Elliptic curves with long arithmetic progressions have large rank
Number Theory
2019-11-01 v1 Complex Variables
Abstract
For any family of elliptic curves over the rational numbers with fixed -invariant, we prove that the existence of a long sequence of rational points whose -coordinates form a non-trivial arithmetic progression implies that the Mordell-Weil rank is large, and similarly for -coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as R\'emond's quantitative extension of results of Faltings.
Cite
@article{arxiv.1910.14485,
title = {Elliptic curves with long arithmetic progressions have large rank},
author = {Natalia Garcia-Fritz and Hector Pasten},
journal= {arXiv preprint arXiv:1910.14485},
year = {2019}
}