Weil group
Encyclopedia
In mathematics, a Weil group, introduced by , 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 a profinite group 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/ (where the superscript c denotes the commutator subgroup
).
For more details about Weil groups see or or .
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
corresponding to the fundamental class uE/F in H2(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.
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.
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.
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 adele
s of F.
Absolute Galois group
In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is unique up to isomorphism...
of a local
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
or global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
, used in class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
. For such a field F, its Weil group is a profinite group 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/ (where the superscript c denotes the commutator subgroup
Commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group....
).
For more details about Weil groups see or or .
Weil group of a class formation
The Weil group of a class formationClass formation
In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. They were invented by Emil Artin and John Tate.-Definitions:...
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
Langlands program
The Langlands program is a web of far-reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. It was proposed by ....
.
If E/F is a normal layer, then the (relative) Weil group WE/F of E/F is the extension
- 1 → AF → WE/F → E/F → 1
corresponding to the fundamental class uE/F in H2(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.
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.Weil–Deligne group
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 . In this extension the Weil group acts on theadditive 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.
Langlands group
Robert LanglandsRobert Langlands
Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...
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 adele
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
s of F.