Expected Mordell-Weil rank heuristics through Sato-Tate, Birch and Swinnerton-Dyer conjectures

TL;DR

This paper develops heuristic predictions for the average Mordell-Weil rank of elliptic curves over number fields by leveraging the Birch and Swinnerton-Dyer and Sato-Tate conjectures, exploring their implications and relations.

Contribution

It introduces a novel heuristic framework connecting BSD and Sato-Tate conjectures to expected ranks, with calculations and questions about their relation to existing models.

Findings

Heuristic predictions for average Mordell-Weil ranks.

Calculations in specific cases illustrating the approach.

Open questions about relations to existing rank models.

Abstract

We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjectures. We do calculations in some cases and raise questions about their relations, if any, with the predictions of various average rank models that have been considered.