MordellWeil

The Mordell–Weil theorem is a central result in Diophantine geometry and the theory of abelian varieties over number fields. It states that for any abelian variety A defined over a number field K, the group A(K) of K-rational points is finitely generated. In particular, for an elliptic curve E defined over Q, the group E(Q) of rational points is finitely generated and has the structure E(Q) ≅ Z^r ⊕ T, where r ≥ 0 is the Mordell–Weil rank and T is a finite torsion subgroup. The finite generation implies a well-defined notion of rank and allows a decomposition into a free part of rank r and a finite part.

Historically, Louis Mordell established the finite generation for elliptic curves over Q in 1922, and André Weil extended the result to all abelian varieties over number fields in the late 1920s. The rank r is called the Mordell–Weil rank and measures the size of the free part of E(Q). The free part, equipped with the Néron–Tate canonical height, forms a lattice of rank r, known as the Mordell–Weil lattice.

Generalizations extend the theorem to abelian varieties beyond elliptic curves and to number fields other than Q. Computationally, determining E(Q) involves descent methods and height calculations to bound r and to search for generators of the free part. The Mordell–Weil theorem is closely connected to conjectures such as the Birch and Swinnerton-Dyer conjecture, which relates the rank to the order of vanishing of the L-function of E at s = 1.