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.