Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Selmer ranks of quadratic twists of elliptic curves with partial rational two-torsion
HTML articles powered by AMS MathViewer

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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G05
  • Retrieve articles in all journals with MSC (2010): 11G05
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