Let $k$ be a field, $\overline{k}$ its algebraic closure, and $A$ an abelian variety defined over $k$, of dimension $g$. For each integer $m \ge 1$, let $A_m$ denote the group of elements $a \in A(\overline{k})$ such that $ma = 0$. let $l$ be a prime number different from the characteristic of $k$, and let $T_l(A)$ denote the projective limit of the groups $A_{l^n}$ with respect to the maps $A_{l^{n + 1}} \to A_{l^n}$ which are induced by multiplication by $l$. It is well known that $T_l(A)$ is a free module of rank $2g$ over the ring $\mathbb{Z}_l$ of $l$-adic integers. The group $G = \text{Gal}(\overline{k}/k)$ operators on $T_l(A)$.

Let $A'$ and $A^{\prime\prime}$ be abelian varieties defined over $k$. The group $\text{Hom}_k(A', A^{\prime\prime})$ of homomorphisms of $A'$ into $A^{\prime\prime}$ defined over $k$ is $\mathbb{Z}$-free, and the canonical map$$\mathbb{Z}_\ell \otimes \text{Hom}_k(A', A^{\prime\prime}) \to \text{Hom}_G(T_l(A'), T_l(A^{\prime\prime}))\tag*{$(1)$}$$is injective.

Theorem (Tate, 1966).If $k$ is finite, the map $(1)$ is bijective.

**Question.** Could anybody supply an outline of Tate's proof of this theorem and contribute their intuition as to why such a result is true?