A Galois module is a -module for a Galois group ; i.e. it is an abelian group on which a Galois group acts in a way compatible with the abelian group structure.
If is a vector space then this is a linear representation of and one speaks of Galois representation.
The category of -modules is equivalent to the category of modules over the group ring .
As always is the case, a group action can equivalently be written as . This is why Galois modules are frequently called Galois representations.
Let be a Galois extension of a number field .
Then the ring of integers of this extension is a Galois module of .
(see also Hilbert-Speiser theorem?)
Let be a prime number. Let be the absolute Galois group of a number field . Then a morphism of groups
is called an -adic representation of . Here is either a unit dimensional vector space over the algebraic closure or a finitely generated module over the integral closure .
In particular the -adic Tate-module is of this kind.
(-adic Tate module) Let be a prime number. Let be an abelian group. The -adic Tate module is defined to be the limit
i.e. it is the limit over the directed diagram . Here the kernel of the multiplication-with- map is called -torsion of .
Let denote the separable closure of . Let be the group of roots of unity of in . Then the -adic Tate-module of the absolute Galois group is called the -adic Tate module of or the -adic cyclotomic character of .
It is equivalently the Tate-module of the multiplicative group scheme .
The Tate-module is endowed with the structure of a -module by .
(-adic Tate module of an abelian variety)
Let be a prime number. Let be an abelian variety over a field . Let denote the separable closure of . The -valued points of assemble to an abelian group.
Then there are classical results on the rank of the Tate-module : For example if the characteristic of is a prime number we have that is a free module of rank .
A special case of the Tate conjecture can be formulated via Tate-modules:
Let be finitely generated over its prime field of characteristic . Let be two abelian varieties over . Then the conjecture states that
If is a finite field or a number field the conjecture is true.
(l-adic cohomology of a smooth variety)
Let be a prime number. Let be a smooth variety? over a field of characteristic prime to . Let denote the separable closure of .
The -adic cohomology in degree is defined to be the directed limit . It is a Galois module where the action is given by pullback.
More specifically, given it acts on the via the second factor. This is an isomorphism, since is an automorphism, and hence on cohomology is an isomorphism.
Note that since we have a equivalence , we have that the l-adic Tate module is a special case of the -adic cohomology.
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.