In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve
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Heights
In order to obtain a reasonable notion of 'average', one must be able to count elliptic curves
for some integers
It can then be shown that the number of elliptic curves
Average rank
We denote by
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
Bhargava and Shankar showed that the average rank of elliptic curves is bounded above by
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
This shows that the rank of
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 + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429In 2009, Elkies discovered a curve with a rank of exactly 19:
y2 + xy + y = x3 − x2 + 31368015812338065133318565292206590792820353345x + 302038802698566087335643188429543498624522041683874493555186062568159847