0 Ju l 2 00 4 CM points on products of Drinfeld modular curves Florian Breuer

The André-Oort conjecture for products of Drinfeld modular curves

Let Z = X1×· · ·×Xn be a product of Drinfeld modular curves. We characterize those algebraic subvarieties X ⊂ 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.

Special subvarieties of Drinfeld modular varieties

We explore an analogue of the André-Oort conjecture for subvarieties of Drinfeld modular varieties. The conjecture states that a subvariety X of a Drinfeld modular variety contains a Zariski-dense set of complex multiplication (CM) points if and only if X is a “special” subvariety (i.e. X is defined by requiring additional endomorphisms). We prove this conjecture in two cases. Firstly when X co...

Torsion bounds for elliptic curves and Drinfeld modules

We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L : K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Rütsche, respectively. Our approach...

Introduction to Drinfeld Modules

(1) Explicit class field theory for global function fields (just as torsion of Gm gives abelian extensions of Q, and torsion of CM elliptic curves gives abelian extension of imaginary quadratic fields). Here global function field means Fp(T ) or a finite extension. (2) Langlands conjectures for GLn over function fields (Drinfeld modular varieties play the role of Shimura varieties). (3) Modular...

Heights of CM Points on Complex AÆne Curves