Birch and Swinnerton-Dyer conjecture

conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–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. It is known as one of the most challenging mathematical problems.

An example of an elliptic curve

BackgroundEdit

An elliptic curve is a function of the form  , where   and   are rational numbers.

The   values that satisfy the equation are called solutions of the elliptic curve. Sometimes these solutions are both rational, both irrational, or one rational and the other irrational. It turns out that the set of rational solutions is an abelian group. This means that given rational points   and  , there is a way to produce the sum  , and this is another rational point.

In 1922, Louis Mordell proved that the set of rational solutions is a finitely generated abelian group. This means that any rational point   can be written as a finite combination of certain generating points. For example, the points   and   are generators of the rational points of  .  This means that all rational points on it, even very complicated points, can be written in terms of these two points. For example, we have:

 

ReferencesEdit