The regulator dominates the rank

TL;DR

This paper establishes a lower bound for the regulator of the Mordell-Weil group of elliptic curves over global function fields, linking it to the rank and height, inspired by properties of number field regulators.

Contribution

It introduces a novel lower bound for the Mordell-Weil regulator in function fields, extending the analogy with number field regulators and their dominance over ranks.

Findings

Lower bound increases with rank and height

Connects regulator bounds to properties of elliptic curves over function fields

Provides insights into the structure of Mordell-Weil groups

Abstract

After noticing that the regulator of a number field dominates the rank of its group of units, we bound from below the regulator of the Mordell-Weil group of elliptic curves over global function fields of characteristic $p\geq5$. The lower bound is an increasing function of the rank and of the height.