Application of the method of Dem'yanenko-Manin to certain families of curves of genus 2 or 3
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 the ordinary and the canonical height of points on an elliptic curve. Silverman is cited for that, and for the special CM-curves (see above), the estimate can be made independent of the curve (that is, of \(a\in \mathbb{N})\) as was shown by Grigorov and Rizov (in an unpublished manuscript). Note that the first estimates were obtained independently by Dem'yanenko and Zimmer. We remark that, in her dissertation, written in 1999 at the University of Saarbrücken, \textit{Susanne Schmitt} has determined the rank and even the whole Mordell-Weil group of any elliptic curve over \(\mathbb{Q}\) modulo the truth of the Birch and Swinnerton-Dyer conjecture. The author mentions this conjecture in connection with the above special CM-curves \(y^2=x^3-a^2x\), \(a\in\mathbb{N}\), some of which then have rank one over \(\mathbb{Q}\). For proving the independence of morphisms, a criterion is used taken from a paper of Cassels, where he establishes Dem'yanenko's method in a weakened setting by referring to his own ideas.
algebraic curve of genus greater than one
scientific article; zbMATH DE number 1310088
scientific article