Galois module

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 in representation theory, but can also be used as a synonym for "G"-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
Examples.
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 "Iw" denote its 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 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(), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q("ζ") where
In fact all the subfields of the cyclotomic fields 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 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 "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.
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 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 ℓ-adic 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 "GK" of "K", now called Artin representations. These are the continuous finite-dimensional linear representations of "GK" 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 "L"-functions.
Because of the incompatibility of the profinite topology on "GK" and the usual (Euclidean) topology on complex vector spaces, the 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 where "M" is either a finite-dimensional vector space over ℓ (the algebraic closure of the ℓ-adic numbers Qℓ) or a finitely generated ℓ-module (where ℓ is the integral closure of Zℓ in ℓ). The first examples to arise were the 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 formula_1. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of 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:
Representations of the Weil group.
If "K" is a local or global field, the theory of class formations attaches to "K" its Weil group "WK", a continuous group homomorphism , and an isomorphism of topological groups
where "CK" is "K"× or the idele class group "IK"/"K"× (depending on whether "K" is local or global) and 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
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 ℓ) and Weil–Deligne representations of "WK" over 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.