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

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 ours. The present result might be a first instance of a direct connection at the proof level 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.