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 $ a prime integer. When E has at least one k-rational point of exact order $, 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 $ = 2 this generalizes a result of Naccarato (2021) which applies for k = 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 $-ranks of class groups in certain extensions, which is derived using relative genus theory.