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On the φ-Selmer groups of the elliptic curves y 2 = x 3Dx

Published online by Cambridge University Press:  09 September 2016

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
Corresponding

Abstract

We study the variation of the φ-Selmer groups of the elliptic curves y 2 = x 3Dx 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
Copyright
Copyright © Cambridge Philosophical Society 2016 

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On the φ-Selmer groups of the elliptic curves y 2 = x 3Dx
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