information:classification
The structure of rational points

Rational points can be constructed by the method of tangents and secants detailed above, starting with a finite number of rational points. More precisely the Mordell–Weil theorem states that the group E (Q) is a finitely generated (abelian) group. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups.

The proof of the theorem involves two parts. The first part shows that for any integer m > 1, the quotient group E (Q)/ mE (Q) is finite (this is the weak Mordell–Weil theorem). Second, introducing a height function h on the rational points E (Q) defined by h (P) = 0 and h (P) = log max(| p |, | q |) if P (unequal to the point at infinity P) has as abscissa the rational number x = p / q (with coprime p and q). This height function h has the property that h (mP) grows roughly like the square of m. Moreover, only finitely many rational points with height smaller than any constant exist on E.

The proof of the theorem is thus a variant of the method of infinite descent and relies on the repeated application of Euclidean divisions on E: let P ∈ E (Q) be a rational point on the curve, writing P as the sum 2 P + Q where Q is a fixed representant of P in E (Q)/2 E (Q), the height of P is about 1 / 4 of the one of P (more generally, replacing 2 by any m > 1, and 1 / 4 by 1 / m). Redoing the same with P, that is to say P = 2 P + Q, then P = 2 P + Q, etc. finally expresses P as an integral linear combination of points Q and of points whose height is bounded by a fixed constant chosen in advance: by the weak Mordell–Weil theorem and the second property of the height function P is thus expressed as an integral linear combination of a finite number of fixed points.

The theorem however doesn't provide a method to determine any representatives of E (Q)/ mE (Q).

The rank of E (Q), that is the number of copies of Z in E (Q) or, equivalently, the number of independent points of infinite order, is called the rank of E. The Birch and Swinnerton-Dyer conjecture is concerned with determining the rank. One conjectures that it can be arbitrarily large, even if only examples with relatively small rank are known. The elliptic curve with the currently largest exactly-known rank is

It has rank 20, found by Noam Elkies and Zev Klagsbrun in 2020. Curves of rank higher than 20 have been known since 1994, with lower bounds on their ranks ranging from 21 to 29, but their exact ranks are not known and in particular it is not proven which of them have higher rank than the others or which is the true "current champion".

As for the groups constituting the torsion subgroup of E (Q), the following is known: the torsion subgroup of E (Q) is one of the 15 following groups (a theorem due to Barry Mazur): Z / N Z for N = 1, 2,..., 10, or 12, or Z /2 Z × Z /2 N Z with N = 1, 2, 3, 4. Examples for every case are known. Moreover, elliptic curves whose Mordell–Weil groups over Q have the same torsion groups belong to a parametrized family.