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I think after reading the title one may understand the intention of me, this question is concerned about the Elliptic curves having a Complex Multiplication.

I have been reading many theorems, ( celebrated papers of Zagier and Kolyvagin ). In majority, much of the proofs I came across, considers Elliptic Curves with Complex Multiplication. And many results have been discovered in that direction. I know what is meant by Complex Multiplication and endomorphism rings. But if some one asks for the reason behind such privilege enjoyed by the elliptic curves with CM ( Complex Multiplication ), what are the precise things one can tell ?

To put in other way, how come the proofs are discovered about Elliptic curves with CM are discovered so easily and why not for the other case ? ( Is there some bird's eye view ? )

Thank you.

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    I too got the same idea as the modular forms are said to have beautiful symmetry, but does this symmetry imply something ? @fretty2012-02-27
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    So, why can't you post a nice answer, combining all comments together into a beautiful elegant account. Anyway thanks a lot, for helping me. But is there any measure on this case ? ( like haar measure ) on the corresponding groups. @fretty2012-02-27
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    Actually, I didn't think I knew that much about CM! I will turn it into an answer.2012-02-27
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    In fact I had a glance about your blog, your intention is awesome, but have you done something to improve it ? @fretty2012-02-27
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    Well I started it last year as a spur of the moment thing. I made a few posts of things that interest me and was planning on getting round to doing some more interesting stuff (elliptic curves, class field theory, modular forms) but never really got round to it. Now I don't really have much time to update it (and tbh noone visits it anyway!) but I may start to update it now that I know someone is watching :p.2012-02-27
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    Do you know Srilakshmi Krishnamoorthy ? , anyway do you know about Hasse-Local Global principle ? , can you see [this](http://math.stackexchange.com/questions/97655/local-global-principle-and-the-cassels-statement) if you are not busy ? , and can you add something you know, I would be very happy, and debt with you. I think you can add something interesting to the blog, automatically the response will come @fretty2012-02-27
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    I only know the motivation behind the local-global principle. I have never studied it in full. However it doesn't work in general...it does for quadratic forms though.2012-02-27
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    In a certain sense, elliptic curves with CM are "induced" from simpler objects. The $L$-function of a CM elliptic curve is a Hecke $L$-function (a "$GL(1)$" object on the automorphic side), whereas the $L$-function of a non-CM elliptic curve is the $L$-function of a cusp form (a "$GL(2)$" object). More-or-less.2012-02-27

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