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

2016/07/26 by Ziyang Gao, Gao, Ziyang
Mathematics · #11G18 #11J81 #14G35 #14K10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1607.07843

openalex publication_date 2016/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Ax–Schanuel for the j-function

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