Newton polygons of $L$-functions of polynomials $x^d+ax^{d-1}$ with $p\equiv-1\bmod d$

TL;DR

This paper determines the slopes of the $q$-adic Newton polygons of $L$-functions for specific polynomials over finite fields when the prime $p$ satisfies a congruence condition, using Dwork's trace formula and Zhu's theorem.

Contribution

It explicitly computes Newton polygon slopes for polynomials $x^d+ax^{d-1}$ over finite fields under certain prime conditions, extending previous understanding.

Findings

Explicit slopes of $L$-function Newton polygons are obtained.

Results hold for primes larger than a bound depending on $d$ and $ ext{log}_p q$.

Uses Dwork's trace formula and Zhu's rigid transform theorem.

Abstract

For prime $p\equiv-1\bmod d$ and $q$ a power of $p$, we obtain the slopes of the $q$-adic Newton polygons of $L$-functions of $x^d+ax^{d-1}\in \mathbb{F}_q[x]$ with respect to finite characters $\chi$ when $p$ is larger than an explicit bound depending only on $d$ and $\log_p q$. The main tools are Dwork's trace formula and Zhu's rigid transform theorem.