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.
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?)
($l$-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
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.
($l$-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
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 $A$.
(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 $z(a_n)_n=((z\; modulo\; p^n)a_n)_n$.
($l$-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
If $k$ is a finite field or a number field the conjecture is true.
(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.
Every Galois representation induces an Artin L-function.
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
See also at function field analogy.
Last revised on September 24, 2014 at 15:38:36. See the history of this page for a list of all contributions to it.