---
tag: NC4B8KC479
kind: definition
name: Tate module
msc: [14K15, 11G10]
defines: 4CSMQFTGD8
uses: [082GHNPJ53, 0DBWXQRXQJ, V6TJAFZ4MD, R7W03TXRS9, PGKWBJDTKX]
sources:
  - url: https://en.wikipedia.org/wiki/Tate_module
---

## Definition

Let $G$ be an [[abelian group|082GHNPJ53]], written additively, and let $\ell$ be a prime number.
For $n \ge 1$, let $G[\ell^n] = \set{x \in G : \ell^n x = 0}$, a subgroup of the [[torsion subgroup|0DBWXQRXQJ]].
Multiplication by $\ell$ maps $G[\ell^{n+1}]$ to $G[\ell^n]$.
The *$\ell$-adic Tate module* of $G$ is the inverse limit
$$T_\ell(G) = \varprojlim_n G[\ell^n]$$
along these maps.
It is a module over the [[$\ell$-adic integers|V6TJAFZ4MD]] $\Z_\ell$.

Let $A$ be an [[abelian variety|R7W03TXRS9]] over a field $k$, and let $k^{s}$ be a separable closure of $k$.
The Tate module of $A$ is $T_\ell(A) = T_\ell(A(k^{s}))$.

## Remarks

If $A$ has dimension $g$ and $\ell$ is not the characteristic of $k$, then $T_\ell(A)$ is a free $\Z_\ell$-module of rank $2g$.
The [[absolute Galois group|PGKWBJDTKX]] $\Gal(k^{s}/k)$ acts on each $A[\ell^n]$, and so continuously on $T_\ell(A)$.

Example: let $G = \mu_{\ell^\infty}$ be the group of roots of unity of $\ell$-power order in $\C^{*}$.
Then $G[\ell^n]$ is cyclic of order $\ell^n$, and the maps $\zeta \mapsto \zeta^\ell$ are onto.
So $T_\ell(G) \cong \Z_\ell$, and this module is written $\Z_\ell(1)$.

Degenerate example: $G = \Z$ has no nonzero torsion, so every $G[\ell^n]$ is $0$ and $T_\ell(\Z) = 0$.
The Tate module only sees the $\ell$-power torsion of $G$.