Galois representation

In general, let K be any field. Write K¯ for a separable closure of K, and GK for the absolute Galois group Gal(K¯/K) of K. Let A be a (HausdorffPlanetmathPlanetmath) AbelianMathworldPlanetmath topological groupMathworldPlanetmath. Then an (A-valued) Galois representationMathworldPlanetmath for K is a continuousPlanetmathPlanetmath homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


where we endow GK with the Krull topology, and where Aut(A) is the group of continuous automorphismsPlanetmathPlanetmath of A, endowed with the compact-open topologyMathworldPlanetmath. One calls A the representation space for ρ.

The simplest case is where A=n, the group of n×1 column vectors with complex entries. Then Aut(n)=GLn(), and we have what is usually called a complex representation of degree n. In the same manner, letting A=Fn, with F any field (such as or a finite fieldMathworldPlanetmath 𝔽q) we obtain the usual definition of a degree n representation over F.

There is an alternate definition which we should also mention. Write [GK] for the group ringMathworldPlanetmath of GK with coefficients in . Then a Galois representation for K is simply a continuous [GK]-module A (i.e. the action of GK on A is given by a continuous homomorphism ρ). In other words, all the information in a representation ρ is preserved in considering the representation space A as a continuous [GK]-module. The equivalence of these two definitions is as described in the entry for the group algebraMathworldPlanetmath.

When A is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, the continuity requirement is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the action of [GK] on A naturally extending to a [[GK]]-module structureMathworldPlanetmath on A. The notation [[GK]] denotes the completed group ring:


where G is any profinite group, and H ranges over all normal subgroupsMathworldPlanetmath of finite index.

A notation we will be using often is the following. Suppose G is a group, ρ:GAut(A) is a representation and HG a subgroupMathworldPlanetmathPlanetmath. Then we let


the subgroup of A fixed pointwise by H.

Given a Galois representation ρ, let G0=kerρ. By the fundamental theorem of infinite Galois theory, since G0 is a closed normal subgroup of GK, it corresponds to a certain normal subfieldMathworldPlanetmath of K¯. Naturally, this is the fixed field of G0, and we denote it by K(ρ). (The notation becomes better justified after we view some examples.) Notice that since ρ is trivial on G0=Gal(K¯/K(ρ)), it factors through a representation


which is faithful. This property characterizes K(ρ).

In the case A=n or A=n, the so-called “no small subgroups” argument implies that the image of GK is finite.

For a first application of definition, we say that ρ is discrete if for all aA, the stabilizerMathworldPlanetmath of a in GK is open in GK. This is the case when A is given the discrete topology, such as when A is finite and Hausdorff. The stabilizer of any aA fixes a finite extensionPlanetmathPlanetmathPlanetmathPlanetmath of K, which we denote by K(a). One has that K(ρ) is the union of all the K(a).

As a second application, suppose that the image ρ(GK) is Abelian. Then the quotientPlanetmathPlanetmath GK/G0 is Abelian, so G0 contains the commutator subgroupMathworldPlanetmath of GK, which means that K(ρ) is contained in Kab, the maximal Abelian extensionMathworldPlanetmathPlanetmath of K. This is the case when ρ is a characterPlanetmathPlanetmathPlanetmath, i.e. a 1-dimensional representation over some (commutativePlanetmathPlanetmath unital) ring,


Associated to any field K are two basic Galois representations, namely those with representation spaces A=L and A=L×, for any normal intermediate field KLK¯, with the usual action of the Galois groupMathworldPlanetmath on them. Both of these representations are discrete. The additivePlanetmathPlanetmath representation is rather simple if L/K is finite: by the normal basis theorem, it is merely a permutation representation on the normal basis. Also, if L=K¯ and xK¯, then K(x), the field obtained by adjoining x to K, agrees with the fixed field of the stabilizer of x in GK. This motivates the notation “K(a)” introduced above.

By contrast, in general, L× can become a rather complicated object. To look at just a piece of the representation L×, assume that L contains the group μm of m-th roots of unityMathworldPlanetmath, where m is prime to the characteristicPlanetmathPlanetmathPlanetmath of K. Then we let A=μm. It is possible to choose an isomorphismMathworldPlanetmathPlanetmath of Abelian groups μm/m, and it follows that our representation is ρ:GK(/m)×. Now assume that m has the form pn, where p is a prime not equal to the characteristic, and set An=μpn. This gives a sequenceMathworldPlanetmathPlanetmath of representations ρn:GK(/pn)×, which are compatible with the natural maps (/pn+1)×(/pn)×. This compatibility allows us to glue them together into a big representation


called the p-adic cyclotomic representation of K. This representation is often not discrete. The notation Tp𝔾m will be explained below.

This example may be generalized as follows. Let B be an Abelian algebraic group defined over K. For each integer n, let Bn=B(K¯)[pn] be the set of K¯-rational points whose order divides pn. Then we define the p-adic Tate module of B via


It acquires a natural Galois action from the ones on the Bn. The two most commonly treated examples of this are the cases B=𝔾m (the multiplicative groupMathworldPlanetmath, giving the cyclotomic representation above) and B=E, an elliptic curveMathworldPlanetmath defined over K.

The last thing which we shall mention about generalities is that to any Galois representation ρ:GKAut(A), one may associateMathworldPlanetmath the Galois cohomology groups Hn(K,ρ), more commonly written Hn(K,A), which are defined to be the group cohomologyMathworldPlanetmathPlanetmath of GK (computed with continuous cochains) with coefficients in A.

Galois representations play a fundamental role in algebraic number theoryMathworldPlanetmath, as many objects and properties related to global fieldsMathworldPlanetmath and local fieldsMathworldPlanetmath may be determined by certain Galois representations and their properties. We shall describe the local case first, and then the global case.

Let K be a local field, by which we mean the fraction field of a complete DVR with finite residue fieldMathworldPlanetmath. We write vK for the normalized valuationMathworldPlanetmathPlanetmath, 𝒪K for the associated DVR, 𝔪K for the maximal idealMathworldPlanetmath of 𝒪K, kK=𝒪K/𝔪K for the residue field, and for the characteristic of kK.

Let L/K be a finite Galois extensionMathworldPlanetmath, and define vL, 𝒪L, 𝔪L, and kL accordingly. There is a natural surjection Gal(L/K)Gal(kL/kK). We call the kernel of this map the inertia group, and write it I(L/K)=ker(Gal(L/K)Gal(kL/kK)). Further, the p-Sylow subgroup of I(L/K) is normal, and we call it the wild ramification group, and denote it by W(L/K). One calls I/W the tame ramification group.

It happens that the formation of these group is compatible with extensions L/L/K, in that we have surjections I(L/K)I(L/K) and W(L/K)W(L/K). This lets us define WKIKGK to be the inverse limitsMathworldPlanetmathPlanetmath of the subgroups W(L/K)I(L/K)Gal(L/K), L as usual ranging over all finite Galois extensions of K in K¯.

Let ρ be a Galois representation for K with representation space A. We say that ρ is unramified if the inertia group IK acts trivially on A, or in other words IKkerρ or AIK=A. Otherwise we say it is ramified. Similarly, we say that ρ is (at most) tamely ramified if the wild ramification group acts trivially, or WKkerρ, or AWK=A; and if not we say it is wildly ramified.

We let Kur=K¯IK be the maximal unramified extension of K, and Ktame=K¯WK be the maximal tamely ramified extension of K.

Unramified or tamely ramified extensions are usually much easier to study than wildly ramified extensions. In the unramified case, it results from the fact that GK/IKGk(K)^ is pro-cyclic. Thus an unramified representation is completely determined by the action of ρ(σ) for a topological generatorPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath σ of GK/IK. (Such a σ is often called a Frobenius elementMathworldMathworld.)

Given a finite extensionMathworldPlanetmath L/K, one defines the inertia degree fL/K=[kL:kK] and the ramification degree eL/K=[vL(L×):vL(K×)] as usual. Then in the Galois case one may recover them as fL/K=[Gal(L/K):I(L/K)] and eL/K=#I(L/K). The tame inertia degree, which is the non-p-part of eL/K, is equal to [I(L/K):W(L/K)], while the wild inertia degree, which is the p-part of eL/K, is equal to #W(L/K).

One finds that the inertia and ramification properties of L/K may be computed from the ramification properties of the Galois representation 𝒪L.

We now turn to global fields. We shall only treat the number fieldMathworldPlanetmath case. Thus we let K be a finite extension of , and write 𝒪K for its ring of integersMathworldPlanetmath. For each place v of K, write Kv for the completion of K with respect to v. When v is a finite place, we write simply v for its associated normalized valuation, 𝒪v for 𝒪Kv, 𝔪v for 𝔪Kv, kv for kKv, and (v) for the characteristic of kv.

For each place v, fix an algebraic closureMathworldPlanetmath K¯v of Kv. Furthermore, choose an embeddingMathworldPlanetmath K¯K¯v. This choice is equivalent to choosing an extension of v to all of K¯×, and to choosing an embedding GKvGK. We denote the image of this last embedding by GvGK; it is called a decomposition group at v. Sitting inside Gv are two groups, Iv and Wv, corresponding to the inertia and wild ramification subgroups IKv and WKv of GKv; we call the images Iv and Wv the inertia group at v and the wild ramification group at v, respectively.

For a Galois representation ρ:GKAut(A) and a place v, it is profitable to consider the restricted representation ρv=ρ|Gv. One calls ρ a global representation, and ρv a local representation. We say that ρ is ramified or tamely ramified (or not) at v if ρv is (or isn’t). The Tchebotarev density theorem implies that the corresponding Frobenius elements σvGv are dense in GK, so that the union of the Gv is dense in GK. Therefore, it is reasonable to try to reduce questions about ρ to questions about all the ρv independently. This is a manifestation of Hasse’s local-to-global principle.

Given a global Galois representation with representation space pn which is unramified at all but finitely many places v, it is a goal of number theoryMathworldPlanetmath to prove that it arises naturally in arithmetic geometry (namely, as a subrepresentation of an étale cohomology group of a motive), and also to prove that it arises from an automorphic form. This can only be shown in certain special cases.

Title Galois representation
Canonical name GaloisRepresentation
Date of creation 2013-03-22 13:28:21
Last modified on 2013-03-22 13:28:21
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 18
Author alozano (2414)
Entry type Definition
Classification msc 11R32
Classification msc 11R04
Classification msc 11R34
Related topic InverseLimit
Defines tame inertia group
Defines wild inertia group
Defines tame inertia degree
Defines wild inertia degree
Defines discrete module
Defines Tate module
Defines cyclotomic representation
Defines Galois cohomology