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

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 $e=d-1$.