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THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

Published online by Cambridge University Press:  19 September 2019

SARA CHECCOLI
Affiliation:
Institut Fourier, 100 rue des Maths, BP74 38402 Saint-Martin-d’Hères Cedex, France; sara.checcoli@univ-grenoble-alpes.fr
FRANCESCO VENEZIANO
Affiliation:
Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri, 3, I-56100 Pisa, Italy; francesco.veneziano@sns.it
EVELINA VIADA
Affiliation:
Mathematisches Institut, Georg-August-Universität, Bunsenstraße 3-5, D-37073, Göttingen, Germany; evelina.viada@math.ethz.ch

Abstract

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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.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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