Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.

Quotes

 * The BSD Conjecture has its natural context within the larger scope of modern algebraic geometry and number theory.


 * Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite.


 * Thanks to their conjecture, Birch and Swinnerton-Dyer are two names that (to mathematicians) are as inextricably linked as the names of Laurel and Hardy, although many have been tricked into believing that there are in fact three mathematicians behind the conjecture - Birch, Swinnerton and Dyer. Birch, with his rather bumbling manner, plays Stan Laurel to Swinnerton-Dyer's rather dour Oliver Hardy.


 * This remarkable conjecture relates the behaviour of a function L at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite.