Birch and Swinnerton-Dyer conjecture

Let E be an elliptic curveMathworldPlanetmath over , and let L(E,s) be the L-series attached to E.

Conjecture 1 (Birch and Swinnerton-Dyer).
  1. 1.

    L(E,s) has a zero at s=1 of order equal to the rank of E().

  2. 2.

    Let R=rank(E()). Then the residue of L(E,s) at s=1, i.e. lims1(s-1)-RL(E,s) has a concrete expression involving the following invariants of E: the real period, the Tate-Shafarevich group, the elliptic regulator and the Neron model of E.

J. Tate said about this conjecture: “This remarkable conjecture relates the behavior of a function L at a point where it is not at present known to be defined to the order of a group (Sha) which is not known to be finite!” The precise statement of the conjecture asserts that:



  • R is the rank of E/.

  • Ω is either the real period or twice the real period of a minimal model for E, depending on whether E() is connected or not.

  • |Sha| is the order of the Tate-Shafarevich group of E/.

  • Reg(E/) is the regulator of E().

  • |Etors()| is the number of torsion points on E/ (including the point at infinity O).

  • cp is an elementary local factor, equal to the cardinality of E(p)/E0(p), where E0(p) is the set of points in E(p) whose reductionPlanetmathPlanetmath modulo p is non-singularPlanetmathPlanetmath in E(𝔽p). Notice that if p is a prime of good reduction for E/ then cp=1, so only cp1 only for finitely many primes p. The number cp is usually called the Tamagawa number of E at p.

The following is an easy consequence of the B-SD conjecture:

Conjecture 2 (Parity Conjecture).

The root number of E, denoted by w, indicates the parity of the rank of the elliptic curve, this is, w=1 if and only if the rank is even.

There has been a great amount of research towards the B-SD conjecture. For example, there are some particular cases which are already known:

Theorem 1 (Coates, Wiles).

Suppose E is an elliptic curve defined over an imaginary quadratic fieldMathworldPlanetmath K, with complex multiplicationMathworldPlanetmath by K, and L(E,s) is the L-series of E. If L(E,1)0 then E(K) is finite.


  • 1 Claymath Institute, Description,
  • 2 J. Coates, A. Wiles, On the Conjecture of Birch and Swinnerton-Dyer, Inv. Math. 39, 223-251 (1977).
  • 3 Keith Devlin, The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, 189 - 212, Perseus Books Group, New York (2002).
  • 4 James Milne, Elliptic Curves, course notes.
  • 5 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 6 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
Title Birch and Swinnerton-Dyer conjecture
Canonical name BirchAndSwinnertonDyerConjecture
Date of creation 2013-03-22 13:49:46
Last modified on 2013-03-22 13:49:46
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 16
Author alozano (2414)
Entry type Conjecture
Classification msc 14H52
Synonym BS-D conjecture
Related topic EllipticCurve
Related topic RegulatorOfAnEllipticCurve
Related topic MordellCurve
Related topic ArithmeticOfEllipticCurves
Defines Birch and Swinnerton-Dyer conjecture
Defines parity conjecture