In the early 1960s, B. Birch and H. P. F. Swinnerton-Dyer conjectured that if a given Elliptic Curve has an infinite number of solutions, then the associated -function has value 0 at a certain fixed point. 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 (Coates-Wiles Theorem), but they were unable to prove the converse. V. Kolyvagin extended this result to modular curves.

**References**

Cipra, B. ``Fermat Prover Points to Next Challenges.'' *Science* **271**, 1668-1669, 1996.

Ireland, K. and Rosen, M. ``New Results on the Birch-Swinnerton-Dyer Conjecture.''
§20.5 in *A Classical Introduction to Modern Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 353-357, 1990.

Mazur, B. and Stevens, G. (Eds.). *-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture.*
Providence, RI: Amer. Math. Soc., 1994.

© 1996-9

1999-05-26