# 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[citation needed] 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,[1] but it is widely believed that such curves are rare. Indeed, Goldfeld [2] and later KatzSarnak [3] 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 ${\displaystyle E(\mathbb {Q} )}$ is a finitely generated abelian group, thus ${\displaystyle E(\mathbb {Q} )\cong E(\mathbb {Q} )_{tors}\times \mathbb {Z} ^{r}}$ where ${\displaystyle E(\mathbb {Q} )_{tors}}$ 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 ${\displaystyle E/\mathbb {Q} }$ 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

${\displaystyle E:y^{2}=x^{3}+Ax+B}$

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

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

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

## Average rank

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

${\displaystyle \lim _{X\rightarrow \infty }{\frac {\sum _{H(E(A,B))\leq X}r(E)}{\sum _{H(E(A,B))\leq X}1}}.}$

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 [4] showed that, conditioned on the Birch–Swinnerton-Dyer conjecture and the Generalized Riemann hypothesis that one can obtain an upper bound of ${\displaystyle 2.3}$ for the average rank. Heath-Brown showed [5] that one can obtain an upper bound of ${\displaystyle 2}$, still assuming the same two conjectures. Finally, Young showed [6] that one can obtain a bound of ${\displaystyle 25/14}$; still assuming both conjectures.

Bhargava and Shankar showed that the average rank of elliptic curves is bounded above by ${\displaystyle 1.5}$ [7] and ${\displaystyle 1.17}$ [8] without assuming either the Birch–Swinnerton-Dyer conjecture or the Generalized Riemann Hypothesis. This is achieved by computing the average size of the ${\displaystyle 2}$-Selmer and ${\displaystyle 3}$-Selmer groups of elliptic curves ${\displaystyle E/\mathbb {Q} }$ 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 ${\displaystyle E/\mathbb {Q} }$. Denote by ${\displaystyle E(\mathbb {Q} )}$ the Mordell-Weil group of rational points on the elliptic curve ${\displaystyle E}$, ${\displaystyle \operatorname {Sel} _{p}(E)}$ the ${\displaystyle p}$-Selmer group of ${\displaystyle E}$, and let Ш${\displaystyle {}_{E}[p]}$ denote the ${\displaystyle p}$-part of the Tate–Shafarevich group of ${\displaystyle E}$. Then we have the following exact sequence

${\displaystyle 0\rightarrow E(\mathbb {Q} )/pE(\mathbb {Q} )\rightarrow \operatorname {Sel} _{p}(E)\rightarrow }$ Ш ${\displaystyle {}_{E}[p]\rightarrow 0.}$

This shows that the rank of ${\displaystyle \operatorname {Sel} _{p}(E)}$, also called the ${\displaystyle p}$-Selmer rank of ${\displaystyle E}$, defined as the non-negative integer ${\displaystyle s}$ such that ${\displaystyle \#\operatorname {Sel} _{p}(E)=p^{s}}$, is an upper bound for the Mordell-Weil rank ${\displaystyle r}$ of ${\displaystyle E(\mathbb {Q} )}$. Therefore, if one can compute or obtain an upper bound on ${\displaystyle p}$-Selmer rank of ${\displaystyle E}$, 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:[1]

y2 + xy + y = x3x220067762415575526585033208209338542750930230312178956502x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429

In 2009, Elkies discovered a curve with a rank of exactly 19:[1]

y2 + xy + y = x3x2 + 31368015812338065133318565292206590792820353345x + 302038802698566087335643188429543498624522041683874493555186062568159847

## References

1. ^ a b c Dujella, Andrej. "History of elliptic curves rank records". Retrieved 3 August 2016.
2. ^ D. Goldfeld, Conjectures on elliptic curves over quadratic fields, in Number Theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math. 751, Springer-Verlag, New York, 1979, pp. 108–118. MR0564926. Zbl 0417.14031. doi:10.1007/BFb0062705.
3. ^ N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., 1999. MR1659828. Zbl 0958.11004.
4. ^ A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109 (1992), 445–472. MR1176198. Zbl 0783.14019. doi:10.1007/BF01232033.
5. ^ D. R. Heath-Brown, The average analytic rank of elliptic curves, Duke Math. J. 122 (2004), 591–623. MR2057019. Zbl 1063.11013. doi:10.1215/S0012-7094-04-12235-3.
6. ^ M. P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (2006), 205–250. MR2169047. Zbl 1086.11032. doi:10.1090/S0894-0347-05-00503-5.
7. ^ M. Bhargava and A. Shankar, Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves, Annals of Mathematics 181 (2015), 191–242 doi:10.4007/annals.2015.181.1.3
8. ^ M. Bhargava and A. Shankar, Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0, Annals of Mathematics 181 (2015), 587–621 doi:10.4007/annals.2015.181.2.4