On the distribution of integer points on curves of genus zero
type: Item
instance of: scholarly article
title: On the distribution of integer points on curves of genus zero
zbMATH Open document ID: 0979.11037
DOI: 10.1016/S0304-3975(99)00189-9
author: Joseph H. Silverman
published in: Theoretical Computer Science
publication date: 2000-06-04T00:00:00Z
review text: Let \(C\subset \mathbb{A}^n\) be a geometrically irreducible affine curve of genus 0 defined over \(\mathbb{Z}\), and let \(C(\mathbb{Z})\) be the set of integral points on \(C\). Let \(\overline{C}\) be the Zariski closure of \(C\) in \(\mathbb{P}^{n}\) and \(C_{\infty} = (\overline{C}\setminus C)(\mathbb{C})\). We denote by \(\Sigma_{\infty}\) the set of discrete valuation rings of the function field of \(C\) at infinity (i.e. the discrete valuation rings which are above a local ring of \(\overline{C}\) at a point of \(C_{\infty}\)). The main result of the paper under review is as follows: Theorem A. The set \(C(\mathbb{Z})\) is infinite if and only if one of the following two conditions is satisfied: (a) \(C_\infty\) consists of one point and \(C(\mathbb{Z})\) contains at least one non-singular point. (b) \(C_\infty\) consists of two points which are conjugate over a real quadratic field and \(C({\mathbf Z})\) contains at least one non-singular point. The part ``only if'' of Theorem A, in the case \(n=2\), is a direct consequence of [\textit{A. Schinzel}, Comment., Pontif. Acad. Sci. 2, No. 20, 1-9 (1969; Zbl 0297.10009)]. Unfortunately, the part ``if'' is not correct, as one can easily see by the following two counterexamples. (1) The equation \(Y^2=(X+1)^2(X^{2}+15)\) defines a rational curve \(C\) with \(C_{\infty}=\{(0\colon 1\colon 0)\}\). Furthermore, the couple \((1,8)\) is a non-singular point on \(C\). Then Theorem A implies that \(C(\mathbb{Z})\) is infinite. If \((x,y)\) is an integer solution of the above equation, then \(x+1\mid y\) and so \(y=(x+1)z\), where \(z\) is an integer. It follows that \(z^{2}- x^{2}=15\), which clearly has finitely many solutions. We deduce that \((X,Y)=(1,\pm 8),(-1,0),(7,\pm 64),(-7,\pm 48)\). Hence part (a) of Theorem A is not correct. (2) Let \(C\) be the rational curve defined by \(f(X,Y)=\) \((X^{2}-2Y^{2})^{2}+XY = 0\). Then \(C_{\infty}=\{(\sqrt{2}\colon 1\colon 0),(-\sqrt{2}\colon 1\colon 0)\}\) and the couple \((…
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 1454439
zbMATH Keywords: integral points
zbMATH Keywords: affine curves
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
cites work: Sur le théorème de Runge
cites work: Q4143433
cites work: Bounds for the minimal solution of genus zero diophantine equations
cites work: Uncomputably large integral points on algebraic plane curves?
cites work: Q3710637
cites work: Q4486753
description: scientific article; zbMATH DE number 1454439