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

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.