A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Uniform bounds for the number of rational points of bounded height on certain elliptic curves

Volume 217 / 2025

Marta Dujella Acta Arithmetica 217 (2025), 309-332 MSC: Primary 11G05; Secondary 11G50 DOI: 10.4064/aa231221-9-10 Published online: 27 December 2024

Abstract

Let $E$ be an elliptic curve defined over a number field $k$, and $\ell $ a prime integer. When $E$ has at least one $k$-rational point of exact order $\ell $, we derive a uniform upper bound $\exp(C\log B/\log\log B)$ for the number of points of $E(k)$ of (exponential) height at most $B$. Here the constant $C = C(k)$ depends on the number field $k$ and is effective. For $\ell = 2$ this generalizes a result of Naccarato (2021) which applies for $k=\mathbb Q$. We follow the methods developed by Bombieri and Zannier (2004) and Naccarato (2021), with the main novelty being the application of Rosen’s result on bounding $\ell $-ranks of class groups in certain extensions, which is derived using relative genus theory.

Authors

  • Marta DujellaDepartment of Mathematics and Computer Science
    University of Basel
    4051 Basel, Switzerland
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image