Valuations of exponential sums and Artin-Schreier curves

Abstract

Let $p$ denote an odd prime. In this paper, we are concerned with the $p$-divisibility of additive exponential sums associated to one variable polynomials over a finite field of characteristic $p$, and with (the very close question of) determining the Newton polygons of some families of Artin-Schreier curves, i.e. $p$-cyclic coverings of the projective line in characteristic $p$. We first give a lower bound on the $p$-divisibility of exponential sums associated to polynomials of fixed degree. Then we show that an Artin-Schreier curve defined over a finite field of characteristic $p$ cannot be supersingular when its genus $g$ has the form $(p-1)\left(i(p^n-1)-1\right)/2$ for some $1\leq i\leq p-1$ and $n\geq 1$ such that $n(p-1)>2$. We also determine the first vertex of the generic Newton polygon of the family of $p$-rank $0$ Artin-Schreier curves of fixed genus, and the associated Hasse polynomial.