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Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois
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  • Research Article
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  • Published: 07 October 2015

Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois

  • Jeremy Rouse1 &
  • David Zureick-Brown2 

Research in Number Theory volume 1, Article number: 12 (2015) Cite this article

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Abstract

We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over \(\mathbb {Q}\). To this end, we compute the ‘arithmetically maximal’ tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.

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Acknowledgments

We thank Jeff Achter, Nils Bruin, Tim Dokchitser, Bjorn Poonen, William Stein, Michael Stoll, Drew Sutherland, and David Zywina for useful conversations and University of Wisconsin-Madison’s Spring 2011 CURL (Collaborative undergraduate research Labs) students (Eugene Yoong, Collin Smith, Dylan Blanchard) for doing initial group theoretical computations. The second author is supported by an NSA Young Investigator grant. We would also like to thank anonymous referees for helpful comments and suggestions that have improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, Wake Forest University, Winston-Salem, 27109, USA

    Jeremy Rouse

  2. David Zureick-Brown, Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    David Zureick-Brown

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  1. Jeremy Rouse
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Correspondence to David Zureick-Brown.

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Rouse, J., Zureick-Brown, D. Elliptic curves over \(\mathbb {Q}\) and 2-adic images of Galois. Res. Number Theory 1, 12 (2015). https://doi.org/10.1007/s40993-015-0013-7

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  • Received: 26 May 2015

  • Accepted: 29 June 2015

  • Published: 07 October 2015

  • DOI: https://doi.org/10.1007/s40993-015-0013-7

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Keywords

  • Conjugacy Class
  • Elliptic Curve
  • Rational Point
  • Maximal Subgroup
  • Elliptic Curf
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