Vol. 315, No. 2, 2021

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

Francesco Veneziano and Evelina Viada

Vol. 315 (2021), No. 2, 477–503
DOI: 10.2140/pjm.2021.315.477
Abstract

Let 𝒞 be an algebraic curve embedded transversally in a power EN 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 EN. 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.

Keywords
Mordell conjecture, subvarieties of products of elliptic curves, diophantine approximation
Mathematical Subject Classification
Primary: 11G50, 14G40
Milestones
Received: 16 June 2020
Revised: 6 September 2021
Accepted: 6 September 2021
Published: 19 January 2022
Authors
Francesco Veneziano
Dipartimento di matematica
Università degli Studi di Genova
16146 Genova (GE)
Italy
Evelina Viada
Mathematisches Institut
Georg-August-Universität
D-37073 Göttingen
Germany