Height function

Summary

A height function measures the arithmetic complexity of a rational or algebraic point, roughly the number of digits needed to write down its coordinates. The basic example is the Weil height on projective space; composing it with a morphism gives heights on any projective variety. The key property, due to Northcott, is that there are only finitely many points of bounded height and bounded degree. Heights drive finiteness proofs such as Mordell's theorem and appear in the Gross–Zagier theorem and the Birch and Swinnerton-Dyer conjecture.