A new unicity theorem and Erdös' problem for polarized semi-abelian varieties
type: Item
instance of: scholarly article
title: A new unicity theorem and Erdös' problem for polarized semi-abelian varieties
zbMATH Open document ID: 1315.14058
author: Pietro Corvaja
author: Junjiro Noguchi
published in: Mathematische Annalen
publication date: 2012-06-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.5066
review text: 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.
reviewed by: Herbert Lange
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14K20
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 6043334
zbMATH Keywords: abelian variety
zbMATH Keywords: semi-abelian variety
MaRDI profile type: Publication
OpenAlex ID: W2152951840
arXiv ID: 0907.5066
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DOI: 10.1007/S00208-011-0692-X
description: scientific article; zbMATH DE number 6043334