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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, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields 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 OL of algebraic integers of L can be considered as an OK[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 elements under G give a free basis for OL over OK. 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 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.[1]


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 representationsUnlike Artin representations, ℓ-adic representations can have infinite image. For example, the image of GQ under the ℓ-adic cyclotomic character is \mathbf{Z}_\ell^\times. ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of Qℓ with C they can be identified with bona fide Artin representations.

Let ℓ be a prime number. An ℓ-adic representation of GK is a continuous group homomorphism ρ : GK → Aut(M) where M is either a finite-dimensional vector space over Q (the algebraic closure of the ℓ-adic numbers Q) or a finitely generated Z-module (where Z is the integral closure of Z in Q). 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 GQ under the ℓ-adic cyclotomic character is \( \mathbf{Z}_\ell^\times \). ℓ-adic representations with finite image are often called Artin representations. Via an isomorphism of Q 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 φ : WK → GK, and an isomorphism of topological groups

\( r_K:C_K\tilde{\rightarrow}W_K^{\text{ab}} \)

where CK is K× or the idele class group IK/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 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 r : WK → AutE(V), where V is a finite-dimensional vector space over E equipped with the discrete topology,
a nilpotent endomorphism N : V → V such that r(w)Nr(w)−1= ||w||N for all w ∈ WK.[2]

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 Qℓ) and Weil–Deligne representations of WK over Qℓ (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

Fröhlich (1983) p.8

Here ||w|| is given by q v(w)
K 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.

References

Kudla, Stephen S. (1994), "The local Langlands correspondence: the non-archimedean case", Motives, Part 2, Proc. Sympos. Pure Math. 55, Providence, R.I.: Amer. Math. Soc., pp. 365–392, ISBN 978-0-8218-1635-6
Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften 323, Berlin: Springer-Verlag, ISBN 978-3-540-66671-4, Zbl 0948.11001, MR 1737196
Tate, John (1979), "Number theoretic background", Automorphic forms, representations, and L-functions, Part 2, Proc. Sympos. Pure Math. 33, Providence, R.I.: Amer. Math. Soc., pp. 3–26, ISBN 978-0-8218-1437-6

Further reading

Snaith, Victor P. (1994), Galois module structure, Fields Institute monographs, Providence, RI: American Mathematical Society, ISBN 0-8218-0264-X, Zbl 0830.11042
Fröhlich, Albrecht (1983), Galois module structure of algebraic integers, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 1, Berlin-Heidelberg-New York-Tokyo: Springer-Verlag, ISBN 3-540-11920-5, Zbl 0501.12012

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