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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
Abstract

We study towers of varieties over a finite field such as y2 = f(xn ) and prove that the characteristic polynomials of the Frobenius on the étale cohomology show a surprising -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 (pn(x))n1 [x] converge -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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