Tate module of linear algebraic group

Question

Let $G$ be a (smooth, connected, geometrically integral) commutative linear algebraic group over $\mathbf F_q$. Just as for abelian varieties, we can define the $\ell$-adic Tate module $$ T_\ell G = \varprojlim_n G(\overline{\mathbf F_q})[\ell^n] . $$ A couple questions:

1. Is $T_\ell G$ (as a representation of $G_{\mathbf F_q}$) a direct sum of copies of (powers of) the cyclotomic character?

2. Even if not, is $T_\ell G$ a free $\mathbf Z_\ell$-module with $\ell$-independent characteristic polynomial of Frobenius?

Answer 1

The basic examples of $G$ are tori, and unipotent groups. The latter have no $\ell$-torsion (if $\ell \neq p$, the char. of $\mathbb F_q$), and so the interesting case is when $G$ is a torus.

A torus is characterized by its associated character lattice $X$, which is a rank $n$ free $\mathbb Z$-module (if $n = \dim G$), with an action of Frobenius via an automorphism of finite order.

The Tate-module will then be isomorphic $X^{\vee}\otimes \mathbb Z_{\ell}(1)$, where $\vee$ denotes the contragredient (so the cocharacter lattice of $G$, is you like), and $\mathbb Z_{\ell}(1)$ denotes the $\ell$-adic cyclotomic character.

So one needn't have just a direct sum of copies of the cyclotomic char.; there is the possibility of having an extra finite-order twist (provided by $X$). But certainly one has independence of $\ell$, since $X$ depends just on $G$ (and not the choice of $\ell$).