Integer points on elliptic curves
type: Item
instance of: scholarly article
title: Integer points on elliptic curves
zbMATH Open document ID: 1311.11048
DOI: 10.1216/RMJ-2014-44-4-1377
published in: Rocky Mountain Journal of Mathematics
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1414760959
reviewed by: Shabnam Akhtari
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 6368771
zbMATH Keywords: integral points on elliptic curves
zbMATH Keywords: Mordell equation
zbMATH Keywords: large sieve
author: Robert C. Vaughan
MaRDI profile type: Publication
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review text: Let \(f(x)\) be a polynomial of degree \(d\) with integer coefficients and \(c\) a non-zero integer. Assume that \(c y^2 - f(x)\) is absolutely irreducible. By \textit{C. L. Siegel}'s theorem [Abh. Preuß. Akad. Wiss., Phys.-Math. Kl. 1929, No. 1, 70 S. (1929; JFM 56.0180.05)], we know that there are only finitely many integral points on the curve \(c y^2 = f(x)\). Siegel's method does not provide any local bound. The author uses \textit{P. X. Gallagher}'s sieve [Mathematika 14, 14--20 (1967; Zbl 0163.04401)] to obtain upper bounds for the number of integral points of bounded size on the curve \(c y^2 = f(x)\). For real numbers \(X_{0}\) and \(X \geq 1\), define \(N_{f}(X ; X_{0})\) to be the number of integral points \((x , y)\) with \(X_{0} < x \leq X_{0} + X\) and \(c y^2 = f(x)\). The author shows that NEWLINE\[NEWLINE N_{f}(X; X_{0})\ll X^{1/2} NEWLINE\]NEWLINE where the implicit constant depends at most on \(d\), the degree of \(f(x)\). In particular, the author concludes that the number of positive integer points on the Mordell equation NEWLINE\[NEWLINE x^3 + y^2 =n, NEWLINE\]NEWLINE where \(n \in \mathbb{N}\), is \(\ll n^{1/6}\).
description: scientific article; zbMATH DE number 6368771