A new unicity theorem and Erdos' problem for polarized semi-abelian varieties

In 1988 Erdös asked if the prime divisors of x n - 1 for all n = 1,2, determine the given integer x; the problem was affirmatively answered by Corrales-Rodrigáñez and Schoof (J Number Theory 64:276-290, 1997) [but a solution could also be deduced from an earlier result of Schinzel (Bull Acad Polon Sci 8:307-309, 2007)] together with its elliptic version. Analogously, Yamanoi (Forum Math 16:749-788, 2004) proved that the support of the pulled-back 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 Noguchi (Forum Math 20:469-503, 2008) we here deal with this problem for semi-abelian varieties; namely, given two polarized semi-abelian varieties (A 1, D 1), (A 2, D 2) and algebraically non-degenerate entire holomorphic curves f i: C → A i, i = 1, 2,we classify the cases when the inclusion holds. We shall remark in §5 that these methods yield an affirmative answer to a question of Lang formulated in 1966. Our answer is more general and more geometric than the original question. Finally, we interpret the main result of Corvaja and Zannier (Invent Math 149:431-451, 2002) to provide an arithmetic counterpart in the toric case. © 2011 Springer-Verlag.