# Tate module

A free $\mathbf Z _ {p}$- module $T ( G)$ associated to a $p$- 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. 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.

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 [4]. Examples are known of non-cyclotomic $\mathbf Z _ {p}$- extensions with $\mu > 0$( see [3]). Even in the case when $\mu = 0$, $T _ {p} ( k _ \infty )$ is not necessarily a free $\mathbf Z _ {p}$- module.

#### References

 [1] J.T. Tate, "-divisible groups" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 158–183 MR0231827 Zbl 0157.27601 [2] I.R. Shafarevich, "The zeta-function" , Moscow (1969) (In Russian) [3] K. Iwasawa, "On the -invariants of -extensions" , Number theory, algebraic geometry and commutative algebra , Kinokuniya (1973) pp. 1–11 MR357371 [4] B. Ferrero, L.C. Washington, "The Iwasawa invariant vanishes for abelian number fields" Ann. of Math. , 109 (1979) pp. 377–395 MR528968 Zbl 0443.12001
How to Cite This Entry:
Tate module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_module&oldid=48951
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article