  For varieties over a finite field $\mathbb F_q$ with "many" automorphisms, we
study the $\ell$-adic properties of the eigenvalues of the Frobenius operator
on their cohomology. The main goal of this paper is to consider towers such as
$y^2 = f(x^{\ell^n})$ and prove that the characteristic polynomials of the
Frobenius on the \'etale cohomology show a surprising $\ell$-adic convergence.
We prove this by proving a more general statement about the convergence of
certain invariants related to a skew-abelian cohomology group. Along the way,
we will prove that many natural sequences $(x_n)_{n\geq 1} \in \mathbb
Z_\ell^{\mathbb N}$ converge $\ell$-adically and give explicit rates of
convergence. In a different direction, we provide a precise criterion for
curves with many automorphisms to be supersingular, generalizing and unifying
many old results.
