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On the φ-Selmer groups of the elliptic curves y2 = x3 − Dx

Published online by Cambridge University Press:  09 September 2016

DANIEL M. KANE  and
JACK A. THORNE
DANIEL M. KANE
Affiliation:
Department of Mathematics, University of California, San Diego, 9500 Gilman Drive #0404, La Jolla, CA 92093-0404. e-mail: dakane@math.ucsd.edu
JACK A. THORNE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, CB3 0WB, U.K. e-mail: thorne@dpmms.cam.ac.uk
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Abstract

We study the variation of the φ-Selmer groups of the elliptic curves y2 = x3 − Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integer D is constrained to lie in a family for which the relative Tamagawa number of the isogeny φ is fixed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 71 - 93
Copyright
Copyright © Cambridge Philosophical Society 2016 

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