algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the

derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pr ...

of ''ℓ''-adic sheaves over the affine line. It was introduced by

Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord P ...

on November 29, 1976 in a letter to

David Kazhdan David Kazhdan ( he, דוד קשדן), born Dmitry Aleksandrovich Kazhdan (russian: Дми́трий Александро́вич Кажда́н), is a Soviet and Israeli mathematician known for work in representation theory. Kazhdan is a 1990 Ma ...

as an analogue of the usual

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...

. It was used by

Gérard Laumon Gérard Laumon (; born 1952) is a French mathematician, best known for his results in number theory, for which he was awarded the Clay Research Award. Life and work Laumon studied at the École Normale Supérieure and Paris-Sud 11 University, Or ...

to simplify Deligne's proof of the

Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. Th ...

References

* * * Algebraic geometry {{abstract-algebra-stub