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THE DISTRIBUTION OF 2-SELMER RANKS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH PARTIAL TWO-TORSION

Part of: Multiplicative number theory Arithmetic algebraic geometry Curves

Published online by Cambridge University Press:  04 May 2015

Zev Klagsbrun  and
Robert J. Lemke Oliver
Zev Klagsbrun
Affiliation:
Center for Communications Research, 4320 Westerra Court, San Diego, CA 92121, U.S.A. email [email protected]
Robert J. Lemke Oliver
Affiliation:
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, U.S.A. email [email protected]
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Abstract

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field $K$ with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion.

MSC classification

Primary: 11G05: Elliptic curves over global fields 11N45: Asymptotic results on counting functions for algebraic and topological structures 14H52: Elliptic curves
Type
Research Article
Information
Mathematika , Volume 62 , Issue 1 , 2016 , pp. 67 - 78
Copyright
Copyright © University College London 2015 

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