A new unicity theorem and Erdös ’ problem for polarized semi - abelian varieties

In 1988 P. Erdös asked if the prime divisors of xn−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 f∗D 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.