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Let $E$ be an elliptic curve defined over a number field $k$ and $\ell$ aprime 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 thenumber of points of $E(k)$ of (exponential) height at most $B$. Here theconstant $C = C(k)$ depends on the number field $k$ and is effective. For $\ell= 2$ this generalizes a result of Naccarato which applies for $k=\mathbb{Q}$.We follow methods previously developed by Bombieri and Zannier and further byNaccarato, with the main novelty being the application of Rosen's result onbounding $\ell$-ranks of class groups in certain extensions, which is derivedusing relative genus theory.

Uniform Bounds for the Number of Rational Points of Bounded Height on Certain Elliptic Curves