Rank of an elliptic curve
In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve formula_1 defined over the field of rational numbers. Mordell's theorem 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. 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 asymptotic sense (see below), 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. Heights. Mordell–Weil's theorem shows formula_2 is a finitely generated abelian group, thus formula_3 where formula_4 is the finite torsion subgroup and r is the rank of the elliptic curve. In order to obtain a reasonable notion of 'average', one must be able to count elliptic curves formula_5 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 formula_5 can be given in terms of a Weierstrass form, that is, we can write for some integers formula_8. Moreover, this model is unique if for any prime number formula_9 such that formula_10 divides formula_11, we have formula_12. We can then assume that formula_13 are integers that satisfy this property and define a height function on the set of elliptic curves formula_14 by It can then be shown that the number of elliptic curves formula_5 with bounded height formula_17 is finite. Average rank. We  denote by formula_18 the Mordell–Weil rank of the elliptic curve formula_5. With the height function formula_17 in hand, one can then define the "average rank" as a limit, provided that it exists: It is not known whether or not this limit exists. However, by replacing the limit with the limit superior, one can obtain a well-defined quantity. Obtaining estimates for this quantity is therefore obtaining upper bounds for the size of the average rank of elliptic curves (provided that an average exists). Upper bounds for the average rank. In the past two decades there has been some progress  made towards the task of finding upper bounds for the average rank. A. Brumer showed that, conditioned on the Birch–Swinnerton-Dyer conjecture and the Generalized Riemann hypothesis that one can obtain an upper bound of formula_22 for the average rank. Heath-Brown showed that one can obtain an upper bound of formula_23, still assuming the same two conjectures. Finally, Young showed that one can obtain a bound of formula_24; still assuming both conjectures. Bhargava and Shankar showed that the average rank of elliptic curves is bounded above by formula_25 and formula_26 without assuming either the Birch–Swinnerton-Dyer conjecture  or the Generalized Riemann Hypothesis. This is achieved by computing the average size of the formula_27-Selmer and formula_28-Selmer groups of elliptic curves formula_5 respectively. Bhargava and Shankar's approach. Bhargava and Shankar's unconditional proof of the boundedness of the average rank of elliptic curves is obtained by using a certain exact sequence involving the Mordell-Weil group of an elliptic curve formula_5. Denote by formula_2 the Mordell-Weil group of rational points on the elliptic curve formula_1, formula_33 the formula_34-Selmer group of formula_1, and let Шformula_36 denote the formula_34-part of the Tate–Shafarevich group of  formula_1. Then we have the following exact sequence formula_39 Ш formula_40 This shows that the "rank" of formula_41, also called the formula_34-Selmer rank of formula_1, defined as the non-negative integer formula_44 such that formula_45, is an upper bound for the Mordell-Weil rank formula_46 of formula_2. Therefore, if one can compute or obtain an upper bound on formula_34-Selmer rank of formula_1, then one would be able to bound the Mordell-Weil rank on average as well. In "Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves", Bhargava and Shankar computed  the 2-Selmer rank of elliptic curves on average. They did so by counting "binary quartic forms", using a method used by Birch and Swinnerton-Dyer in their original computation of the analytic rank of elliptic curves which led to their famous conjecture. Largest known ranks. A common conjecture is that there is no bound on the largest possible rank for an elliptic curve. In 2006, Noam Elkies discovered an elliptic curve with a rank of at least 28: In 2020, Elkies and Zev Klagsbrun discovered a curve with a rank of exactly 20: "x" + Rank of an elliptic