The Explicit Mordell Conjecture for Families of Curves

Abstract

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin-Demjanenko and the analysis of our explicit examples is carried to conclusion.

An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

“…The Proof of Corollary 1.5. In [CVV16], Theorem 6.2, we proved that the curve C is transverse in E 2 and its degree and height are bounded as deg C = 6n + 9, h 2 (C) ≤ 6(2n + 3) (h W (p) + log m + 2C(E)) where h W (p) = h W (1 : p 0 : . .…”

“…In the last years, we have been working to approach the problem with explicit methods aiming to prove new cases of the explicit Mordell Conjecture and to eventually find all the rational points on some curves. In [CVV15] and [CVV16] joint with S. Checcoli and F. Veneziano, we give an explicit bound for the Néron-Tate height of the set of points of rank one on curves of genus at least two in E N where E is without CM. The non CM assumption is technical and we handled there the easier case where the endomorphism ring of E is Z.…”

Explicit height bounds for $K$-rational points on transverse curves in powers of elliptic curves

“…Thus, in the applications, such a dependence must be elaborated on a case-by-case basis with ad hoc strategies and this has been carried out successfully only for some special families of curves with small genus, typically 2 or 3. (We refer to the introduction of [CVV19] for an account on the subject).…”

“…In this article we generalise an explicit method that we introduced in [CVV19] investigating its strength and its limits. More precisely we give a simple formula for the height of the points in the CM and non-CM case and for rank larger than 1.…”

“…In [CVV19] the authors and Checcoli gave a good explicit bound for the Néron-Tate height of the set of points of rank one on algebraic curves of genus at least two in E 2 where E is without CM; the sharpness of the bounds allowed explicit examples to be computed.…”