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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion


Author: Zev Klagsbrun
Journal: Trans. Amer. Math. Soc. 369 (2017), 3355-3385
MSC (2010): Primary 11G05
DOI: https://doi.org/10.1090/tran/6744
Published electronically: October 12, 2016
MathSciNet review: 3605974
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Abstract: This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve $ E$ defined over a number field $ K$ with $ E(K)[2] \simeq \mathbb{Z}/2\mathbb{Z}$. We show that if $ E$ does not have a cyclic 4-isogeny defined over $ K(E[2])$ with kernel containing $ E(K)[2]$, then subject only to constant 2-Selmer parity, each non-negative integer appears infinitely often as the 2-Selmer rank of a quadratic twist of $ E$. If $ E$ has a cyclic 4-isogeny with kernel containing $ E(K)[2]$ defined over $ K(E[2])$ but not over $ K$, then we prove the same result for 2-Selmer ranks greater than or equal to $ r_2$, the number of complex places of $ K$. We also obtain results about the minimum number of twists of $ E$ with rank 0 and, subject to standard conjectures, the number of twists with rank $ 1$, provided $ E$ does not have a cyclic 4-isogeny defined over $ K$.


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Additional Information

Zev Klagsbrun
Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121
Email: zdklags@ccrwest.org

DOI: https://doi.org/10.1090/tran/6744
Received by editor(s): August 9, 2012
Received by editor(s) in revised form: May 7, 2014, March 1, 2015, and May 5, 2015
Published electronically: October 12, 2016
Additional Notes: This paper is based on work conducted by the author as part of his doctoral thesis at UC-Irvine under the direction of Karl Rubin and was supported in part by NSF grants DMS-0457481 and DMS-0757807.
Article copyright: © Copyright 2016 Copyright retained by the Institute for Defense Analyses. Work for hire done under contract with the U.S. Government.

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