Fourier–Deligne transform
Encyclopedia
In algebraic geometry
, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category
of ℓ-adic sheaves
over the affine line. It was introduced by Pierre Deligne
on November 29th, 1976 in a letter to David Kazhdan
as an analogue of the usual Fourier transform
. It was used by to simplify Deligne's proof of the Weil conjectures
.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category
Derived category
In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
of ℓ-adic sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
over the affine line. It was introduced by Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
on November 29th, 1976 in a letter to David Kazhdan
David Kazhdan
David Kazhdan or Každan, Kazhdan, formerly named Dmitry Aleksandrovich Kazhdan , is a Soviet and Israeli mathematician known for work in representation theory.-Life:...
as an analogue of the usual Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
. It was used by to simplify Deligne's proof of the Weil conjectures
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
.