On the variation of Frobenius eigenvalues in a skew-abelian Iwasawa tower

We study towers of varieties over a finite field such as $y^2=f(x^{{\ell }^n})$ and prove that the characteristic polynomials of the Frobenius on the étale 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. The key ingredient is a generalization of Fermat’s little theorem to matrices. Along the way, we will prove that many natural sequences of polynomials ${(p_n(x))}_{n\ge 1}\in {\mathbb{Z}}_{\ell }{[x]}^{\mathbb{N}}$ converge $\ell$-adically and give explicit rates of convergence.