The Elkies Curve has Rank 28 Subject only to GRH
| Autor: | James Weigandt, Zev Klagsbrun, Travis Sherman |
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| Rok vydání: | 2016 |
| Předmět: |
Class (set theory)
Algebra and Number Theory Conjecture Mathematics - Number Theory Rank (linear algebra) Group (mathematics) Applied Mathematics Mathematics::Number Theory 010102 general mathematics Subject (documents) 010103 numerical & computational mathematics 01 natural sciences Combinatorics Computational Mathematics Elliptic curve FOS: Mathematics Cubic field Number Theory (math.NT) 0101 mathematics 11G05 Mathematics |
| DOI: | 10.48550/arxiv.1606.07178 |
| Popis: | 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 We prove similar results for a previously unpublished curve of Elkies having rank 27.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. |
| Databáze: | OpenAIRE |
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