# Spahn Galois module (changes)

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A Galois module is a $G$-module for a Galois group? $G$; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure.

The category of $G$-modules is equivalent to the category of modules? over the group ring $\mathbb{Z}]G]$.

As always is the case, a group representation $G\times A\to A$ can equivalently be written as $G\to Aut(A)$. This is why Galois modules are frequently called Galois representations.

## Properties

###### Proposition

Let $K\hookrightarrow L$ be a Galois extension? of a number field $K$.

Then the ring of integers $O_L$ of this extension is a Galois module of $Gal(K/hookrightarrow L)$.

## Examples

###### Example

($l$-adic representation)

Let $l$ be a prime number. Let $Gal(k\hookrightarrow \overline k)$ be the absolute Galois group? of a number field? $k$. Then a morphism of groups

$Gal(k\hookrightarrow \overline k)\to Aut (M)$

is called an $l$-adic representation of $Gal(k\hookrightarrow \overline k)$. Here $M$ is either a unite dimensional vector space? over the algebraic closure $\overline \mathbb{Q}_l$ or a finitely generated module over the integral closure? $\overline \mathbb{Z}_l$.

In particular the $l$-adic Tate-module is of this kind.

###### Example

($l$-adic Tate module?) Let $l$ be a prime number. Let $A$ be an abelian group. The $l$-adic Tate module is defined to be the limit

$T_l(A)=lim_n \;ker (l^n)$

i.e. it is the limit over the directed diagram? $ker(p^{n+1})\to ker(p^n)$. Here the kernel? $ker(p^n)$ of the multiplication-with-$p^n$ map $p^n:A\to A$ is called $p^n$-torsion? of $A$.

###### Example

(the Tate-module)

Let $k_S$ denote the separable closure of $k$. Let $A$ be the group of roots of unity? of $k_s$ in $k$. Then the $l$-adic Tate-module of the absolute Galois group $Gal(k\hookrightarrow k_s)$ is called the $l$-adic Tate module of $k$ or the $l$-adic cyclotomic character of $k$.

It is equivalently the Tate-module of the multiplicative group scheme $\mu_k$.

The Tate-module is endowed with the structure of a $\mathbb{Z}$-module by $z(a_n)_n=((z\; modulo\; p^n)a_n)_n$.

###### Example

($l$-adic Tate module? of an abelian variety)

Let $l$ be a prime number. Let $G$ be an abelian variety? over a field $k$. Let $k_s$ denote the separable closure of $k$. The $k_s$-valued points of $G$ assemble to an abelian group.

Then there are classical results on the rank? of the Tate-module $T_l(G)$: For example if the characteristic of $k$ is a prime number $p\neq l$ we have that $T_l(G)$ is a free $\mathbb{Z}_l$ module of rank $2dim(G)$.

A special case of the Tate conjecture? can be formulated via Tate-modules:

Let $k$ be finitely generated over its prime field of characteristic $p\neq l$. Let $A,B$ be two abelian varieties over $k$. Then the conjecture states that

$hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))$

If $k$ is a finite field or a number field the conjecture is true.

## Galois modules and quasi finite fields

A Galois module is a $G$-module for a Galois group? $G$; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure.

The category of $G$-modules is equivalent to the category of modules? over the group ring $\mathbb{Z}]G]$.

As always is the case, a group representation $G\times A\to A$ can equivalently be written as $G\to Aut(A)$. This is why Galois modules are frequently called Galois representations.

## Properties

###### Proposition

Let $K\hookrightarrow L$ be a Galois extension? of a number field $K$.

Then the ring of integers $O_L$ of this extension is a Galois module of $Gal(K/hookrightarrow L)$.

## Examples

###### Example

($l$-adic representation)

Let $l$ be a prime number. Let $Gal(k\hookrightarrow \overline k)$ be the absolute Galois group? of a number field? $k$. Then a morphism of groups

$Gal(k\hookrightarrow \overline k)\to Aut (M)$

is called an $l$-adic representation of $Gal(k\hookrightarrow \overline k)$. Here $M$ is either a unite dimensional vector space? over the algebraic closure $\overline \mathbb{Q}_l$ or a finitely generated module over the integral closure? $\overline \mathbb{Z}_l$.

In particular the $l$-adic Tate-module is of this kind.

###### Example

($l$-adic Tate module?) Let $l$ be a prime number. Let $A$ be an abelian group. The $l$-adic Tate module is defined to be the limit

$T_l(A)=lim_n \;ker (l^n)$

i.e. it is the limit over the directed diagram? $ker(p^{n+1})\to ker(p^n)$. Here the kernel? $ker(p^n)$ of the multiplication-with-$p^n$ map $p^n:A\to A$ is called $p^n$-torsion? of $A$.

###### Example

(the Tate-module)

Let $k_S$ denote the separable closure of $k$. Let $A$ be the group of roots of unity? of $k_s$ in $k$. Then the $l$-adic Tate-module of the absolute Galois group $Gal(k\hookrightarrow k_s)$ is called the $l$-adic Tate module of $k$ or the $l$-adic cyclotomic character of $k$.

It is equivalently the Tate-module of the multiplicative group scheme $\mu_k$.

The Tate-module is endowed with the structure of a $\mathbb{Z}$-module by $z(a_n)_n=((z\; modulo\; p^n)a_n)_n$.

###### Example

($l$-adic Tate module? of an abelian variety)

Let $l$ be a prime number. Let $G$ be an abelian variety? over a field $k$. Let $k_s$ denote the separable closure of $k$. The $k_s$-valued points of $G$ assemble to an abelian group.

Then there are classical results on the rank? of the Tate-module $T_l(G)$: For example if the characteristic of $k$ is a prime number $p\neq l$ we have that $T_l(G)$ is a free $\mathbb{Z}_l$ module of rank $2dim(G)$.

A special case of the Tate conjecture? can be formulated via Tate-modules:

Let $k$ be finitely generated over its prime field of characteristic $p\neq l$. Let $A,B$ be two abelian varieties over $k$. Then the conjecture states that

$hom(A,B)\otimes \mathbb{Z}_p\simeq hom(T_l(A),T_l(B))$

If $k$ is a finite field or a number field the conjecture is true.

Bondarko has a few papers about Galois modules over local fields.

Burns is also interested in Galois modules I think

Many things by Snaith et al, for example the book Galois module structure, Fields Inst Monographs 2.

nLab page on Galois module

Last revised on June 12, 2012 at 10:56:18. See the history of this page for a list of all contributions to it.