nLab Galois module

Idea

A Galois module $V$ 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.

If $V$ is a vector space then this is a linear representation of $G$ and one speaks of Galois representation.

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 action $G\times A\to A$ 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 unit 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.

Example

(l-adic cohomology of a smooth variety)

Let $l$ be a prime number. Let $X$ be a smooth variety? over a field $k$ of characteristic prime to $l$. Let $k_s$ denote the separable closure of $k$.

The $l$-adic cohomology in degree $i$ is defined to be the directed limit $lim_n\; H^i_{et}(X_{k_s}, \mathbb{Z}/l^n\mathbb{Z})$. It is a Galois module where the action is given by pullback.

More specifically, given $\sigma\in Gal(k\hookrightarrow k_s)$ it acts on the $X_{k_s}=X\otimes_k k_s$ via the second factor. This is an isomorphism, since $\sigma$ is an automorphism, and hence $\sigma^*$ on cohomology is an isomorphism.

Note that since we have a equivalence $T_l A\simeq H_{et}^1(A_{k_s}, \mathbb{Z}_l)^\vee$, we have that the l-adic Tate module is a special case of the $l$-adic cohomology.

Related concepts

• class field theory

• Every Galois representation induces an Artin L-function.

References

• Wikipedia, Galois module

• Richard Taylor, Galois representations (pdf)

Review of the fact that Galois representations encode local systems are are hence analogs in arithmetic geometry of flat connections in differential geometry includes

• Tom Lovering, Étale cohomology and Galois Representations, 2012 (pdf)

See also at function field analogy.