## Duke Mathematical Journal

### Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties

David Helm

#### 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 $X_{\tau_0,i}$ 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 $G^{\prime}$, which is isomorphic to $G$ at all finite places but not isomorphic to $G$ at archimedean places. More precisely, each cycle $X_{\tau_0,i}$ has a natural desingularization ${\tilde X}_{\tau_0,i}$, which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety $X^{\prime}$ associated to $G^{\prime}$. We exploit this relationship to construct an injection of the étale cohomology of $X^{\prime}$ into that of $X$. This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of $G^{\prime}$ 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

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

#### 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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