Galois theory, functional Lindemann-Weierstrass, and Manin maps

TL;DR

This paper advances the understanding of semiabelian varieties over algebraic closures of function fields by proving new Ax-Lindemann type results and generalizing the kernel theorem for abelian varieties using Galois theory of differential equations.

Contribution

It introduces novel Ax-Lindemann type theorems for semiabelian varieties and extends the kernel theorem for abelian varieties over algebraic closures of function fields.

Findings

New Ax-Lindemann type results for semiabelian varieties

Generalization of the kernel theorem for abelian varieties

Application of Galois theory of logarithmic differential equations

Abstract

We prove several new results of Ax-Lindemann type for semiabelian varieties over the algebraic closure K of C(t), making heavy use of the Galois theory of logarithmic differential equations. Using related techniques, we also give a generalization of the theorem of the kernel for abelian varieties over K.