I study various arithmetico-geometric properties of Shimura varieties. Shimura varieties can be viewed as higher dimensional analogs (in fact generalisations) of the classical modular curves. They can also be viewed as ``non-abelian'' analogs of abelian varieties. A lot of theorems and conjectures about Shimura varieties are inspired by these analogies.

One problem of particular interest to me is the Andre-Oort conjecture on the distribution of special points in Shimura varieties. Besides its intrinsic interest, some special cases of this conjecture have important applications to the transcendence theory of hypergeometric functions.

I am also interested in the study of rational points on some Shimura varieties over local and global fields.