The Elkies Curve has Rank 28 Subject only to GRH

TL;DR

This paper proves, under GRH, that certain elliptic curves have Mordell-Weil ranks matching their analytic ranks, providing evidence for the Birch and Swinnerton-Dyer conjecture at very high ranks, using class group computations.

Contribution

It establishes, subject to GRH, the exact Mordell-Weil ranks of specific high-rank elliptic curves and related class groups, advancing understanding of the BSD conjecture for high-rank curves.

Findings

Elliptic curve from Elkies has rank 28 under GRH.

Class group of a specific cubic field has 2-rank 22.

Class group of a totally real cubic field has 2-rank 20.

Abstract

In 2006, Elkies presented an elliptic curve with 28 independent rational points. We prove that subject to GRH, this curve has Mordell-Weil rank equal to 28 and analytic rank at most 28. We prove similar results for a previously unpublished curve of Elkies having rank 27. We also prove that subject to GRH, certain specific elliptic curves have Mordell-Weil ranks 20, 21, 22, 23, and 24. This complements the work of Jonathan Bober, who proved this claim subject to both the Birch and Swinnerton-Dyer rank conjecture and GRH. This gives some new evidence that the Birch and Swinnerton-Dyer rank conjecture holds for elliptic curves over Q of very high rank. Our results about Mordell-Weil ranks are proven by computing the 2-ranks of class groups of cubic fields associated to these elliptic curves. As a consequence, we also succeed in proving that, subject to GRH, the class group of a particular cubic field has 2-rank equal to 22 and that the class group of a particular totally real cubic field has 2-rank equal to 20.