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.