Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion
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- by Zev Klagsbrun PDF
- Trans. Amer. Math. Soc. 369 (2017), 3355-3385
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 $K$.References
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Additional Information
- Zev Klagsbrun
- Affiliation: Center for Communications Research, 4320 Westerra Court, San Diego, California 92121
- MR Author ID: 1016010
- Email: zdklags@ccrwest.org
- 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.
- © Copyright 2016 Copyright retained by the Institute for Defense Analyses. Work for hire done under contract with the U.S. Government.
- Journal: Trans. Amer. Math. Soc. 369 (2017), 3355-3385
- MSC (2010): Primary 11G05
- DOI: https://doi.org/10.1090/tran/6744
- MathSciNet review: 3605974