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In 1976, Coates and Wiles showed that Elliptic Curves with Complex
Multiplication having an infinite number of solutions have 
-functions which are zero at the relevant fixed point.
This is a special case of the Swinnerton-Dyer Conjecture.
References
Cipra, B. ``Fermat Prover Points to Next Challenges.'' Science 271, 1668-1669, 1996.