Mordell–Weil theorem

Mordell–Weil theorem
Name	Mordell–Weil theorem
Subject	Number theory
Introduced	1922
Authors	Louis Mordell, André Weil
Field	Arithmetic geometry
Statement	The group of rational points on an abelian variety over a number field is finitely generated.

Mordell–Weil theorem The Mordell–Weil theorem asserts that for an abelian variety defined over a number field the group of rational points is a finitely generated abelian group. This foundational result in arithmetic geometry connects Diophantine problems, algebraic geometry, and algebraic number theory, and it underpins work in conjectures and theorems by Gerd Faltings, Andrew Wiles, Pierre Deligne, John Tate, and Serge Lang.

Statement of the theorem

Let A be an abelian variety defined over a number field K; the theorem states that the group A(K) of K-rational points is a finitely generated abelian group. The statement generalizes Louis Mordell's 1922 theorem for elliptic curves over the rational field Q and was extended by André Weil to higher-dimensional abelian varieties and arbitrary number fields, forming a bridge between the work of Richard Dedekind, Ernst Kummer, and later developments by Helmut Hasse and Kurt Hensel.

Historical background and motivation

The origin traces to Mordell's 1922 proof that E(Q) for an elliptic curve E over Q is finitely generated, influenced by classical Diophantine studies stemming from Pierre de Fermat and the Diophantus tradition. Weil's generalization in the 1920s built on the theory of abelian varieties and the concepts developed by André Weil (mathematician), drawing on the work of Oscar Zariski, Federigo Enriques, and the contemporaneous progress in algebraic geometry by Kunihiko Kodaira and Federico Gauss. Later formal refinements and applications invoked tools from class field theory associated with Emil Artin and John Tate, while consequences informed major achievements including the proof of the Taniyama–Shimura conjecture connection used by Wiles in the proof of Fermat's Last Theorem.

Sketch of proof and key ideas

Weil's strategy combines the theory of heights, descent, and the geometry of abelian varieties. One constructs a height function on A(K) using ideas from André Weil and height machine techniques influenced by Alexander Grothendieck and Jean-Pierre Serre, then applies a descent argument akin to classical Selmer group calculations related to work by E. S. Selmer and Birch and Swinnerton-Dyer conjecture heuristics. Key ingredients include the finiteness of the n-Selmer group for each integer n, reductions mod primes studied by Hasse, control of torsion via Mazur-type results over Q and extensions by Ken Ribet and Barry Mazur, and the use of Galois cohomology developed by John Tate and Jean-Pierre Serre. Modern expositions emphasize descent combined with properties of the Néron model introduced by André Néron and intersection-theoretic inputs from Oscar Zariski and Alexander Grothendieck.

Consequences and applications

Finiteness of A(K) up to torsion enables structural classification of rational points on elliptic curves and higher-dimensional varieties, impacting the resolution of Diophantine equations studied by Pierre de Fermat, Diophantus, and methods used by Sophie Germain and Dirichlet. The theorem underlies the formulation of the Birch and Swinnerton-Dyer conjecture for elliptic curves over Richard Dedekind's number fields and is essential in the proof strategies for cases of the Shafarevich conjecture and the finiteness results of Faltings (formerly the Mordell conjecture). Practical applications appear in algorithmic number theory as developed by John Cremona, Henri Cohen, and cryptographic constructions influenced by Neal Koblitz and Victor Miller.

Generalizations and related results

Extensions include finiteness statements for rational points on semi-abelian varieties and the refinement by Faltings proving finiteness of rational points on curves of genus greater than one, drawing on work by Paul Vojta and connections to Arakelov theory developed by Suren Arakelov and Gerd Faltings. Analogues over function fields were established by Lang–Néron and by subsequent contributions from Michael Stoll and Zoé Chatzidakis, while torsion classifications over number fields connect to results by Loïc Merel and Barry Mazur. Cohomological frameworks by Jean-Pierre Serre and John Tate link the theorem to the study of Galois representations investigated by Richard Taylor and Michael Harris.

Examples and computations

For elliptic curves over Q, explicit computation of E(Q) uses descent methods, the computation of Selmer groups, and algorithms from John Cremona's tables; canonical examples include curves used in the proof of cases of the Taniyama–Shimura conjecture and curves studied by Neal Koblitz for cryptographic parameters. The rank calculation for elliptic curves features in databases curated by The L-functions and Modular Forms Database collaborators including William Stein and Noam Elkies, while classical examples traced to Louis Mordell illustrate both torsion subgroups classified by Barry Mazur and infinite-rank phenomena investigated in families by Connor Skinner and Bjorn Poonen.

Category:Theorems in number theory