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*^{s}) (the *K*^{s}-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*.

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*[*p*^{n}] is the *p*^{n} torsion of *A* (i.e. the kernel of the multiplication-by-*p*^{n} map), and the inverse limit is over positive integers *n* with transition morphisms given by the multiplication-by-*p* map *A*[*p*^{n}^{+1}] → *A*[*p*^{n}]. Thus, the Tate module encodes all the *p*-power torsion of *A*. It is equipped with the structure of a **Z**_{p}-module via

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

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 2*d*, 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 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

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
)
.

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 *K*_{m} denote the extension by *p*^{m}-power roots of unity,
K
^
the union of the *K*_{m} and *A*^{(p)} the maximal unramified abelian *p*-extension of
K
^
. Let

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

λ
m
+
μ
p
m
+
κ
.
The Ferrero–Washington theorem states that μ is zero.