# 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.

## Definition

Given an abelian group A and a prime number p, the p-adic Tate module of A is
    $T_{p}(A)={\underset {\longleftarrow }{\lim }}A[p^{n}]$
where A[p$^n$] is the p$^n$ torsion of A (i.e. the kernel of the multiplication-by-p$^n$ map), and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A[p$^{n+1}$] → A[p$^n$]. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Z$_p$-module via
    $z(a_{n})_{n}=((z{\text{ mod }}p^{n})a_{n})_{n}.$

## Examples
### The Tate module

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.

### The Tate module of an abelian variety

Given an abelian variety G over a field K, the K$^s$-valued points of G are an abelian group. The p-adic Tate module T$_p$(G) of G is a Galois representation (of the absolute Galois group, G$_K$, of K).
Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then T$_p$(G) is a free module over Z$_p$ of rank 2d, where d is the dimension of G. In the other case, it is still free, but the rank may take any value from 0 to d (see for example Hasse–Witt matrix).
In the case where p is not equal to the characteristic of K, the p-adic Tate module of G is the dual of the étale cohomology $H_{\text{et}}^{1}(G\times _{K}K^{s},\mathbf {Z} _{p})$.
A special case of the Tate conjecture can be phrased in terms of Tate modules. Suppose K is finitely generated over its prime field (e.g. a finite field, an algebraic number field, a global function field), of characteristic different from p, and A and B are two abelian varieties over K. The Tate conjecture then predicts that
    $\mathrm {Hom} _{K}(A,B)\otimes \mathbf {Z} _{p}\cong \mathrm {Hom} _{G_{K}}(T_{p}(A),T_{p}(B))$
where Hom$_K$(A, B) is the group of morphisms of abelian varieties from A to B, and the right-hand side is the group of G$_K$-linear maps from T$_p$(A) to T$_p$(B). The case where K is a finite field was proved by Tate himself in the 1960s. Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".
In the case of a Jacobian over a curve C over a finite field k of characteristic prime to p, the Tate module can be identified with the Galois group of the composite extension
    $k(C)\subset {\hat {k}}(C)\subset A^{(p)}\$
where ${\hat {k}}$ is an extension of k containing all p-power roots of unity and A$^{(p)}$ is the maximal unramified abelian p-extension of ${\hat {k}}(C)$.

## Tate module of a number field

The description of the Tate module for the function field of a curve over a finite field suggests a definition for a Tate module of an algebraic number field, the other class of global field, introduced by Kenkichi Iwasawa. For a number field K we let K$_m$ denote the extension by p$^m$-power roots of unity, ${\hat {K}}$ the union of the K$_m$ and A$^{(p)}$ the maximal unramified abelian p-extension of ${\hat {K}}$. Let
    $T_{p}(K)=\mathrm {Gal} (A^{(p)}/{\hat {K}})\ .$
Then T$_p$(K) is a pro-p-group and so a Z$_p$-module. Using class field theory one can describe T$_p$(K) as isomorphic to the inverse limit of the class groups C$_m$ of the K$_m$ under norm.
Iwasawa exhibited T$_p$(K) as a module over the completion Z$_p$[[T]] and this implies a formula for the exponent of p in the order of the class groups C$_m$ of the form
    $\lambda m+\mu p^{m}+\kappa \ .$
The Ferrero–Washington theorem states that μ is zero.