The Andre-Oort conjecture for products of Drinfeld modular curves

Abstract

Let $Z=X_1\times ...\times X_n$ be a product of Drinfeld modular curves. We characterize those algebraic subvarieties $X\subset Z$ containing a Zariski-dense set of CM points, i.e. points corresponding to $n$-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic $p$ analogue of a special case of the André-Oort conjecture. We follow closely the approach used by Bas Edixhoven in characteristic zero, see math.NT/0302138. Note that in this paper we assume that the characteristic $p$ is odd, and we only treat the case of Drinfeld $F_q[T]$-modules.

LaTeX, 28 pages. Minor corrections made. To appear in J. reine. Angew. Math