The Elkies curve has rank 28 subject only to GRH
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- by Zev Klagsbrun, Travis Sherman and James Weigandt;
- Math. Comp. 88 (2019), 837-846
- DOI: https://doi.org/10.1090/mcom/3348
- 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 We prove a similar result for a previously unpublished curve of Elkies having rank 27 as well.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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Bibliographic Information
- Zev Klagsbrun
- Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121
- MR Author ID: 1016010
- Email: zdklags@ccrwest.org
- Travis Sherman
- Affiliation: 3208 Riva Ridge Ct, Bowie, Maryland 20721
- Email: glaisher@hotmail.com
- James Weigandt
- Affiliation: ICERM, Brown University, Providence, Rhode Island 02903
- Address at time of publication: P.O. Box 4671, Sidney, Ohio 45365
- MR Author ID: 1175569
- Email: havepenwillfigure@gmail.com
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 837-846
- MSC (2010): Primary 11-04, 11G05, 11Y40, 14G05, 14M52
- DOI: https://doi.org/10.1090/mcom/3348
- MathSciNet review: 3882286