Birch and Swinnerton-Dyer conjecture

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curve</a>. It is an open problem in the field of <a href="/facts/Number_theory/zhoa7zrc">number theory</a> and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians <a href="/facts/Bryan_John_Birch/QVWLhz0w">Bryan John Birch</a> and <a href="/facts/Peter_Swinnerton-Dyer/D66SILlt">Peter Swinnerton-Dyer</a>, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven.
The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve E over a <a href="/facts/Number_field/e0K0TXSh">number field</a> K to the behaviour of the <a href="/facts/Hasse%25E2%2580%2593Weil_L-function/1THWBhB2">Hasse–Weil L-function</a> L(E, s) of E at s = 1. More specifically, it is conjectured that the <a href="/facts/Rank_of_an_abelian_group/zo0y8Mj4">rank</a> of the <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> E(K) of points of E is the order of the zero of L(E, s) at s = 1. The first non-zero coefficient in the <a href="/facts/Taylor_expansion/M0aTuftH">Taylor expansion</a> of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K (Wiles 2006).
The conjecture was chosen as one of the seven <a href="/facts/Millennium_Prize_Problems/WWSB5ddp">Millennium Prize Problems</a> listed by the <a href="/facts/Clay_Mathematics_Institute/9sCIUpD7">Clay Mathematics Institute</a>, which has offered a $1,000,000 prize for the first correct proof.

Birch and Swinnerton-Dyer conjecture open-in-new

Birch and Swinnerton-Dyer conjecture