subvarieties of Drinfeld modular varieties Florian

The André-Oort conjecture for Shimura varieties over C states that any subvariety X of a Shimura variety S/C containing a Zariski-dense set of special points must be of Hodge type. For an up-to-date overview of this topic, and of known results, see Rutger Noot’s Bourbaki talk [18]. The most significant result so far is due to Bas Edixhoven and Andrei Yafaev [9], who prove that any subcurve X ⊂ S containing an infinite set of special points that lie in one Hecke orbit is special. More recently, Yafaev [22] has announced a proof for the case of subcurves X ⊂ S containing any infinite set of special points, assuming the generalized Riemann Hypothesis for CM fields. The aim of the present article is to explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. As this setting is in general easier than that of Shimura varieties over C, it can serve as a convenient testbed, where one might try out new approaches to the original André-Oort conjecture in a more friendly environment. The problem is that much of the machinery needed to even state an analogue of the conjecture in characteristic p is still lacking in the literature, and this article serves to provide some of that machinery. We also prove two special cases of the analogue of the André-Oort conjecture. First we treat the case where the special points all have a certain behaviour above a fixed prime (an analogous result for moduli of abelian varieties was proved by Moonen [16]), and then we look at the case of special points on subcurves of Drinfeld modular varieties, following methods similar to those used by Edixhoven and Yafaev. Our result on subcurves is conditional on a certain intersection hypothesis (Hypothesis 3.7), whereas Yafaev’s proof [22] is conditional on the generalized Riemann Hypothesis (which is known for function fields).