Valuations of exponential sums and Artin-Schreier curves

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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)(i(pⁿ-1)-1)/2 for some 1≤ i≤ p-1 and n≥ 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.