In general, let be any field. Write for a separable closure of , and for the absolute Galois group of . Let be a (Hausdorff) Abelian topological group. Then an (-valued) Galois representation for is a continuous homomorphism
The simplest case is where , the group of column vectors with complex entries. Then , and we have what is usually called a complex representation of degree . In the same manner, letting , with any field (such as or a finite field ) we obtain the usual definition of a degree representation over .
There is an alternate definition which we should also mention. Write for the group ring of with coefficients in . Then a Galois representation for is simply a continuous -module (i.e. the action of on is given by a continuous homomorphism ). In other words, all the information in a representation is preserved in considering the representation space as a continuous -module. The equivalence of these two definitions is as described in the entry for the group algebra.
A notation we will be using often is the following. Suppose is a group, is a representation and a subgroup. Then we let
the subgroup of fixed pointwise by .
Given a Galois representation , let . By the fundamental theorem of infinite Galois theory, since is a closed normal subgroup of , it corresponds to a certain normal subfield of . Naturally, this is the fixed field of , and we denote it by . (The notation becomes better justified after we view some examples.) Notice that since is trivial on , it factors through a representation
which is faithful. This property characterizes .
In the case or , the so-called “no small subgroups” argument implies that the image of is finite.
For a first application of definition, we say that is discrete if for all , the stabilizer of in is open in . This is the case when is given the discrete topology, such as when is finite and Hausdorff. The stabilizer of any fixes a finite extension of , which we denote by . One has that is the union of all the .
As a second application, suppose that the image is Abelian. Then the quotient is Abelian, so contains the commutator subgroup of , which means that is contained in , the maximal Abelian extension of . This is the case when is a character, i.e. a -dimensional representation over some (commutative unital) ring,
Associated to any field are two basic Galois representations, namely those with representation spaces and , for any normal intermediate field , with the usual action of the Galois group on them. Both of these representations are discrete. The additive representation is rather simple if is finite: by the normal basis theorem, it is merely a permutation representation on the normal basis. Also, if and , then , the field obtained by adjoining to , agrees with the fixed field of the stabilizer of in . This motivates the notation “” introduced above.
By contrast, in general, can become a rather complicated object. To look at just a piece of the representation , assume that contains the group of -th roots of unity, where is prime to the characteristic of . Then we let . It is possible to choose an isomorphism of Abelian groups , and it follows that our representation is . Now assume that has the form , where is a prime not equal to the characteristic, and set . This gives a sequence of representations , which are compatible with the natural maps . This compatibility allows us to glue them together into a big representation
called the -adic cyclotomic representation of . This representation is often not discrete. The notation will be explained below.
This example may be generalized as follows. Let be an Abelian algebraic group defined over . For each integer , let be the set of -rational points whose order divides . Then we define the -adic Tate module of via
It acquires a natural Galois action from the ones on the . The two most commonly treated examples of this are the cases (the multiplicative group, giving the cyclotomic representation above) and , an elliptic curve defined over .
The last thing which we shall mention about generalities is that to any Galois representation , one may associate the Galois cohomology groups , more commonly written , which are defined to be the group cohomology of (computed with continuous cochains) with coefficients in .
Galois representations play a fundamental role in algebraic number theory, as many objects and properties related to global fields and local fields may be determined by certain Galois representations and their properties. We shall describe the local case first, and then the global case.
Let be a local field, by which we mean the fraction field of a complete DVR with finite residue field. We write for the normalized valuation, for the associated DVR, for the maximal ideal of , for the residue field, and for the characteristic of .
Let be a finite Galois extension, and define , , , and accordingly. There is a natural surjection . We call the kernel of this map the inertia group, and write it . Further, the -Sylow subgroup of is normal, and we call it the wild ramification group, and denote it by . One calls the tame ramification group.
It happens that the formation of these group is compatible with extensions , in that we have surjections and . This lets us define to be the inverse limits of the subgroups , as usual ranging over all finite Galois extensions of in .
Let be a Galois representation for with representation space . We say that is unramified if the inertia group acts trivially on , or in other words or . Otherwise we say it is ramified. Similarly, we say that is (at most) tamely ramified if the wild ramification group acts trivially, or , or ; and if not we say it is wildly ramified.
We let be the maximal unramified extension of , and be the maximal tamely ramified extension of .
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 is pro-cyclic. Thus an unramified representation is completely determined by the action of for a topological generator of . (Such a is often called a Frobenius element.)
Given a finite extension , one defines the inertia degree and the ramification degree as usual. Then in the Galois case one may recover them as and . The tame inertia degree, which is the non--part of , is equal to , while the wild inertia degree, which is the -part of , is equal to .
One finds that the inertia and ramification properties of may be computed from the ramification properties of the Galois representation .
We now turn to global fields. We shall only treat the number field case. Thus we let be a finite extension of , and write for its ring of integers. For each place of , write for the completion of with respect to . When is a finite place, we write simply for its associated normalized valuation, for , for , for , and for the characteristic of .
For each place , fix an algebraic closure of . Furthermore, choose an embedding . This choice is equivalent to choosing an extension of to all of , and to choosing an embedding . We denote the image of this last embedding by ; it is called a decomposition group at . Sitting inside are two groups, and , corresponding to the inertia and wild ramification subgroups and of ; we call the images and the inertia group at and the wild ramification group at , respectively.
For a Galois representation and a place , it is profitable to consider the restricted representation . One calls a global representation, and a local representation. We say that is ramified or tamely ramified (or not) at if is (or isn’t). The Tchebotarev density theorem implies that the corresponding Frobenius elements are dense in , so that the union of the is dense in . Therefore, it is reasonable to try to reduce questions about to questions about all the independently. This is a manifestation of Hasse’s local-to-global principle.
Given a global Galois representation with representation space which is unramified at all but finitely many places , it is a goal of number theory 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.
|Date of creation||2013-03-22 13:28:21|
|Last modified on||2013-03-22 13:28:21|
|Last modified by||alozano (2414)|
|Defines||tame inertia group|
|Defines||wild inertia group|
|Defines||tame inertia degree|
|Defines||wild inertia degree|