Weil group
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In mathematics, a Weil group, introduced by , is a modification of the
absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' t ...
of a
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
or
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function fi ...
, used in
class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
. 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 In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory &mdash ...
, then the relative Weil group of ''E''/''F'' is ''W''''E''/''F'' = ''WF''/ (where the superscript ''c'' denotes the
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. The commutator subgroup is important because it is the smallest normal ...
). For more details about Weil groups see or or .


Weil group of a class formation

The Weil group of a
class formation In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field t ...
with
fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...
es ''u''''E''/''F'' ∈ ''H''2(''E''/''F'', ''A''''F'') is a kind of modified Galois group, used in various formulations of class field theory, and in particular in the
Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...
. If ''E''/''F'' is a normal layer, then the (relative) Weil group ''W''''E''/''F'' of ''E''/''F'' is the extension :1 → ''A''''F'' → ''W''''E''/''F'' → Gal(''E''/''F'') → 1 corresponding (using the interpretation of elements in the second group cohomology as central extensions) to the fundamental class ''u''''E''/''F'' in ''H''2(Gal(''E''/''F''), ''A''''F''). 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 In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary ...
. A distinguished generator is provided by the
Frobenius automorphism In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphism m ...
. Certain conventions on terminology, such as
arithmetic Frobenius In mathematics, the Frobenius endomorphism is defined in any commutative ring ''R'' that has characteristic ''p'', where ''p'' is a prime number. Namely, the mapping φ that takes ''r'' in ''R'' to ''r'p'' is a ring endomorphism of ''R''. The ...
, trace back to the fixing here of a generator (as the Frobenius or its inverse).


Weil group of a local field

For a local field 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, 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".)


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 ''G''''a'', introduced by . In this extension the Weil group acts on the additive group by : \displaystyle wxw^ = , , w, , x where ''w'' acts on the residue field of order ''q'' as ''a''→''a'', , ''w'', , with , , ''w'', , a power of ''q''. The
local Langlands correspondence In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representation ...
for GL''n'' over ''K'' (now proved) states that there is a natural bijection between isomorphism classes of irreducible admissible representations of GL''n''(''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 representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
s of ''WK'' are used instead. In the archimedean case, the Weil–Deligne group is simply defined to be Weil group.


See also

*
Langlands group In mathematics, the Langlands group is a conjectural group ''L'F'' attached to each local or global field ''F'', that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulatio ...
*
Shafarevich–Weil theorem In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois group In mathematics, in the area of abstract algebra known as Galois theory, the ...


Notes


References

* * * * * * , reprinted in volume I of his collected papers, {{isbn, 0-387-90330-5 Class field theory