Towards the Andre-Oort conjecture for mixed Shimura varieties: The Ax-Lindemann theorem and lower bounds for Galois orbits of special points

We prove in this paper the Ax-Lindemann-Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André-Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of Aⁿ₆ and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.