Tate module

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 ${\displaystyle H_{\text{et}}^1(G{\times }_K{K}^s,{\mathbf{Z}}_p)}$.

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

${\displaystyle {\mathrm{H}\mathrm{o}\mathrm{m}}_K(A,B)\otimes {\mathbf{Z}}_p\cong {\mathrm{H}\mathrm{o}\mathrm{m}}_{G_K}(T_p(A),T_p(B))}$

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

${\displaystyle k(C)\subset \hat{k}(C)\subset A^{(p)}}$

where ${\displaystyle \hat{k}}$ is an extension of k containing all p-power roots of unity and A^(p) is the maximal unramified abelian p-extension of ${\displaystyle \hat{k}(C)}$.

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, ${\displaystyle \hat{K}}$ the union of the K_m and A^(p) the maximal unramified abelian p-extension of ${\displaystyle \hat{K}}$. Let

${\displaystyle T_p(K)=\mathrm{G}\mathrm{a}\mathrm{l}(A^{(p)}/\hat{K}).}$

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

${\displaystyle \lambda m+\mu p^m+\kappa .}$

The Ferrero–Washington theorem states that μ is zero.