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.

Effective estimates for the degrees of maximal special subvarieties

“…In particular, these notions include the condition that H is the generic Mumford-Tate group of X H . By [UY14], Lemma 2.1, when V is contained in S K (G, X + ), we may assume that X + H is contained in X + and α = 1. In this situation, we will say that V is a special subvariety of S K (G, X + ), and we will say that V is defined by (H, X H ) and X + H .…”

Heights of pre-special points of Shimura varieties

“…The André-Oort conjecture is already known under the GRH due to the work of Klingler, Ullmo and Yafaev (see [KY14] and [UY14a]). …”

Hyperbolic Ax–Lindemann theorem in the cocompact case

“…This conjecture has recently been proved under the assumption of the Generalised Riemann Hypothesis for CM fields (see [33] and [13] Part of the strategy consists in establishing a geometric characterisation of special subvarieties of Shimura varieties. This criterion says roughly that subvarieties contained in their images by certain Hecke correspondences are special.…”