Year: 2010
Type: article
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.
Source: Duke Mathematical Journal
Author: David Helm
Institution: The University of Texas at Austin
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1660066638