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

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

Title: The mixed Ax-Lindemann theorem and its applications to the Zilber-Pink conjecture
Author: Gao, Ziyang
Publisher: Mathematical Institute, Faculty of Science, Leiden University/ Département de Mathématiques d'Orsay, Laboratoire de Mathématiques, Université Paris-Sud
Issue Date: 2014-11-24
Keywords: Mixed Shimura varieties
Ax-Lindemann
(weakly) special
Hecke orbits
O-minimal
Abstract: 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.
Description: Promotores: S.J.Edixhoven, E. Ullmo
With Summary in French
With Summary in Dutch
Dissertation Leiden University and Université Paris-Sud
Faculty: Faculteit der Wiskunde en Natuurwetenschappen
Citation: Gao, Z., 2014, Doctoral Thesis, Leiden University
Handle: http://hdl.handle.net/1887/29841
 

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