Application of the method of Dem'yanenko-Manin to certain families of curves of genus 2 or 3
type: Item
instance of: scholarly article
title: Application of the method of Dem'yanenko-Manin to certain families of curves of genus 2 or 3
zbMATH Open document ID: 0941.11027
author: Leopoldo Kulesz
published in: Journal of Number Theory
publication date: 2000-08-10T00:00:00Z
review text: The famous theorem of Faltings proving the Mordell conjecture according to which an algebraic curve of genus greater than one over an algebraic number field \(K\) has at most a finite number of rational points is not effective in the sense that no bounds are given e.g. for the heights of those points. The same is true for the improvements of the verifications of Mordell's conjecture by authors like Bombieri and Vojta. However, in some special cases, there are two earlier, effective methods available, the method of Chabauty and the method of Dem'yanenko-Manin. While the first method was frequently used, for instance by Flynn, Schaefer, Wetherell and others, the same does not hold for the second method. The author of the present paper employs the second method, thus showing its effectiveness, to determine all rational points on the parametrized curves over the rationals \(K=\mathbb{Q}\): \[ ay^2=6(x^2-4)(x^2+2) (x^2+8),\;a\in\mathbb{N}, \] of genus 2 and \[ ay^2=-x(x^2-1)(x+2) (2x+1)(x^2+x+1),\;a\in\mathbb{N}. \] of genus 3. The element \(a\in\mathbb{N}\) is chosen in both cases in such a way that the CM-elliptic curve \(y^2=x^3-ax\) over \(\mathbb{Q}\) has rank one. One then has to exhibit two independent morphisms \(\varphi_1, \varphi_2\) over \(\mathbb{Q}\). For the crux of the method of Dem'yanenko-Manin consists in the statement that if you find \(r\) morphisms \(\varphi_i: {\mathcal C}\to {\mathcal E}\) \((i=1,\dots,r)\) of a curve \({\mathcal C}\) of genus greater than one over a number field \(K\) to an elliptic curve \({\mathcal E}\) over \(K\), if they are independent over \(K\), and if \(r\) is greater than the rank of \({\mathcal C}\) over \(K\), then the cardinality of the set \({\mathcal C}(K)\) is finite. This is one of the first proofs of Mordell's conjecture in a special case. The proof of the results of the present paper is, as opposed to Falting's proof, rather elementary. One of the main ingredients of it is an estimate for the difference between t…
reviewed by: Horst G. Zimmer
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 1310088
zbMATH Keywords: curves
zbMATH Keywords: genus
zbMATH Keywords: morphism
zbMATH Keywords: Faltings theorem
zbMATH Keywords: Mordell conjecture
zbMATH Keywords: algebraic curve of genus greater than one
zbMATH Keywords: method of Demyanenko-Manin
zbMATH Keywords: rational points
zbMATH Keywords: height
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
MaRDI profile type: Publication
full work available at URL: jnth.1998.2339
OpenAlex ID: W1972411183
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DOI: 10.1006/JNTH.1998.2339
description: scientific article; zbMATH DE number 1310088