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 -series 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 -series 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.
See alsoCoates-Wiles Theorem, Elliptic Curve
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ReferencesBirch, B. and Swinnerton-Dyer, H. "Notes on Elliptic Curves. II." J. reine angew. Math. 218, 79-108, 1965.Cipra, B. "Fermat Prover Points to Next Challenges." Science 271, 1668-1669, 1996.Clay Mathematics Institute. "The Birch and Swinnerton-Dyer Conjecture." http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/.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.). p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture. Providence, RI: Amer. Math. Soc., 1994.Wiles, A. "The Birch and Swinnerton-Dyer Conjecture." http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/BSD.pdf.
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Weisstein, Eric W. "Swinnerton-Dyer Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Swinnerton-DyerConjecture.html