Newton slopes for Artin-Schreier-Witt towers

TL;DR

This paper investigates the Newton slopes of zeta functions in Artin-Schreier-Witt towers over finite fields, revealing that for large conductors, these slopes form conductor-independent arithmetic progressions, impacting eigencurve analysis.

Contribution

It establishes the structure of Newton slopes as arithmetic progressions independent of conductor size in Artin-Schreier-Witt towers.

Findings

Newton slopes form arithmetic progressions for large conductors

Slopes are independent of the character’s conductor

Results relate to eigencurve behavior

Abstract

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field and consider the Artin-Schreier-Witt tower defined by $f(x)$; this is a tower of curves $\cdots \to C_m \to C_{m-1} \to \cdots \to C_0 =\mathbb A^1$, with total Galois group $\mathbb Z_p$. 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 form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.