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A free  $  \mathbf Z _ {p} $-
module  $  T ( G) $
associated to a [[P-divisible group| $  p $-
divisible group]]  $  G $
defined over a complete discrete valuation ring  $  R $
of characteristic 0 with residue field  $  k $
of characteristic  $  p $. 
Let  $  G = \{ G _  \nu  , i _  \nu  \} $, 
$  \nu \geq  0 $, 
and  $  T ( G) = \lim\limits _  \leftarrow   G _  \nu  ( \overline{K}\; ) $, 
where  $  \overline{K}\; $
is the algebraic closure of the quotient field  $  K $
of the ring  $  R $; 
the limit is taken with respect to the mappings  $  j _  \nu  :  G _ {\nu + 1 }  \rightarrow G _  \nu  $
for which  $  i _  \nu  \circ j _  \nu  = p $. 
Then  $  T ( G) = \mathbf Z _ {p}  ^ {h} $, 
where  $  h $
is the height of the group  $  G $
and  $  T ( G) $
has the natural structure of a  $  G ( \overline{K}\; /K) $-
module. The functor  $  G \rightarrow T ( G) $
allows one to reduce a number of questions about the group  $  G $
to simpler questions about  $  G ( \overline{K}\; /K) $-
modules.

The Tate module is defined similarly for an [[Abelian variety|Abelian variety]]. Let  $  A $
be an Abelian variety defined over  $  k $, 
and let  $  A _ {p  ^ {n}  } $
be the group of points of order  $  p  ^ {n} $
in  $  A ( \overline{k}\; ) $. 
Then  $  T ( A) $
is defined as  $  \lim\limits _  \leftarrow   A _ {p  ^ {n}  } $. 
The Tate module of a curve  $  X $
is the Tate module of its [[Jacobi variety|Jacobi variety]].

The construction of the module  $  T _ {p} ( X) $
can be extended to number fields. Let  $  K $
be an algebraic number field and let  $  k _  \infty  $
be a  $  \mathbf Z _ {p} $-
extension of the field  $  k $(
an extension with Galois group isomorphic to  $  \mathbf Z _ {p} $). 
For the intermediate field  $  k _ {n} $
of degree  $  p  ^ {n} $
over  $  k $, 
let  $   \mathop{\rm Cl} ( k _ {n} ) _ {p} $
be the  $  p $-
component of the ideal class group of the field  $  k _ {n} $. 
Then  $  T _ {p} ( k _  \infty  ) = \lim\limits _  \leftarrow    \mathop{\rm Cl} ( k _ {n} ) _ {p} $, 
where the limit is taken with respect to norm-mappings  $   \mathop{\rm Cl} ( k _ {m} ) _ {p} \rightarrow  \mathop{\rm Cl} ( k _ {n} ) _ {p} $
for  $  m > n $. 
The module  $  T _ {p} ( k _  \infty  ) $
is characterized by its Iwasawa invariants  $  \lambda $, 
$  \mu $
and  $  \nu $, 
defined by

$$ 
|  \mathop{\rm Cl} ( k _ {n} ) _ {p} |  = \ 
p ^ {e _ {n} } ,
$$

where  $  e _ {n} = \lambda n + \mu p ^ {n + \nu } $
for all sufficiently large  $  n $. 
For cyclotomic  $  \mathbf Z _ {p} $-
extensions the invariant  $  \mu $
is equal to 0. This was also proved for Abelian fields [[#References|[4]]]. Examples are known of non-cyclotomic  $  \mathbf Z _ {p} $-
extensions with  $  \mu > 0 $(
see [[#References|[3]]]). Even in the case when  $  \mu = 0 $, 
$  T _ {p} ( k _  \infty  ) $
is not necessarily a free  $  \mathbf Z _ {p} $-
module.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.T. Tate, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227061.png" />-divisible groups" T.A. Springer (ed.) et al. (ed.) , ''Proc. Conf. local fields (Driebergen, 1966)'' , Springer (1967) pp. 158–183 {{MR|0231827}} {{ZBL|0157.27601}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> K. Iwasawa, "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227062.png" />-invariants of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227063.png" />-extensions" , ''Number theory, algebraic geometry and commutative algebra'' , Kinokuniya (1973) pp. 1–11 {{MR|357371}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B. Ferrero, L.C. Washington, "The Iwasawa invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092270/t09227064.png" /> vanishes for abelian number fields" ''Ann. of Math.'' , '''109''' (1979) pp. 377–395 {{MR|528968}} {{ZBL|0443.12001}} </TD></TR></table>