Spahn Galois module (Rev #2, changes)

Showing changes from revision #1 to #2: Added | ~~Removed~~ | ~~Chan~~ged

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 $Gal(K/hookrightarrow L)$ .

(see also Hilbert-Speiser theorem?)

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 $Gal(K/hookrightarrow L)$ .

(see also Hilbert-Speiser theorem?)

~~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~~

Revision on June 12, 2012 at 10:56:18 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.