The stable property of Newton slopes for general Witt towers

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 $(p^θ,T)$-topology.