An application of the value distribution theory for semi-abelian varieties to problems of Ax-Lindemann and Manin-Mumford types

TL;DR

This paper applies value distribution theory to semi-abelian varieties to prove Ax-Lindemann and Manin-Mumford type theorems, establishing a novel link between complex analysis and arithmetic geometry.

Contribution

It introduces a new approach using value distribution theory to prove classical conjectures for semi-abelian varieties, connecting complex analysis with Diophantine problems.

Findings

Proved an Ax-Lindemann type theorem for semi-abelian varieties.

Established a Manin-Mumford type theorem using value distribution theory.

Suggested a new link between holomorphic maps and arithmetic geometry.

Abstract

The aim of this paper is to prove a theorem of Ax-Lindemann type for complex semi-abelian varieties as an application of a big Picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical Manin-Mumford Conjecture for semi-abelian varieties, which was proved by M. Raynaud 1983, M. Hindry 1988, ....., and Pila-Zannier 2008 by a different method from others, which is most relevant to our's. The present result might be a first instance of a direct connection between the value distribution theory of holomorphic maps and the arithmetic (Diophantine) theory over algebraic number fields, while there have been many analogies between them.