Discretisation for odd quadratic twists

TL;DR

This paper investigates the discretisation problem for odd quadratic twists of elliptic curves, exploring models and data to understand how height and other factors influence the distribution of rational points.

Contribution

It introduces new models and presents data addressing the less understood discretisation problem for odd quadratic twists, extending previous work on even twists.

Findings

Insights into how height affects discretisation in odd quadratic twists

Data supporting proposed models for odd twists

Comparison with predictions from random matrix theory

Abstract

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.