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In [[mathematics]], a '''Galois module''' is a [[G-module|''G''-module]] where ''G'' is the [[Galois group]] of some [[field extension|extension]] of [[Field (mathematics)|fields]]. The term '''Galois representation''' is frequently used when the ''G''-module is a [[vector space]] over a [[field (mathematics)|field]] or a [[free module]] over a [[ring (mathematics)|ring]], but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of [[local field|local]] or [[global field]]s is an important tool in [[number theory]].

==Examples==
*Given a field ''K'', the [[unit group|multiplicative group]] (''Ks'')× of a [[separable closure]] of ''K'' is a Galois module for the [[absolute Galois group]]. Its second cohomology group is [[isomorphic]] to the [[Brauer group]] of ''K'' (by [[Hilbert's theorem 90]], its first [[group cohomology|cohomology group]] is zero).

*If ''X'' is a [[smooth morphism|smooth]] [[proper morphism|proper]] scheme over a field ''K'' then the [[l-adic cohomology|ℓ-adic cohomology]] groups of its [[geometric fibre]] are Galois modules for the absolute Galois group of ''K''.

===Ramification theory===
Let ''K'' be a [[valued field]] (with valuation denoted ''v'') and let ''L''/''K'' be a [[finite extension|finite]] [[Galois extension]] with Galois group ''G''. For an [[extension of a valuation|extension]] ''w'' of ''v'' to ''L'', let ''Iw'' denote its [[inertia group of an extension of valuations|inertia group]]. A Galois module ρ : ''G'' → Aut(''V'') is said to be '''unramified''' if ρ(''Iw'') = {1}.

==Galois module structure of algebraic integers==
In classical [[algebraic number theory]], let ''L'' be a Galois extension of a field ''K'', and let ''G'' be the corresponding Galois group. Then the ring ''O''''L'' of [[algebraic integer]]s of ''L'' can be considered as an ''O''''K''[''G'']-module, and one can ask what its structure is. This is an arithmetic question, in that by the [[normal basis theorem]] one knows that ''L'' is a free ''K''[''G'']-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a '''normal integral basis''', i.e. of α in ''OL'' such that its [[conjugate element]]s under ''G'' give a free basis for ''OL'' over ''O''''K''. This is an interesting question even (perhaps especially) when ''K'' is the [[rational number]] field '''Q'''.

For example, if ''L'' = '''Q'''(√-3), is there a normal integral basis? The answer is yes, as one sees by identifying it with '''Q'''(ζ) where

:ζ = exp(2πi/3).

In fact all the subfields of the [[cyclotomic field]]s for ''p''-th [[roots of unity]] for ''p'' a ''prime number'' have normal integral bases (over '''Z'''), as can be deduced from the theory of [[Gaussian period]]s (the [[Hilbert–Speiser theorem]]). On the other hand the [[Gaussian rational|Gaussian field]] does not. This is an example of a ''necessary'' condition found by [[Emmy Noether]] (''perhaps known earlier?''). What matters here is ''tame'' [[ramification]]. In terms of the [[discriminant of an algebraic number field|discriminant]] ''D'' of ''L'', and taking still ''K'' = '''Q''', no prime ''p'' must divide ''D'' to the power ''p''. Then Noether's theorem states that tame ramification is necessary and sufficient for ''OL'' to be a [[projective module]] over '''Z'''[''G'']. It is certainly therefore necessary for it to be a ''free'' module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.

A classical result, based on a result of [[David Hilbert]], is that a tamely ramified [[abelian number field]] has a normal integral basis. This may be seen by using the [[Kronecker–Weber theorem]] to embed the abelian field into a cyclotomic field.<ref name=F8>Fröhlich (1983) p.8</ref>

==Galois representations in number theory==
Many objects that arise in number theory are naturally Galois representations. For example, if ''L'' is a [[Galois extension]] of a [[number field]] ''K'', the [[ring of integers]] ''OL'' of ''L'' is a Galois module over ''OK'' for the Galois group of ''L''/''K'' (see [[Hilbert–Speiser theorem]]). If ''K'' is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of ''K'' and its study leads to [[local class field theory]]. For [[global class field theory]], the union of the [[idele class group]]s of all finite [[separable extension]]s of ''K'' is used instead.

There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the [[Tate module|ℓ-adic Tate modules]] of [[abelian variety|abelian varieties]].

===Artin representations===
Let ''K'' be a number field. [[Emil Artin]] introduced a class of Galois representations of the absolute Galois group ''GK'' of ''K'', now called '''Artin representations'''. These are the [[continuous function|continuous]] finite-dimensional linear representations of ''GK'' on [[complex vector space]]s. Artin's study of these representations led him to formulate the [[Artin reciprocity law]] and conjecture what is now called the [[Artin conjecture (L-functions)|Artin conjecture]] concerning the [[holomorphy]] of [[Artin L-function|Artin ''L''-functions]].

Because of the incompatibility of the [[profinite topology]] on ''GK'' and the usual (Euclidean) topology on complex vector spaces, the [[image (mathematics)|image]] of an Artin representation is always finite.

===ℓ-adic representations===
Let ℓ be a [[prime number]]. An '''ℓ-adic representation''' of ''GK'' is a continuous [[group homomorphism]] {{nowrap|ρ : ''GK'' → Aut(''M'')}} where ''M'' is either a finite-dimensional vector space over {{overline|'''Q'''}}ℓ (the algebraic closure of the [[p-adic number|ℓ-adic numbers]] '''Q'''ℓ) or a [[finitely generated module|finitely generated]] {{overline|'''Z'''}}ℓ-module (where {{overline|'''Z'''}}ℓ is the [[integral closure]] of '''Z'''ℓ in {{overline|'''Q'''}}ℓ). The first examples to arise were the [[p-adic cyclotomic character|ℓ-adic cyclotomic character]] and the ℓ-adic Tate modules of abelian varieties over ''K''. Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on ℓ-adic cohomology groups of algebraic varieties.

Unlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of ''G'''''Q''' under the ℓ-adic cyclotomic character is <math>\mathbf{Z}_\ell^\times</math>. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of {{overline|'''Q'''}}ℓ with '''C''' they can be identified with ''bona fide'' Artin representations.

===Mod ℓ representations===

These are representations over a finite field of characteristic ℓ. They often arise as the reduction mod ℓ of an ℓ-adic representation.

===Local conditions on representations===

There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name with different meanings. Some of these conditions include:

*Abelian representations. This means that the image of the Galois group in the representations is abelian.
*Absolutely irreducible representations. These remain irreducible over an algebraic closure of the field.
*Barsotti–Tate representations. These are similar to finite flat representations.
*Crystalline representations.
*de Rham representations.
*Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat group scheme.
*Good representations. These are similar to finite flat representations.
*Hodge–Tate representations.
*Irreducible representations. These are irreducible in the sense that the only subrepresentation is the whole space or zero.
*Minimally ramified representations.
*Modular representations. These are representations coming from a modular form.
*Ordinary representations. These are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character ε on the submodule.
*Potentially something representations. This means that the representations restricted to an open subgroup of finite index has some property.
*Reducible representations. These have a proper non-zero sub-representation.
*Semistable representations. These are two dimensional representations related to the representations coming from semistable elliptic curves.
*Tamely ramified representations. These are trivial on the (first) ramification group.
*Unramified representations. These are trivial on the inertia group.
*Wildly ramified representations. These are non-trivial on the (first) ramification group.

==Representations of the Weil group==
If ''K'' is a local or global field, the theory of [[class formation]]s attaches to ''K'' its [[Weil group of a class formation|Weil group]] ''WK'', a continuous group homomorphism {{nowrap|φ : ''WK'' → ''GK''}}, and an [[isomorphism]] of [[topological group]]s
:<math>r_K:C_K\tilde{\rightarrow}W_K^{\text{ab}}</math>
where ''CK'' is ''K''× or the [[idele class group]] ''IK''/''K''× (depending on whether ''K'' is local or global) and {{SubSup|''W''|''K''|ab}} is the [[abelianization]] of the Weil group of ''K''. Via φ, any representation of ''GK'' can be considered as a representation of ''WK''. However, ''WK'' can have strictly more representations than ''GK''. For example, via ''rK'' the continuous complex characters of ''WK'' are in bijection with those of ''CK''. Thus, the absolute value character on ''CK'' yields a character of ''WK'' whose image is infinite and therefore is not a character of ''GK'' (as all such have finite image).

An ℓ-adic representation of ''WK'' is defined in the same way as for ''GK''. These arise naturally from geometry: if ''X'' is a smooth projective variety over ''K'', then the ℓ-adic cohomology of the geometric fibre of ''X'' is an ℓ-adic representation of ''GK'' which, via φ, induces an ℓ-adic representation of ''WK''. If ''K'' is a local field of residue characteristic ''p'' ≠ ℓ, then it is simpler to study the so-called Weil–Deligne representations of ''WK''.

===Weil–Deligne representations===
Let ''K'' be a local field. Let ''E'' be a field of characteristic zero. A '''Weil–Deligne representation''' over ''E'' of ''WK'' (or simply of ''K'') is a pair (''r'', ''N'') consisting of
* a continuous group homomorphism {{nowrap|''r'' : ''WK'' → Aut''E''(''V'')}}, where ''V'' is a finite-dimensional vector space over ''E'' equipped with the [[discrete topology]],
* a [[nilpotent]] [[endomorphism]] {{nowrap|''N'' : ''V'' → ''V''}} such that ''r''(''w'')N''r''(''w'')−1= ||''w''||''N'' for all ''w'' ∈ ''WK''.<ref>Here ||''w''|| is given by {{SubSup|''q''|''K''|''v''(''w'')}} where ''qK'' is the size of the residue field of ''K'' and ''v''(''w'') is such that ''w'' is equivalent to the −''v''(''w'')th power of the (arithmetic) Frobenius of ''WK''.</ref>
These representations are the same as the representations over ''E'' of the [[Weil–Deligne group]] of ''K''.

If the residue characteristic of ''K'' is different from ℓ, [[Grothendieck]]'s [[ℓ-adic monodromy theorem]] sets up a bijection between ℓ-adic representations of ''WK'' (over {{overline|'''Q'''}}ℓ) and Weil–Deligne representations of ''WK'' over {{overline|'''Q'''}}ℓ (or equivalently over '''C'''). These latter have the nice feature that the continuity of ''r'' is only with respect to the discrete topology on ''V'', thus making the situation more algebraic in flavor.

==See also==
*[[Compatible system of ℓ-adic representations]]

==Notes==
{{reflist}}

==References==
*{{Citation
|last=Kudla
|first=Stephen S.
|contribution=The local Langlands correspondence: the non-archimedean case
|title=Motives, Part 2
|pages=365–392
|series=Proc. Sympos. Pure Math.
|volume=55
|publisher=Amer. Math. Soc.
|publication-place=Providence, R.I.
|year=1994
|isbn=978-0-8218-1635-6
}}
*{{Neukirch et al. CNF}}
*{{Citation
|last=Tate
|first=John
|author-link=John Tate
|contribution=Number theoretic background
|url=http://www.ams.org/online_bks/pspum332/
|title=Automorphic forms, representations, and L-functions, Part 2
|pages=3–26
|series=Proc. Sympos. Pure Math.
|volume=33
|publisher=Amer. Math. Soc.
|publication-place=Providence, R.I.
|year=1979
|isbn=978-0-8218-1437-6
}}

==Further reading==
* {{citation | last=Snaith | first=Victor P. | title=Galois module structure | series=Fields Insitute monographs | location=Providence, RI | publisher=[[American Mathematical Society]] | year=1994 | isbn=0-8218-0264-X | zbl=0830.11042 }}
* {{citation | last=Fröhlich | first=Albrecht | authorlink=Albrecht Fröhlich | title=Galois module structure of algebraic integers | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=1 | location=Berlin-Heidelberg-New York-Tokyo | publisher=[[Springer-Verlag]] | year=1983 | isbn=3-540-11920-5 | zbl=0501.12012 }}

[[Category:Algebraic number theory]]
[[Category:Galois theory]]

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