in Project Euclid: 16 November 2010

zbMATH: 1205.11070
MathSciNet: MR2738581
Digital Object Identifier: 10.1215/00127094-2010-061

Subjects:
Primary: 11G18
Secondary: 14G35

Rights: Copyright © 2010 Duke University Press

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Vol.155 • No. 3 • 1 December 2010
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