Mordell conjecture

Mordell conjecture
Renate Schmid · CC BY-SA 2.0 de · source
Name	Mordell conjecture
Proposed	1922
Proposer	Louis Mordell
Proven	1983
Prover	Gerd Faltings
Area	Diophantine geometry
Related	Faltings' theorem, Shafarevich conjecture, Mordell–Weil theorem

Mordell conjecture The Mordell conjecture posited that a smooth projective algebraic curve of genus greater than one defined over a number field has only finitely many rational points. It connected classical problems studied by Pierre de Fermat, Diophantus of Alexandria, Carl Friedrich Gauss, and André Weil to modern frameworks developed by Alexander Grothendieck, Serge Lang, and John Tate. The statement guided decades of research in Number theory, Algebraic geometry, and Arithmetic geometry until its proof by Gerd Faltings.

Statement of the conjecture

Mordell conjectured that for a curve C defined over a number field K with genus g>1, the set C(K) of K-rational points is finite. The formulation built on precedents such as the Mordell–Weil theorem about rational points on abelian varieties and on finiteness statements in the Shafarevich conjecture for abelian varieties. The conjecture can be viewed through the lenses of the Jacobian variety, the Abel–Jacobi map, and notions from Néron models and Arakelov theory.

Historical background and development

Louis Mordell proposed the conjecture in 1922 after earlier work on rational solutions to equations by Diophantus of Alexandria and the modern formalization of curves by Bernhard Riemann and Oscar Zariski. Progress included special cases: Thue's theorem treated binary forms, Siegel's theorem on integral points handled integral points for genus zero and one, and results of Gerd Faltings’ predecessors such as Shafarevich and Igor Shafarevich on finiteness of isomorphism classes. Influential frameworks by Alexander Grothendieck (schemes), Jean-Pierre Serre (Galois representations), and Nicholas Katz (l-adic cohomology) provided tools. Work by Paul Vojta proposed analogies to Nevanlinna theory and contributed conjectural unifications.

Proof by Faltings

In 1983 Gerd Faltings proved the conjecture by establishing several finiteness results for abelian varieties, now consolidated as Faltings' theorem. He proved finiteness of isomorphism classes of principally polarized abelian varieties with fixed invariants, the Tate conjecture for endomorphisms of abelian varieties over number fields, and the finiteness of rational points on subvarieties via the Mordell–Lang conjecture framework. Faltings’ approach used methods from Hodge theory, l-adic representations, and reduction techniques inspired by Shafarevich, together with innovations in handling heights inspired by Arakelov theory of Suren Arakelov and height inequalities developed by David Masser and Joseph H. Silverman.

Consequences and related results

Faltings' proof implied several major consequences: resolution of the Mordell conjecture itself, proof of cases of the Tate conjecture for abelian varieties, and progress toward the Shafarevich conjecture. It influenced proofs and formulations of the Mordell–Lang conjecture by Enrico Bombieri and Serge Lang and later work by Hrushovski on function field analogues. The theorem catalyzed advances in effective approaches, prompting research by Alan Baker on bounds, Michel Raynaud on torsion subgroups, and Loïc Merel on uniform boundedness results. Connections were explored with the Birch and Swinnerton-Dyer conjecture and with conjectures by Paul Vojta relating value distribution and Diophantine approximation.

Examples and explicit cases

Concrete examples include classical curves such as the hyperelliptic curves studied by Joseph-Louis Lagrange and later by Carl Gustav Jacobi; modular curves like X0(N) considered by Atkin and John Coates; and Fermat-type curves linked to results surrounding Fermat's Last Theorem resolved by Andrew Wiles with contributions by Richard Taylor. Explicit finiteness is known for many families: hyperelliptic curves treated by techniques of Baker and Mordell; Thue equations solved using T. N. Shorey and R. Tijdeman methods; and special modular parametrizations studied by Barry Mazur and Ken Ribet which constrain rational points on specific curves.

Analogues and generalizations

Analogues arise in the function field setting where results by Samuel and later refinements by Manin and Igusa established counterparts; the conjecture generalizes to higher-dimensional varieties via conjectures of Lang and Campana. The Mordell–Lang conjecture for subvarieties of semiabelian varieties was proven by Faltings and extended by G. Rémond and E. Hrushovski in model-theoretic contexts. Over finitely generated fields, arithmetic analogues involve work of Frey, Szpiro, and Caporaso–Harris–Mazur relating uniformity questions to conjectures of Bombieri and Lang.

Techniques and methods used

The proof and surrounding work combine algebraic, analytic, and arithmetic techniques: heights from Arakelov theory and Néron–Tate height estimates; l-adic Galois representations studied by Jean-Pierre Serre and Pierre Deligne; moduli of abelian varieties following Mumford and Deligne; deformation and reduction methods developed by Grothendieck and Raynaud; and Diophantine approximation methods initiated by Alan Baker and refined by Enrico Bombieri. Logical and model-theoretic approaches by Hrushovski provided alternate proofs in special contexts, while algorithmic and effective work by Michael Stoll and Bjorn Poonen seeks explicit enumeration of rational points.

Category:Diophantine geometry