Regulators of elliptic curves

TL;DR

This paper investigates the properties of the regulator of elliptic curves over various fields, establishing new bounds and limitations for families of such curves with bounded regulators and ranks.

Contribution

It introduces a new Northcott property for regulators of high-rank elliptic curves over number fields and limits on families over function fields.

Findings

Proves a new Northcott property for elliptic curves of rank ≥ 4 over number fields.

Shows limitations on families of elliptic curves with bounded regulators over function fields.

Raises new questions for further research.

Abstract

We study the regulator of the Mordell-Weil group of elliptic curves over number fields, functions fields of characteristic zero or function fields of characteristic $p>0$. We prove a new Northcott property for the regulator of elliptic curves of rank at least $4$ defined over a number field. In the function field case, we obtain that non-trivial families of elliptic curves with bounded regulators and bounded ranks are limited families. We conclude by asking new questions.