Let $K$ be a field, let $\overline{K}$ be an [algebraic closure](/page/Algebraic%20Closure) of $K$, and let $A/K$ be an abelian variety of dimension $g$. Let $l$ be a prime such that $l \neq \operatorname{char}(K)$, with the convention that this condition is automatic when $\operatorname{char}(K)=0$. For every integer $n \geq 1$, the finite abelian group of geometric $l^n$-torsion points satisfies

\begin{align*} A[l^n](\overline{K}) \cong (\mathbb{Z}/l^n\mathbb{Z})^{2g}. \end{align*}

Consequently, the $l$-adic Tate module

\begin{align*} T_lA := \varprojlim_{n} A[l^n](\overline{K}), \end{align*}

where the transition maps are multiplication by $l$, is a free $\mathbb{Z}_l$-module of rank $2g$.