  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.
