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 $\mathcal{𝒞}$ 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 $\mathcal{𝒞}$ 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.

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