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Groups associated with conics

Published online by Cambridge University Press:  01 August 2016

Shailesh A. Shirali
Shailesh A. Shirali*
Affiliation:
Rishi Valley School, Rishi Valley 517 352, Andhra Pradesh, Indiae-mail: Shailesh.Shirali@gmail.com
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Extract

The fact that an abelian group can be associated with an irreducible cubic curve has been known for a long time, and many results of great beauty are known about such groups. Ever since Wiles' proof of Fermat's Last Theorem, some of these results have become well known even outside the mathematical world.

It may come as a surprise to the reader that there is a natural way to associate an abelian group with a much better known and simpler class of curves – the conic sections! In this article we explore some properties of these groups.

Type
Articles
Information
The Mathematical Gazette , Volume 93 , Issue 526 , March 2009 , pp. 27 - 41
Copyright
Copyright © The Mathematical Association 2009

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References

1. Lemmermeyer, Franz, Conies – a poor man’s elliptic curves http://www.fen.bilkent.edu.tr/~franz/publ/conics.pdf+ Google Scholar
2. Lemmermeyer, Franz, Higher descent on Pell conies http://www.fen.bilkent.edu.tr/~franz/publ/pell-c.pdf+ Google Scholar
3. Silverman, Joseph H. & Tate, John, Rational points on elliptic curves. Springer Verlag (1992).Google Scholar