Enlarged mixed Shimura varieties, bi-algebraic system and some Ax type transcendental results
arXiv:1607.07843
Abstract
We develop a theory of enlarged mixed Shimura varieties, putting the universal vectorial bi-extension defined by Coleman into this framework to study some functional transcendental results of Ax type. We study their bi-algebraic systems, formulate the Ax-Schanuel conjecture and explain its relation with the Ax logarithmique theorem and the Ax-Lindemann theorem which we shall prove. We also prove the whole Ax-Schanuel conjecture for the unipotent part. All these bi-algebraic and transcendental results generalize their counterparts for mixed Shimura varieties. In the end we briefly discuss about the Andre-Oort and Zilber-Pink type problems for enlarged mixed Shimura varieties.
Revised the definition of morphisms between enlarged mixed Shimura data, and changed the order of categorical and geometrical comparisons. Corrected the characterization of bi-algebraic subvarieties. Removed Ax-Schanuel for the fiber but kept Ax-Lindemann (with an updated proof)