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Selmer Groups of Elliptic Curves with Complex Multiplication

Published online by Cambridge University Press:  20 November 2018

A. Saikia
A. Saikia*
Affiliation:
McGill University, Montreal, Quebec, H3A 2K6
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Abstract

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Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in $K$. Let $p$ be a rational prime that splits as ${{\mathfrak{p}}_{1}}{{\mathfrak{p}}_{2}}$ in $K$. Let ${{E}_{{{p}^{n}}}}$ denote the ${{p}^{n}}$-division points on $E$. Assume that $F\left( {{E}_{{{p}^{n}}}} \right)$ is abelian over $K\,\text{for}\,\text{all}\,n\,\ge \,0$. This paper proves that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty }$-Selmer group of $E$ over $F\left( {{E}_{{{p}^{\infty }}}} \right)$ is a finitely generated free $\wedge $-module, where $\wedge $ is the Iwasawa algebra of $\text{Gal}\left( F\left( {{E}_{{{p}^{\infty }}}} \right)/F\left( E\mathfrak{p}_{1}^{\infty }{{\mathfrak{p}}_{2}} \right) \right)$ . It also gives a simple formula for the rank of the Pontrjagin dual as a $\wedge $-module.

Keywords

11R2311G05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 1 , 01 February 2004 , pp. 194 - 208
Copyright
Copyright © Canadian Mathematical Society 2004

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