# Galois representation

In general, let $K$ be any field. Write $\overline{K}$ for a separable closure of $K$, and ${G}_{K}$ for the absolute Galois group $\mathrm{Gal}(\overline{K}/K)$ of $K$. Let $A$ be a (Hausdorff^{}) Abelian^{} topological group^{}. Then an ($A$-valued) Galois representation^{} for $K$ is a continuous^{} homomorphism^{}

$$\rho :{G}_{K}\to \mathrm{Aut}(A),$$ |

where we endow ${G}_{K}$ with the Krull topology, and where $\mathrm{Aut}(A)$ is the group of continuous automorphisms^{} of $A$, endowed with the compact-open topology^{}. One calls $A$ the representation space for $\rho $.

The simplest case is where $A={\u2102}^{n}$, the group of $n\times 1$ column vectors with complex entries. Then $\mathrm{Aut}({\u2102}^{n})={\mathrm{GL}}_{n}(\u2102)$, and we have what is usually called a complex representation of degree $n$. In the same manner, letting $A={F}^{n}$, with $F$ any field (such as $\mathbb{R}$ or a finite field^{} ${\mathbb{F}}_{q}$) we obtain the usual definition of a degree $n$ representation over $F$.

There is an alternate definition which we should also mention. Write $\mathbb{Z}[{G}_{K}]$ for the group ring^{} of ${G}_{K}$ with coefficients in $\mathbb{Z}$. Then a Galois representation for $K$ is simply a continuous $\mathbb{Z}[{G}_{K}]$-module $A$ (i.e. the action of ${G}_{K}$ on $A$ is given by a continuous homomorphism $\rho $). In other words, all the information in a representation $\rho $ is preserved in considering the representation space $A$ as a continuous $\mathbb{Z}[{G}_{K}]$-module. The equivalence of these two definitions is as described in the entry for the group algebra^{}.

When $A$ is complete^{}, the continuity requirement is equivalent^{} to the action of $\mathbb{Z}[{G}_{K}]$ on $A$ naturally extending to a $\mathbb{Z}[[{G}_{K}]]$-module structure^{} on $A$. The notation $\mathbb{Z}[[{G}_{K}]]$ denotes the completed group ring:

$$\mathbb{Z}[[G]]=\underset{\u27f5}{lim}\mathbb{Z}[G/H],$$ |

where $G$ is any profinite group, and $H$ ranges over all normal subgroups^{} of finite index.

A notation we will be using often is the following. Suppose $G$ is a group, $\rho :G\to \mathrm{Aut}(A)$ is a representation and $H\subseteq G$ a subgroup^{}. Then we let

$${A}^{H}=\{a\in A\mid \rho (h)a=a,\mathrm{for}\mathrm{all}h\in H\},$$ |

the subgroup of $A$ fixed pointwise by $H$.

Given a Galois representation $\rho $, let ${G}_{0}=\mathrm{ker}\rho $. By the fundamental theorem of infinite Galois theory, since ${G}_{0}$ is a closed normal subgroup of ${G}_{K}$, it corresponds to a certain normal subfield^{} of $\overline{K}$. Naturally, this is the fixed field of ${G}_{0}$, and we denote it by $K(\rho )$. (The notation becomes better justified after we view some examples.) Notice that since $\rho $ is trivial on ${G}_{0}=\mathrm{Gal}(\overline{K}/K(\rho ))$, it factors through a representation

$$\stackrel{~}{\rho}:\mathrm{Gal}(K(\rho )/K)\to \mathrm{Aut}(A),$$ |

which is faithful. This property characterizes $K(\rho )$.

In the case $A={\mathbb{R}}^{n}$ or $A={\u2102}^{n}$, the so-called “no small subgroups” argument implies that the image of ${G}_{K}$ is finite.

For a first application of definition, we say that $\rho $ is discrete if for all $a\in A$, the stabilizer^{} of $a$ in ${G}_{K}$ is open in ${G}_{K}$. This is the case when $A$ is given the discrete topology, such as when $A$ is finite and Hausdorff. The stabilizer of any $a\in A$ fixes a finite extension^{} of $K$, which we denote by $K(a)$. One has that $K(\rho )$ is the union of all the $K(a)$.

As a second application, suppose that the image $\rho ({G}_{K})$ is Abelian. Then the quotient^{} ${G}_{K}/{G}_{0}$ is Abelian, so ${G}_{0}$ contains the commutator subgroup^{} of ${G}_{K}$, which means that $K(\rho )$ is contained in ${K}^{\mathrm{ab}}$, the maximal Abelian extension^{} of $K$. This is the case when $\rho $ is a character^{}, i.e. a $1$-dimensional representation over some (commutative^{} unital) ring,

$$\rho :{G}_{K}\to {\mathrm{GL}}_{1}(A)={A}^{\times}.$$ |

Associated to any field $K$ are two basic Galois representations, namely those with representation spaces $A=L$ and $A={L}^{\times}$, for any normal intermediate field $K\subseteq L\subseteq \overline{K}$, with the usual action of the Galois group^{} on them. Both of these representations are discrete. The additive^{} 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 $L=\overline{K}$ and $x\in \overline{K}$, then $K(x)$, the field obtained by adjoining $x$ to $K$, agrees with the fixed field of the stabilizer of $x$ in ${G}_{K}$. This motivates the notation “$K(a)$” introduced above.

By contrast, in general, ${L}^{\times}$ can become a rather complicated object. To look at just a piece of the representation ${L}^{\times}$, assume that $L$ contains the group ${\mu}_{m}$ of $m$-th roots of unity^{}, where $m$ is prime to the characteristic^{} of $K$. Then we let $A={\mu}_{m}$. It is possible to choose an isomorphism^{} of Abelian groups ${\mu}_{m}\cong \mathbb{Z}/m$, and it follows that our representation is $\rho :{G}_{K}\to {(\mathbb{Z}/m)}^{\times}$. Now assume that $m$ has the form ${p}^{n}$, where $p$ is a prime not equal to the characteristic, and set ${A}_{n}={\mu}_{{p}^{n}}$. This gives a sequence^{} of representations ${\rho}_{n}:{G}_{K}\to {(\mathbb{Z}/{p}^{n})}^{\times}$, which are compatible with the natural maps ${(\mathbb{Z}/{p}^{n+1})}^{\times}\to {(\mathbb{Z}/{p}^{n})}^{\times}$. This compatibility allows us to glue them together into a big representation

$$\rho :{G}_{K}\to \mathrm{Aut}({T}_{p}{\mathbb{G}}_{m})\cong {\mathbb{Z}}_{p}^{\times},$$ |

called the $p$-adic cyclotomic representation of $K$. This representation is often not discrete. The notation ${T}_{p}{\mathbb{G}}_{m}$ 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 ${B}_{n}=B(\overline{K})[{p}^{n}]$ be the set of $\overline{K}$-rational points whose order divides ${p}^{n}$. Then we define the $p$-adic Tate module of $B$ via

$${T}_{p}B=\underset{\u27f5}{lim}{B}_{n}.$$ |

It acquires a natural Galois action from the ones on the ${B}_{n}$. The two most commonly treated examples of this are the cases $B={\mathbb{G}}_{m}$ (the multiplicative group^{}, giving the cyclotomic representation above) and $B=E$, an elliptic curve^{} defined over $K$.

The last thing which we shall mention about generalities is that to any Galois representation $\rho :{G}_{K}\to \mathrm{Aut}(A)$, one may associate^{} the Galois cohomology groups ${H}^{n}(K,\rho )$, more commonly written ${H}^{n}(K,A)$, which are defined to be the group cohomology^{} of ${G}_{K}$ (computed with continuous cochains) with coefficients in $A$.

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 $K$ be a local field, by which we mean the fraction field of a complete DVR with finite residue field^{}. We write ${v}_{K}$ for the normalized valuation^{}, ${\mathcal{O}}_{K}$ for the associated DVR, ${\U0001d52a}_{K}$ for the maximal ideal^{} of ${\mathcal{O}}_{K}$, ${k}_{K}={\mathcal{O}}_{K}/{\U0001d52a}_{K}$ for the residue field, and $\mathrm{\ell}$ for the characteristic of ${k}_{K}$.

Let $L/K$ be a finite Galois extension^{}, and define ${v}_{L}$, ${\mathcal{O}}_{L}$, ${\U0001d52a}_{L}$, and ${k}_{L}$ accordingly. There is a natural surjection $\mathrm{Gal}(L/K)\to \mathrm{Gal}({k}_{L}/{k}_{K})$. We call the kernel of this map the inertia group, and write it $I(L/K)=\mathrm{ker}(\mathrm{Gal}(L/K)\to \mathrm{Gal}({k}_{L}/{k}_{K}))$. 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 ${L}^{\prime}/L/K$, in that we have surjections $I({L}^{\prime}/K)\to I(L/K)$ and $W({L}^{\prime}/K)\to W(L/K)$. This lets us define ${W}_{K}\subset {I}_{K}\subset {G}_{K}$ to be the inverse limits^{} of the subgroups $W(L/K)\subseteq I(L/K)\subseteq \mathrm{Gal}(L/K)$, $L$ as usual ranging over all finite Galois extensions of $K$ in $\overline{K}$.

Let $\rho $ be a Galois representation for $K$ with representation space $A$. We say that $\rho $ is unramified if the inertia group ${I}_{K}$ acts trivially on $A$, or in other words ${I}_{K}\subseteq \mathrm{ker}\rho $ or ${A}^{{I}_{K}}=A$. Otherwise we say it is ramified. Similarly, we say that $\rho $ is (at most) tamely ramified if the wild ramification group acts trivially, or ${W}_{K}\subseteq \mathrm{ker}\rho $, or ${A}^{{W}_{K}}=A$; and if not we say it is wildly ramified.

We let ${K}_{\mathrm{ur}}={\overline{K}}^{{I}_{K}}$ be the maximal unramified extension of $K$, and ${K}_{\mathrm{tame}}={\overline{K}}^{{W}_{K}}$ 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 ${G}_{K}/{I}_{K}\cong {G}_{k(K)}\cong \widehat{\mathbb{Z}}$ is pro-cyclic. Thus an unramified representation is completely determined by the action of $\rho (\sigma )$ for a topological generator^{} $\sigma $ of ${G}_{K}/{I}_{K}$. (Such a $\sigma $ is often called a Frobenius element^{}.)

Given a finite extension^{} $L/K$, one defines the inertia degree ${f}_{L/K}=[{k}_{L}:{k}_{K}]$ and the ramification degree ${e}_{L/K}=[{v}_{L}({L}^{\times}):{v}_{L}({K}^{\times})]$ as usual. Then in the Galois case one may recover them as ${f}_{L/K}=[\mathrm{Gal}(L/K):I(L/K)]$ and ${e}_{L/K}=\mathrm{\#}I(L/K)$. The tame inertia degree, which is the non-$p$-part of ${e}_{L/K}$, is equal to $[I(L/K):W(L/K)]$, while the wild inertia degree, which is the $p$-part of ${e}_{L/K}$, is equal to $\mathrm{\#}W(L/K)$.

One finds that the inertia and ramification properties of $L/K$ may be computed from the ramification properties of the Galois representation ${\mathcal{O}}_{L}$.

We now turn to global fields. We shall only treat the number field^{} case. Thus we let $K$ be a finite extension of $\mathbb{Q}$, and write ${\mathcal{O}}_{K}$ for its ring of integers^{}. For each place $v$ of $K$, write ${K}_{v}$ for the completion of $K$ with respect to $v$. When $v$ is a finite place, we write simply $v$ for its associated normalized valuation, ${\mathcal{O}}_{v}$ for ${\mathcal{O}}_{{K}_{v}}$, ${\U0001d52a}_{v}$ for ${\U0001d52a}_{{K}_{v}}$, ${k}_{v}$ for ${k}_{{K}_{v}}$, and $\mathrm{\ell}(v)$ for the characteristic of ${k}_{v}$.

For each place $v$, fix an algebraic closure^{} ${\overline{K}}_{v}$ of ${K}_{v}$. Furthermore, choose an embedding^{} $\overline{K}\hookrightarrow {\overline{K}}_{v}$. This choice is equivalent to choosing an extension of $v$ to all of ${\overline{K}}^{\times}$, and to choosing an embedding ${G}_{{K}_{v}}\hookrightarrow {G}_{K}$. We denote the image of this last embedding by ${G}_{v}\subset {G}_{K}$; it is called a decomposition group at $v$. Sitting inside ${G}_{v}$ are two groups, ${I}_{v}$ and ${W}_{v}$, corresponding to the inertia and wild ramification subgroups ${I}_{{K}_{v}}$ and ${W}_{{K}_{v}}$ of ${G}_{{K}_{v}}$; we call the images ${I}_{v}$ and ${W}_{v}$ the inertia group at $v$ and the wild ramification group at $v$, respectively.

For a Galois representation $\rho :{G}_{K}\to \mathrm{Aut}(A)$ and a place $v$, it is profitable to consider the restricted representation ${\rho}_{v}={\rho |}_{{G}_{v}}$. One calls $\rho $ a global representation, and ${\rho}_{v}$ a local representation. We say that $\rho $ is ramified or tamely ramified (or not) at $v$ if ${\rho}_{v}$ is (or isn’t). The Tchebotarev density theorem implies that the corresponding Frobenius elements ${\sigma}_{v}\in {G}_{v}$ are dense in ${G}_{K}$, so that the union of the ${G}_{v}$ is dense in ${G}_{K}$. Therefore, it is reasonable to try to reduce questions about $\rho $ to questions about all the ${\rho}_{v}$ independently. This is a manifestation of Hasse’s local-to-global principle.

Given a global Galois representation with representation space ${\mathbb{Z}}_{p}^{n}$ which is unramified at all but finitely many places $v$, 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.

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 |