1,720,978 research outputs found
Lattices and Rational Points
No full textIn this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N - 1 on transverse curves in E N , where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has Q -rank ≤ N - 1 . We also give some explicit examples. This result generalises from rank 1 to rank N - 1 previous results of S. Checcoli, F. Veneziano and the author
An Explicit Manin-Dem’janenko Theorem in Elliptic Curves
No full textAbstractLet be a curve of genus at least 2 embedded in E1 × … × EN, where the Ei are elliptic curves for i = 1, . . . , N. In this article we give an explicit sharp bound for the Néron–Tate height of the points of contained in the union of all algebraic subgroups of dimension < max(), where is the minimal dimension of a translate (resp. of a torsion variety) containing .As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin–Dem’janenko method for curves in products of elliptic curves.</jats:p
On the minimal set for counterexamples to the local-global principle
No full textLet p > 3 be a prime number and let n be a positive integer. We prove that the local-global principle for divisibility by p(n) holds for elliptic curves defined over the rationals. For this, we refine our previous criterion for the validity of the principle. We also give an example that shows that the assumptions of our criterion are necessary. (C) 2014 Elsevier Inc. All rights reserved
A sharp Bogomolov-type bound
No full textWe prove a sharp lower bound for the essential minimum of a nontranslate variety in certain abelian varieties. This uses and generalises a result of Galateau. Our bound is a new step in the direction of an abelian analogue by David and Philippon of a toric conjecture of Amoroso and David and has applications in the framework of anomalous intersections.Swiss National Science Foundation (SNSF
A sharp Bogomolov-type bound
No full textWe prove a sharp lower bound for the essential minimum of a nontranslate variety in certain abelian varieties. This uses and generalises a result of Galateau. Our bound is a new step in the direction of an abelian analogue by David and Philippon of a toric conjecture of Amoroso and David and has applications in the framework of anomalous intersections.Swiss National Science Foundation (SNSF
Explicit height bounds for K-rational points on transverse curves in powers of elliptic curves
No full textLet C be an algebraic curve embedded transversally in a power E-N of an elliptic curve E with complex multiplication. We produce a good explicit bound for the height of all the algebraic points on C contained in the union of all proper algebraic subgroups of E-N. The method gives a totally explicit version of the Manin-Demjanenko theorem in the elliptic case and complements previous results only proved when E does not have complex multiplication
THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES
No full textInternational audienceIn 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
On the Torsion Anomalous Conjecture in CM abelian varieties
No full textInternational audienceThe Torsion Anomalous Conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties. In this work we prove, with an effective method, some cases of the TAC when the ambient variety A has CM, generalising our previous results in products of CM elliptic curves. When V is a curve, we give new results and we deduce some implications on the effective Mordell-Lang Conjecture
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