Let $A$ be an abelian variety defined over a field $K$ of characteristic $p>0$. Let $A[\ell]$ be the group of $\ell$-torsion points, $\ell\neq p$ a prime. Are there positive constants $C, \eta$ depending on $A$ and $K$ only such that $[K(A[\ell]):K]> C \ell^{\eta}$? What about $[K(P):K]> C' \ell^{\eta'}$, $P$ an $\ell$-torsion point? In some cases I know the Lie algebra to be $sp(4,Q\_{\ell}) \oplus Q\_{\ell} Id$ and $sp(6,Q\_{\ell}) \oplus Q\_{\ell} Id$. Is this enough?



lower bound for torsion of abelian varieties
Now let me address your last question. For the sake of simplicity, let us assume that $K$ is a global field of characteristic $p>2$ and the ring $End(A)$ of all endomorphisms of $A$ (over an algebraic closure of $K$) is the ring $Z$ of integers. Then my old results (Math. Notes: 21 (1978), 415--419 and 22 (1978), 493--498) imply that for all but finitely many primes $\ell$ the Galois module $A[\ell]$ is absolutely simple and the Galois group $Gal(K(A[\ell])/K)$ is noncommutative. 


I claim that for all but finitely many primes $\ell$ the order of 
$Gal(K(A[\ell])/K)$ is divisible by $\ell$ and therefore $[K(A[\ell]):K] > \ell$.


Indeed, suppose that for a given $\ell$ the order of 
$Gal(K(A[\ell])/K)$ is not divisible by $\ell$. Let us call such $\ell$ exceptional. Then the natural representation of $Gal(K(A[\ell])/K)$ in $A[\ell]$ can be lifted to characteristic zero, i.e., 
$Gal(K(A[\ell])/K)$ is isomorphic to a (finite) subgroup of $GL(2\dim(A),C)$ where $C$ is the field of complex numbers. Now, by a theorem of Jordan, there exists a positive integer $N$ that depends only on $\dim(A)$ anf such that $Gal(K(A[\ell])/K)$ contains a normal commutative subgroup, whose index does not exceed $N$. This means that $K(A[\ell])/K$ contains a Galois subextension $K\_{0,\ell}/K$ such that $[K\_{0,\ell}:K] \le N$ and the field extension $K(A[\ell])/K\_{0,\ell}$ is abelian.


Let $S$ be the (finite) set of places of bad reduction for $A$. The field extension $K(A[\ell])/K$ is unramified outside $S$; the same is true for the subextension $K\_{0,\ell}/K$. (In the number field case one should add the divisors of $\ell$ but we live in characteristic $p$!). Recall that the Galois extensions of global $K$ with degree $\le N$ and ramification only at $S$ constitute a finite set. Let $L/K$ be the compositum of all such extensions. Clearly, $L$ is also a global field while $L/K$ is a finite Galois extension that contains $K\_{0,\ell}$. In addition, the Galois group $Gal(L(A[\ell])/L)$ is commutative. Now considering $A$ as an abelian variety over $L$ and applying previously mentioned results, we obtain that the set of exceptional primes $\ell$ is finite.


