On a Fiber Conjecture of Wan

Abstract

For a prime $p$ and $p$-power $q$, let $f(x)\in {\mathbb{F}}_q[x]$ with $\text{deg}f$ coprime to $p$. As $\lambda$ varies in $\overline{{\mathbb{F}}_p^{\times }}$, Wan has conjectured that the $p$-adic Newton polygon of the corresponding Artin-Schreier curve given by $\lambda f$ is constant. That is, \[ \textrm{NP}(f) = \textrm{NP}(λf). \] In this paper, we prove this conjecture when $\lambda \in {\mathbb{F}}_p^{\times }$ and provide a detailed counterexample showing it is false in general.