 # Tate module

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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(Ks) (the Ks-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 ) = lim A [ p n ]

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 Zp-module via

z ( a n ) n = ( ( z  mod  p n ) a n ) n .

## 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 (where the choice of p and K are tacitly understood). 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(G) of G is a Galois representation (of the absolute Galois group, GK, of K).

Classical results on abelian varieties show that if K has characteristic zero, or characteristic ℓ where the prime number p ≠ ℓ, then Tp(G) 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 (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 et 1 ( G × K K s , 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

H o m K ( A , B ) Z p H o m G K ( T p ( A ) , T p ( B ) )

where HomK(A, B) 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(A) to Tp(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 ) k ^ ( C ) A ( p )

where k ^ is an extension of k containing all p-power roots of unity and A(p) is the maximal unramified abelian p-extension of 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 Iwasawa. For a number field K we let Km denote the extension by pm-power roots of unity, K ^ the union of the Km and A(p) the maximal unramified abelian p-extension of K ^ . Let

T p ( K ) = G a l ( A ( p ) / K ^ )   .

Then Tp(K) is a pro-p-group and so a Zp-module. Using class field theory one can describe Tp(K) as isomorphic to the inverse limit of the class groups Cm of the Km under norm.

Iwasawa exhibited Tp(K) as a module over the completion Zp[[T]] and this implies a formula for the exponent of p in the order of the class groups Cm of the form

λ m + μ p m + κ   .

The Ferrero–Washington theorem states that μ is zero.

## References

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