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, Ks is the separable closure of K, and A = 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
where A is the pn torsion of A, and the inverse limit is over positive integers n with transition morphisms given by the multiplication-by-p map A → A. Thus, the Tate module encodes all the p-power torsion of A. It is equipped with the structure of a Zp-module via

Examples

''The'' Tate module

When the abelian group A is the group of roots of unity in a separable closure Ks of K, the p-adic Tate module of A is sometimes referred to as the Tate module. It is a free rank one module over Zp with a linear action of the absolute Galois group GK 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 Gm,K over K.

The Tate module of an abelian variety

Given an abelian variety G over a field K, the Ks-valued points of G are an abelian group. The p-adic Tate module Tp of G is a Galois representation.
Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then Tp is a free module over Zp 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.
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.
A special case of the Tate conjecture can be phrased in terms of Tate modules. Suppose K is finitely generated over its prime field, of characteristic different from p, and A and B are two abelian varieties over K. The Tate conjecture then predicts that
where HomK is the group of morphisms of abelian varieties from A to B, and the right-hand side is the group of GK-linear maps from Tp to Tp. 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
where is an extension of k containing all p-power roots of unity and A is the maximal unramified abelian p-extension of.

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 Km denote the extension by pm-power roots of unity, the union of the Km and A the maximal unramified abelian p-extension of. Let
Then Tp is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp as isomorphic to the inverse limit of the class groups Cm of the Km under norm.
Iwasawa exhibited Tp as a module over the completion ZpT and this implies a formula for the exponent of p in the order of the class groups Cm of the form
The Ferrero–Washington theorem states that μ is zero.