Bombieri–Lang conjecture

Bombieri–Lang conjecture is a conjecture in Diophantine geometry proposing that varieties of general type over number fields have only finitely many rational points, generalizing earlier finiteness results for curves and connecting to broad themes in algebraic geometry and arithmetic geometry. It synthesizes ideas from work of Faltings, Mordell, and Manin and has driven research linking Galois representations, Hodge theory, and conjectures of Grothendieck and Tate. The conjecture influences approaches to problems associated with modular forms, elliptic curves, and conjectural correspondences like the Birch and Swinnerton-Dyer conjecture.

Statement

The conjecture asserts that for a smooth projective variety X of general type defined over a number field K, the set X(K) of K-rational points is not Zariski dense; more strongly, X(K) is contained in a proper Zariski-closed subset of X, and in many formulations is expected to be finite after removing a proper closed set of special subvarieties. This statement is framed relative to notions developed by Kodaira, Enriques, and Matsusaka in complex classification theory and builds on finiteness theorems such as the Mordell conjecture proven by Faltings and earlier results of Siegel and Thue.

Motivation and context

Motivation arises from attempts to generalize the finiteness of rational points on curves of genus ≥2 (the Mordell conjecture) and to place arithmetic finiteness theorems in the context of the birational classification of varieties by Kodaira dimension as developed by Iitaka and Miyaoka. The conjecture connects to Lang's broader conjectures on hyperbolicity formulated alongside questions studied by Shafarevich and Bombieri and to analogies with Nevanlinna theory studied by Cartan and Siu. It also relates to conjectures about exceptional sets formulated by Vojta and to height inequalities developed by Northcott and Weil as refined by Silverman.

Evidence and partial results

Evidence includes Faltings's proof of the Mordell conjecture for curves, which establishes finiteness for smooth projective curves of genus ≥2 and thus supplies a key base case. Work of Bombieri, Vojta, and McQuillan obtained partial results for surfaces and special families, while results of Caporaso, Harris, and Mazur connect uniformity questions for rational points to the conjecture. Techniques exploiting moduli spaces and Arakelov theory have produced finiteness statements under additional hypotheses; related progress by Silverman, Zannier, Frey, Hindry, and Raynaud addresses special cases involving irregularity, geometric fibrations, and covers. Conditional implications from conjectures of Langlands and Tate provide heuristic support, and analogues over function fields by Moriwaki and Parshin offer further corroboration along the route pioneered by Grothendieck.

Consequences and applications

If true, the conjecture would imply broad finiteness and non-density results for rational points on high-dimensional varieties, yielding consequences for Diophantine equations studied by Pythagoras-related problems, classical Diophantine families treated by Thue and Siegel, and many parametrized families appearing in the work of Hilbert and Noether. It would constrain arithmetic dynamics questions considered by Silverman and Tate and inform effective approaches to problems influenced by Baker and Bilu. The conjecture would interact with finiteness statements in the theory of moduli of varieties developed by Deligne and Mumford, and would refine understanding of exceptional or special subvarieties as investigated by André and Pink.

Variants and related conjectures

Related formulations include Lang's conjecture on entire curves and complex hyperbolicity connecting to work of Kobayashi and Brody, Vojta's conjectures linking value distribution to Diophantine approximation and inspired by Nevanlinna and Osgood, and the Bombieri–Lang uniformity variants explored by Caporaso and Harris. The function field analogues due to Moriwaki and results of Raynaud connect to the Mordell–Lang conjecture and to the Manin–Mumford conjecture proven by Raynaud and refined by Ullmo and Zhang. Conjectural ties to the ABC conjecture and to the Birch and Swinnerton-Dyer conjecture via heights and regulators further situate Bombieri–Lang within a web of central open problems studied by Granville, Silverman, Goldfeld, and Wiles.

Category:Conjectures in number theory