The stable property of Newton slopes for general Witt towers

TL;DR

This paper investigates the Newton slopes of zeta functions in Artin-Schreier-Witt towers over finite fields, showing they form stable, proportionally changing arithmetic progressions as conductors grow large.

Contribution

It generalizes previous results by proving the stability and proportional change of Newton slopes for a broad class of Witt towers using $(p^{ heta},T)$-topology.

Findings

Newton slopes form unions of arithmetic progressions

Slopes change proportionally with conductor increases

Generalizes previous results beyond roots of unity

Abstract

Any polynomial $f(x)\in\mathbb{Z}_q[x]$ defines a Witt vector $[f]\in W(\mathbb{F}_q[x])$. Consider the Artin-Schreier-Witt tower $y^F-y=[f]$. This is a tower of curves over $\mathbb{F}_q$, with total Galois group $\mathbb{Z}_p$. We want to study the Newton slopes of zeta functions of this tower. We reduce it to the Newton polygons of L-functions associated with characters on the Galois groups. We prove that, when the conductors are large enough, these Newton slopes are unions of arithmetic progressions which are changing proportionally as the conductor increases. This is a generalization of the result of \cite{Da}, where they get the same result in the case the non-zero coefficients of $f(x)$ are roots of unity. To overcome the new difficulty in our process, we apply some $(p^{\theta},T)$-topology.