2023/12/06 by Dujella, Marta
#11G05 #11G50 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2312.03655
Let E be an elliptic curve defined over a number field k and ℓ a prime integer. When E has at least one k-rational point of exact order ℓ, 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 ℓ = 2 this generalizes a result of Naccarato which applies for k=ℚ. We follow methods previously developed by Bombieri and Zannier and further by Naccarato, with the main novelty being the application of Rosen's result on bounding ℓ-ranks of class groups in certain extensions, which is derived using relative genus theory.