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THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 September 2019

SARA CHECCOLI Open the ORCID record for SARA CHECCOLI [Opens in a new window] ,
FRANCESCO VENEZIANO Open the ORCID record for FRANCESCO VENEZIANO [Opens in a new window]  and
EVELINA VIADA
SARA CHECCOLI
Affiliation:
Institut Fourier, 100 rue des Maths, BP74 38402 Saint-Martin-d’Hères Cedex, France; [email protected]
FRANCESCO VENEZIANO
Affiliation:
Collegio Puteano, Scuola Normale Superiore, Piazza dei Cavalieri, 3, I-56100 Pisa, Italy; [email protected]
EVELINA VIADA
Affiliation:
Mathematisches Institut, Georg-August-Universität, Bunsenstraße 3-5, D-37073, Göttingen, Germany; [email protected]

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.

MSC classification

Primary: 11G50: Heights
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e31
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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