Height of rational points on quadratic twists of a given elliptic curve

TL;DR

This paper proposes a conjecture on how the canonical height of the smallest non-torsion rational point varies across quadratic twists of a fixed elliptic curve, highlighting deep unresolved questions in number theory.

Contribution

It introduces a new conjecture about the distribution of minimal rational points' heights on quadratic twists of elliptic curves, with partial results supporting it.

Findings

Partial proofs of the conjecture

Insights into the distribution of rational points

Connections to deep number theory problems

Abstract

We formulate a conjecture about the distribution of the canonical height of the lowest non-torsion rational point on a quadratic twist of a given elliptic curve, as the twist varies. This conjecture seems to be very deep and we can only prove partial results in this direction.