Glossary of Arithmetic and Diophantine Geometry - W

W

Weights

The yoga of weights is a formulation by Alexander Grothendieck of analogies between Hodge theory and l-adic cohomology.

Weil cohomology

The initial idea, later somewhat modified, for proving the Weil conjectures (q.v.), was to construct a cohomology theory applying to algebraic varieties over finite fields that would both be as good as singular homology at detecting topological structure, and have Frobenius mappings acting in such a way that the Lefschetz fixed-point theorem could be applied to the counting in local zeta-functions. For later history see motive (algebraic geometry), motivic cohomology.

Weil conjectures

The Weil conjectures were three highly-influential conjectures of André Weil, made public around 1949, on local zeta-functions. The proof was completed in 1973. Those being proved, there remain extensions of the Chevalley–Warning theorem congruence, which comes from an elementary method, and improvements of Weil bounds, e.g. better estimates for curves of the number of points than come from Weil's basic theorem of 1940. The latter turn out to be of interest for Goppa codes.

Weil distributions on algebraic varieties

André Weil proposed a theory in the 1920s and 1930s on prime ideal decomposition of algebraic numbers in co-ordinates of points on algebraic varieties. It has remained somewhat under-developed.

Weil function

A Weil function on an algebraic variety is a real-valued function defined off some Cartier divisor which generalises the concept of Green's function in Arakelov theory. They are used in the construction of the local components of the Néron–Tate height.

Weil height machine

The Weil height machine is an effective procedure for assigning a height function to any divisor on smooth projective variety over a number field (or to Cartier divisors on non-smooth varieties).