A simplified proof of the André-Oort conjecture for products of modular curves

摘要

In this paper we give a short proof of the André-Oort conjecture for products of modular curves under the Generalised Riemann Hypothesis using only simple Galois-theoretic and geometric arguments. We believe this method represents a strategy for proving the conjecture for a general Shimura variety under GRH without using ergodic theory. We also demonstrate a short proof of the Manin-Mumford conjecture for Abelian varieties using similar arguments.