Darmon’s points and quaternionic Shimura varieties

The field of moduli of quaternionic multiplication on abelian varieties

We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties. Published in Intern. J. Math. M. Sc. 52 (2004), 2795-2808.

Parametrizing Shimura Subvarieties of A1 Shimura Varieties and Related Geometric Problems

This paper gives a complete parametrization of the commensurability classes of totally geodesic subspaces of irreducible arithmetic quotients of Xa,b = (H ) × (H). A special case describes all Shimura subvarieties of type A1 Shimura varieties. We produce, for any n ≥ 1, examples of manifolds/Shimura varieties with precisely n commensurability classes of totally geodesic submanifolds/Shimura sub...

Cohomology of quaternionic Shimura varieties at parahoric levels

On the Zeta Function of Quaternionic Shimura Varieties

Shimura Curves Embedded in Igusa’s Threefold

Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus QO of quaternionic multiplication byO in the moduli spaceAg of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe QO as a union of Atkin-Lehner quotients of Shimura varieties and we comput...