A New Unicity Theorem and Erdos' Problem for Polarized Semi-Abelian Varieties

TL;DR

This paper establishes a unicity theorem for polarized semi-abelian varieties, showing how the support of pull-backed divisors by entire curves determines the varieties, extending previous results from abelian to semi-abelian cases.

Contribution

It generalizes unicity results from abelian to semi-abelian varieties using recent Nevanlinna theory advances and explores an arithmetic analogue via recurrence sequences.

Findings

Support of pull-backed divisors determines the semi-abelian variety in certain cases.

Classification of when divisor supports inclusion implies equality of varieties.

Application of Nevanlinna theory and recurrence sequences to complex and arithmetic problems.

Abstract

In 1988 P. Erd\"os 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-Rodorig\'a\~nez and R. Schoof in 1997 together with its elliptic version. Analogously, K. Yamanoi 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 $f: \C \to A$ essentially determines the pair $(A, D)$. By making use of a recent theorem of Noguchi-Winkelmann-Yamanoi in Nevanlinna theory, 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 entire non-degenerate holomorphic curves $f_i:\C\to A_i$, $i=1,2$, we classify the cases when the inclusion $\supp f_1^*D_1\subset \supp f_2^* D_2$ holds. We also apply a result of Corvaja-Zannier on linear recurrence sequences to prove an arithmetic counterpart.