Enlarged mixed Shimura varieties, bi-algebraic system and some Ax type transcendental results

TL;DR

This paper develops a theory of enlarged mixed Shimura varieties, establishing bi-algebraic systems and proving Ax-Schanuel type transcendental results, including the full conjecture for the unipotent part, generalizing previous results for mixed Shimura varieties.

Contribution

It introduces enlarged mixed Shimura varieties and proves the Ax-Schanuel conjecture for their unipotent parts, extending known results to a broader framework.

Findings

Proved Ax-Schanuel conjecture for unipotent parts.

Established bi-algebraic systems for enlarged mixed Shimura varieties.

Connected transcendental results with classical Ax theorems.

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.