# 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*) (the

^{s}*K*-valued points of

^{s}*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*[*p ^{n}*] is the

*p*torsion of

^{n}*A*(i.e. the kernel of the multiplication-by-

*p*map), and the inverse limit is over positive integers

^{n}*n*with transition morphisms given by the multiplication-by-

*p*map

*A*[

*p*

^{n+1}] →

*A*[

*p*]. Thus, the Tate module encodes all the

^{n}*p*-power torsion of

*A*. It is equipped with the structure of a

**Z**

_{p}-module via

## Examples

*The* Tate module

*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*of

_{K}*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*, of

_{K}*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 2*d*, 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 .

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

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.[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

where is an extension of *k* containing all *p*-power roots of unity and *A*^{(p)} is the maximal unramified abelian *p*-extension of .[5]

## 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, the union of the *K*_{m} and *A*^{(p)} the maximal unramified abelian *p*-extension of . Let

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.[5]

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

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

## See also

## Notes

- Murty 2000, Proposition 13.4
- Murty 2000, §13.8
- Tate 1966
- Faltings 1983
- Manin & Panchishkin 2007, p. 245
- Manin & Panchishkin 2007, p. 246

## References

- Faltings, Gerd (1983), "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern",
*Inventiones Mathematicae*,**73**(3): 349–366, doi:10.1007/BF01388432 - Hazewinkel, Michiel, ed. (2001) [1994], "Tate module",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Murty, V. Kumar (2000),
*Introduction to abelian varieties*, CRM Monograph Series,**3**, American Mathematical Society, ISBN 978-0-8218-1179-5 - Section 13 of Rohrlich, David (1994), "Elliptic curves and the Weil–Deligne group", in Kisilevsky, Hershey; Murty, M. Ram (eds.),
*Elliptic curves and related topics*, CRM Proceedings and Lecture Notes,**4**, American Mathematical Society, ISBN 978-0-8218-6994-9 - Tate, John (1966), "Endomorphisms of abelian varieties over finite fields",
*Inventiones Mathematicae*,**2**: 134–144, doi:10.1007/bf01404549, MR 0206004