Slopes for higher rank Artin-Schreier-Witt Towers

TL;DR

This paper investigates the Newton slopes of zeta functions in higher rank Artin-Schreier-Witt towers over finite fields, demonstrating their asymptotic arithmetic progression structure and extending previous rank one results.

Contribution

It extends the analysis of Newton slopes and spectral properties from rank one to higher rank Artin-Schreier-Witt towers, revealing their asymptotic behavior.

Findings

Newton slopes form a finite union of arithmetic progressions for large conductors

Spectral halo property is established for higher rank towers

Extends previous rank one results to higher rank cases

Abstract

We fix a monic polynomial $\bar f(x) \in \mathbb{F}_q[x]$ over a finite field of characteristic $p$, and consider the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower defined by $\bar f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb{A}^1$, whose Galois group is canonically isomorphic to $\mathbb{Z}_{p^\ell}$, the degree $\ell$ unramified extension of $\mathbb{Z}_p$, which is abstractly isomorphic to $(\mathbb{Z}_p)^\ell$ as a topological group. We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function asymptotically form a finite union of arithmetic progressions. As a corollary, we prove the spectral halo property of the spectral variety associated to the $\mathbb{Z}_{p^{\ell}}$-Artin-Schreier-Witt tower. This extends the main result in [DWX] from rank one case $\ell=1$ to the higher rank case $\ell\geq 1$.