TOPICS
Search

Swinnerton-Dyer Conjecture

The Birch-Swinnerton-Dyer conjecture states that the rank of an elliptic curve E over the rationals equals the order of vanishing of its Hasse-Weil L-series at s=1,

rankE(Q)=ord_(s=1)L(E,s).

Its refined form also gives the leading coefficient of L(E,s) at s=1 in terms of arithmetic invariants of E, including the order of the Tate-Shafarevich group; in particular, it predicts that this group is finite.

Coates and Wiles (1977) proved one direction for elliptic curves with complex multiplication (Coates-Wiles theorem). Building on the Gross-Zagier formula, Kolyvagin (1989, 1990) proved the rank equality and finiteness of the Tate-Shafarevich group for modular elliptic curves of analytic rank 0 or 1.

See also

Coates-Wiles Theorem, Elliptic Curve, Gross-Zagier Formula, Hasse-Weil L-Function, Tate-Shafarevich Group

Explore with Wolfram|Alpha

References

Birch, 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." https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/.Gross, B. H. and Zagier, D. B. "Heegner Points and Derivatives of L-Series." Invent. Math. 84, 225-320, 1986. https://doi.org/10.1007/BF01388809.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.Kolyvagin, V. A. "Finiteness of E(Q) and Sha(E,Q) for a Subclass of Weil Curves." Math. USSR-Izv. 32, 523-541, 1989. https://doi.org/10.1070/IM1989v032n03ABEH000779.Kolyvagin, V. A. "Euler Systems." In The Grothendieck Festschrift, Vol. II. Boston, MA: Birkhäuser, pp. 435-483, 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." https://www.claymath.org/wp-content/uploads/2022/05/birchswin.pdf.

Referenced on Wolfram|Alpha

Swinnerton-Dyer Conjecture

Cite this as:

Weisstein, Eric W. "Swinnerton-Dyer Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Swinnerton-DyerConjecture.html

Subject classifications