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Title:
Ax-Schanuel for Shimura Varieties
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Abstract:
Abstract: (joint with N.Mok, J.Pila) Shimura varieties (S) are uniformized by symmetric spaces (H), and the uniformization map Pi:H --> S is quite transcendental. Understanding the interaction of this map with the two algebraic structures is of particular interest in arithmetic, as it is a necessary ingredient for the modern approaches to the Andre-Oort and Zilber-Pink conjectures, as well
As a new approac to Shafarevich type theorems about integral points on varieties. We establish an analogue of the Ax-Schanuel theorem in this context, which essentially says that any atypical algebraic relations between subvarieties in H and S are governed by Shimura subvarieties.
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