On the Newton polygons of twisted $L$-functions of binomials

TL;DR

This paper investigates the Newton polygons of twisted $L$-functions of binomials over finite fields, providing bounds and conditions for their coincidence, and conjectures about their behavior for large primes.

Contribution

It introduces new bounds for Newton polygons of twisted $L$-functions of binomials and proposes a conjecture on their behavior for sufficiently large primes.

Findings

Lower bounds for Newton polygons when p > (d-e)(2d-1)

Coincidence of Newton polygons under certain divisibility conditions

Conjecture that conditions hold for large enough p

Abstract

Let $\chi$ be an order $c$ multiplicative character of a finite field and $f(x)=x^d+\lambda x^e$ a binomial with $(d,e)=1$. We study the twisted classical and $T$-adic Newton polygons of $f$. When $p>(d-e)(2d-1)$, we give a lower bound of Newton polygons and show that they coincide if $p$ does not divide a certain integral constant depending on $p\bmod {cd}$. We conjecture that this condition holds if $p$ is large enough with respect to $c,d$ by combining all known results and the conjecture given by Zhang-Niu. As an example, we show that it holds for $e=d-1$.