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Valuations of exponential sums and Artin-Schreier curves
Let denote an odd prime. In this paper, we are concerned with the
-divisibility of additive exponential sums associated to one variable
polynomials over a finite field of characteristic , and with (the very close
question of) determining the Newton polygons of some families of Artin-Schreier
curves, i.e. -cyclic coverings of the projective line in characteristic .
We first give a lower bound on the -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 cannot be supersingular
when its genus has the form for some
and such that . We also determine the
first vertex of the generic Newton polygon of the family of -rank
Artin-Schreier curves of fixed genus, and the associated Hasse polynomial.Comment: 19 page