Birch and Swinnerton-Dyer conjecture
Let be an elliptic curve over , and let be the L-series attached to .
Conjecture 1 (Birch and Swinnerton-Dyer).
J. Tate said about this conjecture: “This remarkable conjecture relates the behavior of a function 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:
is the rank of .
is the order of the Tate-Shafarevich group of .
is the http://planetmath.org/node/RegulatorOfAnEllipticCurveelliptic regulator of .
The following is an easy consequence of the B-SD conjecture:
Conjecture 2 (Parity Conjecture).
The root number of , denoted by , indicates the parity of the rank of the elliptic curve, this is, 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).
- 1 Claymath Institute, Description, http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/online.
- 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, http://www.jmilne.org/math/CourseNotes/math679.htmlonline 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|
|Date of creation||2013-03-22 13:49:46|
|Last modified on||2013-03-22 13:49:46|
|Last modified by||alozano (2414)|
|Defines||Birch and Swinnerton-Dyer conjecture|