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The aim of this paper is to prove a theorem of Ax-Lindemann type for complexsemi-abelian varieties as an application of a big Picard theorem proved by theauthor in 1981, and then apply it to prove a theorem of classical Manin-MumfordConjecture 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 instanceof a direct connection between the value distribution theory of holomorphicmaps and the arithmetic (Diophantine) theory over algebraic number fields,while there have been many analogies between them.

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