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The Elkies curve has rank 28 subject only to GRH

Authors: Zev Klagsbrun, Travis Sherman and James Weigandt
Journal: Math. Comp. 88 (2019), 837-846
MSC (2010): Primary 11-04, 11G05, 11Y40, 14G05, 14M52
Published electronically: May 17, 2018
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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 We prove a similar result for a previously unpublished curve of Elkies having rank 27 as well.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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Additional Information

Zev Klagsbrun
Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121

Travis Sherman
Affiliation: 3208 Riva Ridge Ct, Bowie, Maryland 20721

James Weigandt
Affiliation: ICERM, Brown University, Providence, Rhode Island 02903
Address at time of publication: P.O. Box 4671, Sidney, Ohio 45365

Received by editor(s): June 23, 2017
Received by editor(s) in revised form: November 6, 2017, and December 5, 2017
Published electronically: May 17, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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