Bombieri–Lang conjecture

The Bombieri–Lang conjecture predicts that, over a number field, rational points on every variety of general type are not Zariski dense.

Variety of general type: An algebraic variety whose canonical divisor is big, capturing a broad class of geometrically complex varieties. This is the class of varieties to which the conjecture applies.
Weil height: A function that measures the arithmetic complexity of algebraic points relative to chosen projective coordinates. Height estimates help turn geometric restrictions into bounds on rational points.
Enrico Bombieri: An Italian mathematician known for major contributions to number theory, including work on rational and integral points. His conjectures helped shape the arithmetic side of the prediction.
Weak Lang conjecture: A conjectural statement that rational points on a variety of general type are not Zariski dense. This common name isolates the rational-point prediction from stronger uniformity claims.
Zariski topology: A topology on an algebraic variety in which closed sets are algebraic subsets. The conjecture measures rational points by density in this topology.
Faltings's theorem: A theorem stating that a curve of genus greater than one over a number field has finitely many rational points. It proves the conjectured conclusion for curves of genus greater than one.
Serge Lang: An influential mathematician whose conjectures connected diophantine arithmetic with the geometry of varieties. His predictions supplied the broader geometric framework.
Uniformity conjecture: A conjecture asserting uniform bounds on rational points across families of curves or varieties. Uniform versions would strengthen pointwise predictions and constrain entire families.
Rational point: A point on an algebraic variety whose coordinates lie in the specified ground field. These are the points whose distribution the conjecture constrains.
Northcott's theorem: A finiteness theorem asserting that bounded-height algebraic points of bounded degree form a finite set. It converts bounded arithmetic complexity into actual finiteness.

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Manin conjectureRelated: It frames the contrast between sparse points on general-type varieties and abundant points on Fano varieties.