Tate module

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Not to be confused with Hodge–Tate module. ‹See Tfd›

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[edit]

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[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\text{ mod }p^n)a_n)_n.

Examples[edit]

The Tate module[edit]

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[edit]

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.[1] 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^1_{\text{et}}(G\times_KK^s,\mathbf{Z}_p).

A special case of the Tate conjecture can be phrased in terms of Tate modules.[2] 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 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.[3] Gerd Faltings proved the case where K is a number field in his celebrated "Mordell paper".[4]

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).[5]

Tate module of a number field[edit]

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, \hat K the union of the Km and A(p) the maximal unramified abelian p-extension of \hat K. Let

T_p(K) = \mathrm{Gal}(A^{(p)}/\hat 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.[5]

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

 \lambda m + \mu p^m + \kappa \ .

The Ferrero–Washington theorem states that μ is zero.[6]

See also[edit]

Notes[edit]

References[edit]