On effectivity and uniformity in a result of André–Oort type

The André–Oort Conjecture (AOC) states that the irreducible components of the Zariski closure of a set of special points in a Shimura variety are special subvarieties. Here, a special variety is an irreducible component of the image of a sub-Shimura variety by a Hecke correspondence. In our talk, we plan to discuss a rarely known approach to the André–Oort Conjecture (AOC) that goes back to Yves André himself. Before the recent model-theoretic proofs of the AOC in certain cases by Pila André's proof was the only known unconditional proof of the AOC for a non-trivial Shimura variety. In our talk, we spot light on some recent improvements and additions to André's techniques, which enable us to give an effective proof of the AOC in the case of a product of two modular curves. Furthermore, we discuss the aspect of uniform bounds on the number of special points on a non-special curve in some detail.