Birch and Swinnerton-Dyer conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function. It has been proved only in special cases (2004).On the basis of numerical evidence gathered from about 1960 by Bryan Birch and Peter Swinnerton-Dyer, in one of the most important early applications of electronic computing, it appeared that the rank was in all likelihood predicted by the order of the zero of the L-function associated to E by a recipe of Hasse and Weil for its L-series, at a specified point. At that time even the analytic continuation to the point wasn't established.
The details remain highly technical, and the Birch and Swinnerton-Dyer conjecture is now a prize problem in its full generality, though important cases have been proved. In summary:
- initial work concentrated on the case (complex multiplication) where the analytic continuation could be proved;
- this connected (via results of Deuring) the theory with earlier work of Hecke on his theory of L-functions constructed with characters more general than Dirichlet's;
- the exact value in the case of rank 0 needed accounting for, and turned out to involve invariants of E studied by Cassels, Tate, Shafarevich and others;
- the exact constant in the functional equation was conjecturally linked to the parity of the rank.
External Links
- Mathworld, http://mathworld.wolfram.com/Swinnerton-DyerConjecture.html
- Clay Math Institute, http://www.claymath.org/Millennium_Prize_Problems/Birch_and_Swinnerton-Dyer_Conjecture/
- PlanetMath.org, http://planetmath.org/encyclopedia/BirchAndSwinnertonDyerConjecture.html