Rigidity in Elliptic Curve Local-Global Principles
Number Theory
2023-06-09 v3
Abstract
We study the rigidity of the local conditions in two well-known local-global principles for elliptic curves over number fields. In particular, we consider a local-global principle for torsion due to Serre and Katz, and one for isogenies due to Sutherland. For each of these local-global principles, we prove that if an elliptic curve over a number field is such that it fails to satisfy the local condition for at least one prime ideal of of good reduction, then can satisfy the local condition at no more than 75% of prime ideals. We also give for (conjecturally) all elliptic curves over the rationals without complex multiplication, the densities of primes that satisfy the local conditions mentioned above.
Cite
@article{arxiv.2005.05881,
title = {Rigidity in Elliptic Curve Local-Global Principles},
author = {Jacob Mayle},
journal= {arXiv preprint arXiv:2005.05881},
year = {2023}
}