A new unicity theorem and Erdös' problem for polarized semi-abelian varieties
Erdős' problem is the question whether the prime divisors of \(x^n- 1\) for all positive integers \(n\) determine the given integer \(x\). \textit{K. Yamanoi} proved in [Forum Math. 16, No. 5, 749--788 (2004; Zbl 1073.32007)] an analogue result for abelian varieties, namely that the support of the pullback divisor \(f^*D\) of an ample divisor \(D\) on an abelian variety \(A\) by an algebraically nondegenerate holomorphic curve \(f: \mathbb{C}\to A\) essentially determines the polarized abelian variety \((A, D)\). The present paper studies the same question the case of a semi-abelian variety: Let \((A_i,D_i)\), \(i=1,2\) be two pairs of semi-abelian varieties with reduced divisors. Suppose that nondegenerate holomorphic curves \(f_i: \mathbb{C}\to A_i\) are given. The main result of the paper is a criterion, in terms of \(f^*_iD_i\), \(i=1,2\), for the existence of a finite étale morphism \(A_1\to A_2\) commuting with the maps \(f_i\). Some application to arithmetical questions are given.
scientific article; zbMATH DE number 6043334
scientific article