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

G., Asvin

doi:10.2140/ant.2023.17.2151

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

Asvin G.

Vol. 17 (2023), No. 12, 2151–2179

DOI: 10.2140/ant.2023.17.2151

Abstract

We study towers of varieties over a finite field such as $y^{2} = f (x^{ℓ^{n}})$ and prove that the characteristic polynomials of the Frobenius on the étale cohomology show a surprising $ℓ$ -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 ${(p_{n} (x))}_{n \geq 1} \in ℤ_{ℓ} {[x]}^{ℕ}$ converge $ℓ$ -adically and give explicit rates of convergence.

Keywords

Iwasawa theory, $L$-functions over finite fields

Mathematical Subject Classification

Primary: 11R23

Secondary: 11G20

Milestones

Received: 30 March 2022

Revised: 12 January 2023

Accepted: 20 March 2023

Published: 8 October 2023

Authors

Asvin G.
Department of Mathematics University of Wisconsin-Madison WI United States

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Vol. 17, No. 12, 2023