Rank of an elliptic curve

Short description: Number of independent rational basis points with infinite order

In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve [math]\displaystyle{ E }[/math] 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 that [math]\displaystyle{ E(\mathbb{Q}) }[/math] is a finitely generated abelian group, thus [math]\displaystyle{ E(\mathbb{Q})\cong E(\mathbb{Q})_{\mathrm{tors}} \times \mathbb{Z}^r }[/math] where [math]\displaystyle{ E(\mathbb{Q})_{\mathrm{tors}} }[/math] 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 [math]\displaystyle{ E/\mathbb{Q} }[/math] 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 [math]\displaystyle{ E/\mathbb{Q} }[/math] can be given in terms of a Weierstrass form, that is, we can write

[math]\displaystyle{ E: y^2 = x^3 + Ax + B }[/math]

for some integers [math]\displaystyle{ A, B }[/math]. Moreover, this model is unique if for any prime number [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ p^4 }[/math] divides [math]\displaystyle{ A }[/math], we have [math]\displaystyle{ p^6 \nmid B }[/math]. We can then assume that [math]\displaystyle{ A,B }[/math] are integers that satisfy this property and define a height function on the set of elliptic curves [math]\displaystyle{ E/\mathbb{Q} }[/math] by

[math]\displaystyle{ H(E) = H(E(A,B)) = \max\{4|A|^3, 27B^2\}. }[/math]

It can then be shown that the number of elliptic curves [math]\displaystyle{ E/\mathbb{Q} }[/math] with bounded height [math]\displaystyle{ H(E) }[/math] is finite.

Average rank

We denote by [math]\displaystyle{ r(E) }[/math] the Mordell–Weil rank of the elliptic curve [math]\displaystyle{ E/\mathbb{Q} }[/math]. With the height function [math]\displaystyle{ H(E) }[/math] in hand, one can then define the "average rank" as a limit, provided that it exists:

[math]\displaystyle{ \lim_{X \rightarrow \infty} \frac{\sum_{H(E(A,B)) \leq X} r(E)}{\sum_{H(E(A,B)) \leq X} 1}. }[/math]

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 [math]\displaystyle{ 2.3 }[/math] for the average rank. Heath-Brown showed that one can obtain an upper bound of [math]\displaystyle{ 2 }[/math], still assuming the same two conjectures. Finally, Young showed that one can obtain a bound of [math]\displaystyle{ 25/14 }[/math]; still assuming both conjectures.

Bhargava and Shankar showed that the average rank of elliptic curves is bounded above by [math]\displaystyle{ 1.5 }[/math] and [math]\displaystyle{ \frac{7}{6} }[/math] without assuming either the Birch–Swinnerton-Dyer conjecture or the Generalized Riemann Hypothesis. This is achieved by computing the average size of the [math]\displaystyle{ 2 }[/math]-Selmer and [math]\displaystyle{ 3 }[/math]-Selmer groups of elliptic curves [math]\displaystyle{ E/\mathbb{Q} }[/math] 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 [math]\displaystyle{ E/\mathbb{Q} }[/math]. Denote by [math]\displaystyle{ E(\mathbb{Q}) }[/math] the Mordell-Weil group of rational points on the elliptic curve [math]\displaystyle{ E }[/math], [math]\displaystyle{ \operatorname{Sel}_p(E) }[/math] the [math]\displaystyle{ p }[/math]-Selmer group of [math]\displaystyle{ E }[/math], and let Ш[math]\displaystyle{ {}_E[p] }[/math] denote the [math]\displaystyle{ p }[/math]-part of the Tate–Shafarevich group of [math]\displaystyle{ E }[/math]. Then we have the following exact sequence

[math]\displaystyle{ 0 \rightarrow E(\mathbb{Q})/p E(\mathbb{Q}) \rightarrow \operatorname{Sel}_p(E) \rightarrow }[/math] Ш [math]\displaystyle{ {}_E[p] \rightarrow 0. }[/math]

This shows that the rank of [math]\displaystyle{ \operatorname{Sel}_p(E) }[/math], also called the [math]\displaystyle{ p }[/math]-Selmer rank of [math]\displaystyle{ E }[/math], defined as the non-negative integer [math]\displaystyle{ s }[/math] such that [math]\displaystyle{ \# \operatorname{Sel}_p(E) = p^s }[/math], is an upper bound for the Mordell-Weil rank [math]\displaystyle{ r }[/math] of [math]\displaystyle{ E(\mathbb{Q}) }[/math]. Therefore, if one can compute or obtain an upper bound on [math]\displaystyle{ p }[/math]-Selmer rank of [math]\displaystyle{ E }[/math], 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:

y² + xy + y = x³ − x² − 20067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429

In 2020, Elkies and Zev Klagsbrun discovered a curve with a rank of exactly 20:

y² + xy + y = x³ − x² -

244537673336319601463803487168961769270757573821859853707x + 961710182053183034546222979258806817743270682028964434238957830989898438151121499931