#### 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 $\mathsc{𝒞}$ 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 $\mathsc{𝒞}$ 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