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 where
The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture.
Indeed, Goldfeld[2] and later Katz–Sarnak[3] conjectured that in a suitable asymptotic sense (see below), the rank of elliptic curves should be 1/2 on average.
In order to obtain a reasonable notion of 'average', one must be able to count elliptic curves
This requires the introduction of a height function on the set of rational elliptic curves.
are integers that satisfy this property and define a height function on the set of elliptic curves
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
Bhargava and Shankar showed that the average rank of elliptic curves is bounded above by
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
In Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves,[7] 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.
It is in general an open problem whether the rank of all elliptic curves over a fixed field K is bounded by a number
According to Park et al. Néron in 1950 held the existence of an absolute bound
Honda in 1960 conjectured for a general abelian variety A defined over
In 1966 Cassels, 1974 Tate and 1982 Mestre expressed their disbelief in such a bound
which itself is unbounded for varying E. In 2016 Park et al. introduced a new random model drawing on analogies to the Cohen-Lenstra heuristics for class groups of number fields and the Keating-Snaith heuristics based on random matrix theory for L-functions.
Their model was geared along the known results on distribution of elliptic curves in low ranks and their Tate-Shafarevich groups.
The model makes further predictions on upper bounds which are consistent with all currently known lower bounds from example families of elliptic curves in special cases (such as restrictions on the type of torsion groups).
For K a general number field the same model would predict the same bound, which however cannot hold.
They attribute the failure of their model in this case to the existence of elliptic curves E over general number fields K which come from base change of a proper subfield
should hold, however Park et al. also show the existence of a number field K such that
are finite for every number field K (Park et al. even state it is plausible) it is not clear which modified heuristics would predict correct values, let alone which approach would prove such bounds.
As of 2024 there is no consensus among the experts if the rank of an elliptic curve should be expected to be bounded uniformly only in terms of its base number field or not.
For the question of boundedness of ranks of elliptic curves over some field K to make sense, one needs a Mordell-Weil-type theorem over that field that guarantees finite generation for the group K-rational points.
one has an infinite filtration where the successive quotients are finite groups of a well classified structure.
But for general K there is no universal analog in place of the rank that is an interesting object of study.
A common conjecture is that there is no bound on the largest possible rank for an elliptic curve over
[1] It was shown[12] that under GRH it has exactly rank 28: Many other examples of (families of) elliptic curves over