On the variation of the Frobenius in a non abelian Iwasawa tower

Abstract

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 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. Along the way, we will prove that many natural sequences $(x_n{)}_{n\ge 1}\in {\mathbb{Z}}_{\ell }^{\mathbb{N}}$ 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.

Accepted version

Cited by in corpus (1)

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