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. The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture. It is widely believed that there is no maximum rank for an elliptic curve, and it has been shown that there exist curves with rank as large as 28, but it is widely believed that such curves are rare. Indeed, Goldfeld and later Katz–Sarnak conjectured that in a suitable sense, the rank of elliptic curves should be 1/2 on average. In other words, 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.

In order to obtain a reasonable notion of 'average', one must be able to count elliptic curves ${\displaystyle E/\mathbb {Q} }$ 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 ${\displaystyle E/\mathbb {Q} }$ can be given in terms of a Weierstrass form, that is, we can write

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