Rank of an elliptic curve

In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve ${\displaystyle E}$ defined over the field of rational numbers or more generally a number field K. Mordell's theorem (generalized to arbitrary number fields by André Weil) says the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subset of the rational points on the curve, from which all further rational points may be generated. If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is the rank of the curve.

In mathematical terms the set of K-rational points is denoted E(K) and Mordell's theorem can be stated as the existence of an isomorphism of abelian groups

${\displaystyle E(K)\cong \mathbb {Z} ^{r}\oplus E(K)_{\text{tors}},}$

where ${\displaystyle E(K)_{\text{tors}}}$ is the torsion group of E, for which comparatively much is known, and ${\displaystyle r\in \mathbb {Z} _{\geq 0}}$ is a nonnegative integer called the rank of ${\displaystyle E}$ (over K).

The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture. There is currently no consensus among the experts on whether one should expect the ranks of elliptic curves over ${\displaystyle \mathbb {Q} }$ to be bounded or not. It has been shown that there exist curves with rank at least 29, but it is widely believed that such curves are rare. Indeed, Goldfeld and later Katz–Sarnak conjectured that in a suitable asymptotic sense (see below), the rank of elliptic curves should be 1/2 on average. An even stronger conjecture is that half of all elliptic curves should have rank 0 (meaning that the infinite part of its Mordell–Weil group is trivial) and the other half should have rank 1; all remaining ranks consist of a total of 0% of all elliptic curves over ${\displaystyle \mathbb {Q} }$.