Mordell–Weil group

Mordell–Weil group
Name	Mordell–Weil group
Field	Number theory, Algebraic geometry
Introduced by	Louis Mordell, André Weil
Year	1922–1928
Objects	Elliptic curve, Abelian variety
Key concepts	Rational point, Group law, Rank (mathematics), Torsion subgroup

Mordell–Weil group The Mordell–Weil group is the abelian group formed by rational points of an elliptic curve or more generally an abelian variety defined over a number field or a function field. It combines arithmetic properties studied by Louis Mordell and structural results developed by André Weil, and it plays a central role in conjectures and results involving the Birch and Swinnerton-Dyer conjecture, Faltings' theorem, Tate conjecture, Modularity theorem, and Iwasawa theory.

Definition and basic properties

For an elliptic curve E defined over a number field K, the set E(K) of K-rational points carries a natural abelian group structure given by a geometric group law; this group is called the Mordell–Weil group of E over K. The same construction applies to an abelian variety A over K, producing A(K) as an abelian group intimately linked to arithmetic invariants studied by Carl Friedrich Gauss-era arithmetic, Ernst Kummer, Goro Shimura, and modern researchers such as Andrew Wiles, Richard Taylor, and Jean-Pierre Serre. Basic invariants of the Mordell–Weil group include the rank (the free part) and the torsion subgroup, both of which enter into statements of the Birch and Swinnerton-Dyer conjecture and influence techniques from descent and Galois cohomology.

Mordell–Weil theorem

The Mordell–Weil theorem asserts that for an abelian variety A over a number field K, the group A(K) is finitely generated. Historically, Louis Mordell proved finiteness for elliptic curves over Q and André Weil extended the result to general abelian varietys over number fields. The theorem underpins later breakthroughs by Gerd Faltings (formerly Faltings' theorem), John Tate's investigations into Tate modules, and structural analyses by Barry Mazur and John Coates. Proofs combine height function finiteness, Galois descent, and weak Mordell–Weil theorem techniques related to Kummer theory and Selmer group computations.

Height functions and canonical height

Height functions measure arithmetic complexity of rational points and are essential in proving finite generation. The naive height on projective space, introduced in work influenced by Diophantus and formalized by Heinrich Weber-era arithmetic, is refined to the canonical (Néron–Tate) height by André Néron and John Tate. The canonical height quadratic form on an elliptic curve or abelian variety gives control of growth under multiplication-by-n maps and simplifies descent arguments; it is used in computations involving the Regulator (number theory), the L-series appearing in the BSD conjecture, and height bounds in Baker's theorem-type results. Modern explicit height pairings are computed using techniques from Arakelov theory, Néron models, and analytic input such as the Gross–Zagier theorem.

Structure and group law examples

For an elliptic curve E over a field, the geometric chord-and-tangent construction gives the group law; classical examples include curves used by Diophantus and later studied by Bernhard Riemann and Niels Henrik Abel. Concrete Mordell–Weil groups appear in examples like the congruent number curves studied by Fermat-era problems and modern work of Joseph Silverman, John Tate, and Noam Elkies. Structure theorems decompose A(K) as a direct sum of a free abelian group of finite rank and a finite torsion subgroup; explicit computations of generators have been performed by SageMath-derived projects, John Cremona's tables, and databases maintained by LMFDB contributors like J. E. Cremona and William Stein.

Descent methods and computing the rank

Descent methods reduce the problem of determining A(K) to finite computations in Selmer groups and Sha-related obstructions; foundational ideas trace to Fermat and were systematized by Kummer, Weil, and later by Coates and Wiles. Two-descent and higher n-descent, combined with explicit Galois cohomology calculations and use of local-global principles such as Hasse principle instances, yield bounds on the rank. Algorithmic advances by John Cremona, Noam Elkies, Bjorn Poonen, and Nils Bruin apply techniques from computational algebraic geometry, elliptic integrals, and modular methods; these feed into rank records and heuristics advanced by Goldfeld and Heath-Brown.

Torsion subgroup and Mazur's theorem

The torsion subgroup of E(K) comprises points of finite order. For E over Q, Barry Mazur proved a classification theorem now known as Mazur's theorem, listing possible torsion structures, building on work by Mazur, Ken Ribet, Gerd Faltings, and modular curve techniques originating with Yutaka Taniyama and Goro Shimura and crystallized by the Modularity theorem proved by Andrew Wiles and Richard Taylor. Generalizations to other number fields involve results by Kamienny, Merel, and Parent, with uniform boundedness results proven by Loïc Merel combining modular curve arguments and Hecke operators.

Applications and generalizations (abelian varieties)

The Mordell–Weil group influences major topics: the Birch and Swinnerton-Dyer conjecture links the rank to the order of vanishing of the L-function; Faltings' theorem uses arithmetic of abelian varieties to prove finiteness of rational points on curves of genus >1; Iwasawa theory studies growth of Mordell–Weil ranks in towers of fields; Diophantine geometry applies height inequalities and Vojta's conjecture analogies. Generalizations include points on Jacobians of curves, rational points on K3 surfaces via abelian varieties, and noncommutative analogues in motivic contexts developed by researchers such as Grothendieck, Deligne, and Minhyong Kim. Computational databases like LMFDB and software systems such as PARI/GP, Magma (computer algebra), and SageMath provide extensive data on Mordell–Weil groups supporting ongoing research by mathematicians at institutions including Princeton University, Harvard University, and École Normale Supérieure.

Category:Elliptic curves