Swinnerton-Dyer Conjecture

In the early 1960s, B. Birch and H. P. F. Swinnerton-Dyer conjectured that if a given Elliptic Curve has aninfinite number of solutions, then the associated -function has value 0 at a certain fixed point. In 1976, Coates andWiles showed that elliptic curves with Complex multiplication having an infinite number ofsolutions have -functions which are zero at the relevant fixed point (Coates-Wiles Theorem), but they were unableto 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.

• Batch
• Bateman Function
• Batrachion
• Bauer's Identical Congruence
• Bauer's Theorem
• Bauspiel
• Bayes' Formula
• Bayesian Analysis
• Bayes' Theorem
• Bays' Shuffle
• Beam Detector
• Bean Curve
• Beast Number
• Beatty Sequence
• Beauzamy and Dégot's Identity
• Bee
• Behrens-Fisher Test
• Behrmann Cylindrical Equal-Area Projection
• Bei
• Bell Curve
• Bell Number
• Bellows Conjecture
• Bell Polynomial
• Bell Triangle
• Beltrami Differential Equation