On the connected components of Shimura varieties for CM unitary groups in odd variables

P -adic Uniformization of Unitary Shimura Varieties

Introduction Let Γ ⊂ PGUd−1,1(R) 0 be a torsion-free cocompact lattice. Then Γ acts on the unit ball B ⊂ C by holomorphic automorphisms. The quotient Γ\B is a complex manifold, which has a unique structure of a complex projective variety XΓ (see [Sha, Ch. IX, §3]). Shimura had proved that when Γ is an arithmetic congruence subgroup, XΓ has a canonical structure of a projective variety over some...

Nonhomeomorphic conjugates of connected Shimura varieties

We show that conjugation by an automorphism of C may change the topological fundamental group of a locally symmetric variety over C. As a consequence, we obtain a large class of algebraic varieties defined over number fields with the property that different embeddings of the number field into C give complex varieties with nonisomorphic fundamental groups. Let V be an algebraic variety over C, a...

Special Cycles on Unitary Shimura Varieties I. Unramified Local Theory

A relation between a generating series constructed from arithmetic cycles on an integral model of a Shimura curve and the derivative of a Siegel Eisenstein series of genus 2 was established by one of us in [9]. There, the hope is expressed that such a relation should hold in greater generality for integral models of Shimura varieties attached to orthogonal groups of signature (2, n − 2) for any...

Rankin-selberg L-functions and Cycles on Unitary Shimura Varieties

Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties

Let G be a unitary group over a totally real field, and X 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 ...