Tate module

In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, K^s is the separable closure of K, and A = G(K^s) (the K^s-valued points of G). In this case, the Tate module of A is equipped with an action of the absolute Galois group of K, and it is referred to as the Tate module of G.

Given an abelian group A and a prime number p, the p-adic Tate module of A is

where A[pⁿ] is the pⁿ torsion of A (i.e. the kernel of the multiplication-by-pⁿ map), and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A[pⁿ⁺¹] → A[pⁿ]. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Z_p-module via

When the abelian group A is the group of roots of unity in a separable closure K^s of K, the p-adic Tate module of A is sometimes referred to as the Tate module (where the choice of p and K are tacitly understood). It is a free rank one module over Z_p with a linear action of the absolute Galois group G_K of K. Thus, it is a Galois representation also referred to as the p-adic cyclotomic character of K. It can also be considered as the Tate module of the multiplicative group scheme G_m,K over K.

...
Wikipedia