# Newton slopes for twisted Artin--Schreier--Witt Towers

## TL;DR

This paper investigates the $p$-adic Newton slopes of $L$-functions associated with twisted Artin--Schreier--Witt towers, showing they form arithmetic progressions when the character's conductor is sufficiently large.

## Contribution

It establishes the pattern of Newton slopes as arithmetic progressions for a broad class of twisted $L$-functions in the context of Artin--Schreier--Witt towers.

## Key findings

- Newton slopes form arithmetic progressions for large conductor
- Results apply to twisted $L$-functions over finite fields
- Provides new insights into $p$-adic properties of $L$-functions

## Abstract

We fix a monic polynomial $f(x) \in \mathbb F_q[x]$ over a finite field of characteristic $p$ of degree relatively prime to $p$. Let $a\mapsto \omega(a)$ be the Teichm\"uller lift of $\mathbb F_q$, and let $\chi:\mathbb{Z}\to \mathbb C_p^\times$ be a finite character of $\mathbb Z_p$. The $L$-function associated to the polynomial $f$ and the so-called twisted character $\omega^u\times \chi$ is denoted by $L_f(\omega^u,\chi,s)$. We prove that, when the conductor of the character is large enough, the $p$-adic Newton slopes of this $L$-function form arithmetic progressions.