Glossary of Arithmetic and Diophantine Geometry - C

C

Canonical height

The canonical height on an abelian variety is a height function that is a distinguished quadratic form. See Néron–Tate height.

Chabauty's method

Chabauty's method, based on p-adic analytic functions, is a special application but capable of proving cases of the Mordell conjecture for curves whose Jacobian's rank is less than its dimension. It developed ideas from Thoralf Skolem's method for an algebraic torus. (Other older methods for Diophantine problems include Runge's method.)

Coates–Wiles theorem

The Coates–Wiles theorem states that an elliptic curve with complex multiplication by an imaginary quadratic field of class number 1 and positive rank has L-function with a zero at s=1. This is a special case of the Birch and Swinnerton-Dyer conjecture.

Crystalline cohomology

Crystalline cohomology is a p-adic cohomology theory in characteristic p, introduced by Alexander Grothendieck to fill the gap left by étale cohomology which is deficient in using mod p coefficients in this case. It is one of a number of theories deriving in some way from Dwork's method, and has applications outside purely arithmetical questions.