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Arithmetic of Elliptic Curves Upon Quadratic Extension Author s Kenneth Kramer Source Transactions of the American Mathematical Society Vol 264 No 1 Mar 1981 pp 121 135 Published by American Mathematical Society Stable URL http www jstor org stable 1998414 Accessed 01 09 2009 14 50 Your use of the JSTOR archive indicates your acceptance of JSTOR s Terms and Conditions of Use available at http www jstor org page info about policies terms jsp JSTOR s Terms and Conditions of Use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the JSTOR archive only for your personal non commercial use Please contact the publisher regarding any further use of this work Publisher contact information may be obtained at http www jstor org action showPublisher publisherCode ams Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission JSTOR is a not for profit organization founded in 1995 to build trusted digital archives for scholarship We work with the scholarly community to preserve their work and the materials they rely upon and to build a common research platform that promotes the discovery and use of these resources For more information about JSTOR please contact support jstor org American Mathematical Society is collaborating with JSTOR to digitize preserve and extend access to Transactions of the American Mathematical Society http www jstor org TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 264 Number 1 March 1981 ARITHMETIC OF ELLIPTIC CURVES UPON QUADRATIC EXTENSION BY KENNETH KRAMER1 ABSTRACT This paper is a study of variations in the rank of the Mordell Weil group of an elliptic curve E defined over a number field F as one passes to quadratic extensions K of F Let S K be the Selmer group for multiplication by 2 on E K In analogy with genus theory we describe

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