In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve E {\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
E ( K ) ≅ Z r ⊕ E ( K ) tors , {\displaystyle E(K)\cong \mathbb {Z} ^{r}\oplus E(K)_{\text{tors}},}
where E ( K ) tors {\displaystyle E(K)_{\text{tors}}} is the torsion group of E, for which comparatively much is known, and r ∈ Z ≥ 0 {\displaystyle r\in \mathbb {Z} _{\geq 0}} is a nonnegative integer called the rank of E {\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 Q {\displaystyle \mathbb {Q} } to be bounded or not. It has been shown that there exist curves with rank at least 31, 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 Q {\displaystyle \mathbb {Q} } .
Heights In order to obtain a reasonable notion of 'average', one must be able to count elliptic curves E / Q {\displaystyle E/\mathbb {Q} } somehow. This requires the introduction of a height function on the set of rational elliptic curves. To define such a function, recall that a rational elliptic curve E / Q {\displaystyle E/\mathbb {Q} } can be given in terms of a Weierstrass form, that is, we can write
E : y 2 = x 3 + A x + B {\displaystyle E:y^{2}=x^{3}+Ax+B}
for some integers A , B {\displaystyle A,B} . Moreover, this model is unique if for any prime number p {\displaystyle p} such that p 4 {\displaystyle p^{4}} divides A {\displaystyle A} , we have p 6 ∤ B {\displaystyle p^{6}\nmid B} . We can then assume that A , B {\displaystyle A,B} are integers that satisfy this property and define a height function on the set of elliptic curves E / Q {\displaystyle E/\mathbb {Q} } by
H ( E ) = H ( E ( A , B ) ) = max { 4 | A | 3 , 27 B 2 } . {\displaystyle H(E)=H(E(A,B))=\max\{4|A|^{3},27B^{2}\}.}
It can then be shown that the number of elliptic curves E / Q {\displaystyle E/\mathbb {Q} } with bounded height H ( E ) {\displaystyle H(E)} is finite.
Average rank We denote by r ( E ) {\displaystyle r(E)} the Mordell–Weil rank of the elliptic curve E / Q {\displaystyle E/\mathbb {Q} } . With the height function H ( E ) {\displaystyle H(E)} in hand, one can then define the "average rank" as a limit, provided that it exists:
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