2 00 9 A New Unicity Theorem and Erdös ’ Problem for Polarized Semi - Abelian Varieties ∗

In 1988 P. Erdös asked if the prime divisors of x−1 for all n = 1, 2, . . . determine the given integer x; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof [2] in 1997 together with its elliptic version. Analogously, K. Yamanoi [14] proved in 2004 that the support of the pull-backed divisor fD of an ample divisor on an abelian variety A by an algebraically non-degenerate entire holomorphic curve f : C → A essentially determines the pair (A,D). By making use of the main theorem of [10] we here deal with this problem for semi-abelian varieties: namely, given two polarized semi-abelian varieties (A1,D1), (A2,D2) and entire non-degenerate holomorphic curves fi : C → Ai, i = 1, 2, we classify the cases when the inclusion Suppf 1 D1 ⊂ Suppf ∗ 2 D2 holds. We also apply the main result of [4] to prove an arithmetic counterpart.