Universiteit Leiden Universiteit Leiden

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Proefschrift

The mixed Ax-Lindemann theorem and its applications to the Zilber-Pink conjecture

Promotores: Prof.dr. S.J. Edixhoven, Prof.dr. E. Ullmo (University Paris-Sud)

Auteur Z. Gao
Datum
Links Thesis in Leiden Repository

The Zilber-Pink conjecture is a common generalization of the Andre-Oort and the Mordell-Lang conjectures. In this dissertation, we study its sub-conjectures: Andre-Oort, which predicts that a subvariety of a mixed Shimura variety having dense intersection with the set of special points is special; and Andre-Pink-Zannier which predicts that a subvariety of a mixed Shimura variety having dense intersection with a generalized Hecke orbit is weakly special. One of the main results of this dissertation is to prove the Ax-Lindemann theorem, a generalization of the functional analogue of the classical Lindemann-Weierstrass theorem, in its most general form. Another main result is to prove the Andre-Oort conjecture for a large class of mixed Shimura varieties: unconditionally for any product of the Poincare bundles over A6 and under GRH for all mixed Shimura varieties of abelian type. As for Andre-Pink-Zannier, we prove several cases when the ambient mixed Shimura variety is the universal family of abelian varieties: for the generalized Hecke orbit of a special point; for any subvariety contained in an abelian scheme over a curve and the generalized Hecke orbit of a torsion point of a fiber; for curves and the generalized Hecke orbit of an algebraic point.