In mathematics, a Weil group, introduced by André Weil, is a modification of the absolute Galois group of a local or global field, used in class field theory is a finite extension, then the relative Weil group denotes the commutator subgroup).

Class formation

The Weil group of a class formation with fundamental classes is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program.

is the extension

corresponding (using the interpretation of elements in the second group cohomology. The Weil group of the whole formation is defined to be the inverse limit.

to the abelianization of the Weil group.

Archimedean local field

of non-zero complex numbers of the non-zero quaternions.

Finite field

For finite fields the Weil group is infinite cyclic. A distinguished generator is provided by the Frobenius automorphism. Certain conventions on terminology, such as arithmetic Frobenius, trace back to the fixing here of a generator (as the Frobenius or its inverse).

Local field

, the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).

-adic fields the Weil group is a dense subgroup of the absolute Galois group, and consists of all elements whose image in the Galois group of the residue field is an integral power of the Frobenius automorphism.

More specifically, in these cases, the Weil group does not have the subspace topology, but rather a finer topology. This topology is defined by giving the inertia subgroup its subspace topology and imposing that it be an open subgroup of the Weil group. (The resulting topology is "locally profinite".)

Function field

(function fields), the Weil group is the subgroup of the absolute Galois group of elements that act as a power of the Frobenius automorphism on the constant field (the union of all finite subfields).

Number field

For number fields there is no known "natural" construction of the Weil group without using cocycles to construct the extension. The map from the Weil group to the Galois group is surjective, and its kernel is the connected component of the identity of the Weil group, which is quite complicated.

Weil–Deligne group

The Weil–Deligne group scheme (or simply Weil–Deligne group, introduced by Pierre Deligne. In this extension the Weil group acts on the additive group by

.

The local Langlands correspondence states that there is a natural bijection.

, or is simply done away with and Weil–Deligne representations are used instead.

In the archimedean case, the Weil–Deligne group is simply defined to be Weil group.

See also

• Langlands group

• Shafarevich–Weil theorem

Notes

References