Enlarged mixed Shimura varieties, bi-algebraic system and some Ax type transcendental results
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.
Keywords
Cite
@article{arxiv.1607.07843,
title = {Enlarged mixed Shimura varieties, bi-algebraic system and some Ax type transcendental results},
author = {Ziyang Gao},
journal= {arXiv preprint arXiv:1607.07843},
year = {2018}
}
Comments
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)