 Galois module

In mathematics, a Galois module is a Gmodule where G is the Galois group of some extension of fields. The term Galois representation is frequently used when the Gmodule is a vector space over a field or a free module over a ring, but can also be used as a synonym for Gmodule. The study of Galois modules for extensions of local or global fields is an important tool in number theory.
Contents
Examples
 Given a field K, the multiplicative group (K^{s})^{×} 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 cohomology group is zero).
 If X is a smooth proper scheme over a field K then the ℓadic 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 Galois extension with Galois group G. For an extension w of v to L, let I_{w} denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(I_{w}) = {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 integers 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 O_{L} such that its conjugate elements under G give a free basis for O_{L} 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 fields for pth roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem). On the other hand the 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 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 O_{L} 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.
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 O_{L} of L is a Galois module over O_{K} 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 groups of all finite separable extensions 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 ℓadic Tate modules of abelian varieties.
Artin representations
Let K be a number field. Emil Artin introduced a class of Galois representations of the absolute Galois group G_{K} of K, now called Artin representations. These are the continuous finitedimensional linear representations of G_{K} on complex vector spaces. Artin's study of these representations led him to formulate the Artin reciprocity law and conjecture what is now called the Artin conjecture concerning the holomorphy of Artin Lfunctions.
Because of the incompatibility of the profinite topology on G_{K} and the usual (Euclidean) topology on complex vector spaces, the image of an Artin representation is always finite.
ℓadic representations
Let ℓ be a prime number. An ℓadic representation of G_{K} is a continuous group homomorphism ρ : G_{K} → Aut(M) where M is either a finitedimensional vector space over Q_{ℓ} (the algebraic closure of the ℓadic numbers Q_{ℓ}) or a finitely generated Z_{ℓ}module (where Z_{ℓ} is the integral closure of Z_{ℓ} in Q_{ℓ}). The first examples to arise were the ℓadic cyclotomic character and the ℓadic Tate modules of abelian varieties over K.
Unlike Artin representations, ℓadic representations can have infinite image. For example, the image of G_{Q} under the ℓadic cyclotomic character is . ℓadic representations with finite image are often called Artin representations. Via an isomorphism of Q_{ℓ} with C they can be identified with bona fide Artin representations.
For an extension L/K of global fields with Galois group G, a finite place v of K, and place w of L above v, a Galois module ρ : G → Aut(V) is said to be unramified at w if ρ(I_{w}) = {1} (where I_{w} is the inertia subgroup of w). Equivalently, the extension L^{ker(ρ)}/K is unramified at w. If ρ is not unramified at w it is called ramified at w.
Representations of the Weil group
If K is a local or global field, the theory of class formations attaches to K its Weil group W_{K}, a continuous group homomorphism φ : W_{K} → G_{K}, and an isomorphism of topological groups
where C_{K} is K^{×} or the idele class group I_{K}/K^{×} (depending on whether K is local or global) and W ab
K is the abelianization of the Weil group of K. Via φ, any representation of G_{K} can be considered as a representation of W_{K}. However, W_{K} can have strictly more representations than G_{K}. For example, via r_{K} the continuous complex characters of W_{K} are in bijection with those of C_{K}. Thus, the absolute value character on C_{K} yields a character of W_{K} whose image is infinite and therefore is not a character of G_{K} (as all such have finite image).An ℓadic representation of W_{K} is defined in the same way as for G_{K}. These arise naturally from geometry: if X is a smooth projective variety over K, then the ℓadic cohomology of the geometric fibre of X is an ℓadic representation of G_{K} which, via φ, induces an ℓadic representation of W_{K}. If K is a local field of residue characteristic p ≠ ℓ, then it is simpler to study the socalled Weil–Deligne representations of W_{K}.
Weil–Deligne representations
Let K be a local field. Let E be a field of characteristic zero. A Weil–Deligne representation over E of W_{K} (or simply of K) is a pair (r, N) consisting of
 a continuous group homomorphism r : W_{K} → Aut_{E}(V), where V is a finitedimensional vector space over E equipped with the discrete topology,
 a nilpotent endomorphism N : V → V such that r(w)Nr(w)^{−1}= wN for all w ∈ W_{K}.^{[1]}
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 ℓadic monodromy theorem sets up a bijection between ℓadic representations of W_{K} (over Q_{ℓ}) and Weil–Deligne representations of W_{K} over 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
Notes
 ^ Here w is given by q v(w)
K where q_{K} 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 W_{K}.
References
 Kudla, Stephen S. (1994), "The local Langlands correspondence: the nonarchimedean case", Motives, Part 2, Proc. Sympos. Pure Math., 55, Providence, R.I.: Amer. Math. Soc., pp. 365–392, ISBN 9780821816356
 Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Berlin: SpringerVerlag, ISBN 9783540666714, MR1737196
 Tate, John (1979), "Number theoretic background", Automorphic forms, representations, and Lfunctions, Part 2, Proc. Sympos. Pure Math., 33, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 9780821814376, http://www.ams.org/online_bks/pspum332/
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