Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture

Abstract

In this paper we develop a strategy and some technical tools for proving the André-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.