In mathematics, a Weil group, introduced by Weil (1951), is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field F, its Weil group is generally denoted WF. There also exists "finite level" modifications of the Galois groups: if E/F is a finite extension, then the relative Weil group of E/F is WE/F = WF/W c
E (where the superscript c denotes the commutator subgroup).
Weil group of a class formation
The Weil group of a class formation with fundamental classes uE/F ∈ H2(E/F, AF) is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the Langlands program.
If E/F is a normal layer, then the (relative) Weil group WE/F of E/F is the extension
- 1 → AF → WE/F → Gal(E/F) → 1
corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class uE/F in H2(Gal(E/F), AF). The Weil group of the whole formation is defined to be the inverse limit of the Weil groups of all the layers G/F, for F an open subgroup of G.
The reciprocity map of the class formation (G, A) induces an isomorphism from AG to the abelianization of the Weil group.
Weil group of an archimedean local field
For archimedean local fields the Weil group is easy to describe: for C it is the group C× of non-zero complex numbers, and for R it is a non-split extension of the Galois group of order 2 by the group of non-zero complex numbers, and can be identified with the subgroup C× ∪ j C× of the non-zero quaternions.
Weil group of a 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).
Weil group of a local field
For local of characteristic p > 0, 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).
For p-adic fields the Weil group is a dense subgroup of the absolute Galois group, consisting 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" one.)
Weil group of a function field
For global fields of characteristic p>0 (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).
Weil group of a 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.
The Weil–Deligne group scheme (or simply Weil–Deligne group) W′K of a non-archimedean local field, K, is an extension of the Weil group WK by a one-dimensional additive group scheme Ga, introduced by Deligne (1973, 8.3.6). In this extension the Weil group acts on the additive group by
where w acts on the residue field of order q as a→aq||w||.
The local Langlands correspondence for GLn over K (now proved) states that there is a natural bijection between isomorphism classes of irreducible admissible representations of GLn(K) and certain n-dimensional representations of the Weil–Deligne group of K.
The Weil–Deligne group often shows up through its representations. In such cases, the Weil–Deligne group is sometimes taken to be WK × SL(2,C) or WK × SU(2,R), or is simply done away with and Weil–Deligne representations of WK are used instead.
In the archimedean case, the Weil–Deligne group is simply defined to be Weil group.
Robert Langlands introduced a conjectural group LF attached to each local or global field F, coined the Langlands group of F by Robert Kottwitz, that satisfies properties similar to those of the Weil group. In Kottwitz's formulation, the Langlands group should be an extension of the Weil group by a compact group. When F is local, LF is the Weil–Deligne group of F, but when F is global, the existence of LF is still conjectural. The Langlands correspondence for F is a "natural" bijection between the irreducible n-dimensional complex representations of LF and, in the local case, the irreducible admissible representations of GLn(F), in the global case, the cuspidal automorphic representations of GLn(AF), where AF denotes the adeles of F.
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