The Elkies curve has rank 28 subject only to GRH

Klagsbrun, Zev; Sherman, Travis; Weigandt, James

doi:10.1090/mcom/3348

The Elkies curve has rank 28 subject only to GRH
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Math. Comp. 88 (2019), 837-846 Request permission

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. We prove a similar result for a previously unpublished curve of Elkies having rank 27 as well.

Our work complements work of Bober and Booker and Dwyer that can be used to obtain these same results subject to both GRH and the BSD conjecture. This provides new evidence that the rank portion of the BSD conjecture holds for elliptic curves over $\mathbb {Q}$ 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 $20$.

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