Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer g ≥ 2, we prove that the number of torsion points lying on a smooth complex algebraic curve of genus g embedded into its Jacobian is uniformly bounded. Complementing other recent work of Dimitrov, Gao, and Habegger, we obtain a rather uniform version of the Mordell-Lang conjecture as well. In particular, the number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian.

A result of the second-named author states that there are only finitely many CM-elliptic curves over C whose j-invariant is an algebraic unit. His proof depends on Duke's Equidistribution Theorem and is hence noneffective. In this article, we give a completely effective proof of this result. To be precise, we show that every singular modulus that is an algebraic unit is associated with a CM-elliptic curve whose endomorphism ring has discriminant less than 10 15 . Through further refinements and computerassisted arguments, we eventually rule out all remaining cases, showing that no singular modulus is an algebraic unit. This allows us to exhibit classes of subvarieties in C n not containing any special points.

Using transcendence theory we prove the André-Oort conjecture in case of the Shimura variety A 2 C . It is well known that this result implies the André-Oort conjecture for a product of two arbitrary modular curves. In contrast to all previous proofs we obtain a result that is at once effective and unconditional.

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We establish an effective version of the André-Oort conjecture for linear subspaces of Y (1) n C ≈ A n C . Apart from the trivial examples provided by weakly special subvarieties, this yields the first algebraic subvarieties in a Shimura variety of dimension > 1 whose CM-points can be (theoretically) determined.