8 p - DENSITY , EXPONENTIAL SUMS AND ARTIN - SCHREIER CURVES

In this paper we define the p-density of a finite subset D ⊂ Nr , and show that it gives a good lower bound for the p-adic valuation of exponential sums over finite fields of characteristic p. We also give an application: when r = 1, the p-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in D. 0. Introduction This paper deals with the p-adic valuation of exponential sums over a finite field of characteristic p. A classical result in this field is Stickelberger’s theorem on the valuation of Gauss sums [12]. It is closely related to the question of the existence of rational points on an algebraic variety over a finite field. This question was raised by E. Artin, then solved by Chevalley, and precised by Warning [14], leading to the celebrated Chevalley-Warning theorem. During the 60s, the emergence of Dwork’s ideas allowed improvements on this result, first by Ax [2], then by Katz [4]. Both based their result on a suitable estimation for the p-adic valuation of exponential sums, and their bounds only depends on the degrees of the polynomials defining the variety. More recently, Adolphson and Sperber have improved these results, considering more precisely the monomials appearing in the polynomial [1]. They look at its Newton polyhedron, i.e. the convex closure of the exponents effectively appearing in it. All these bounds are independent of the prime p, and are optimal in the following sense: if we fix degrees (resp. a Newton polyhedron), then for any p we can find a (system of) polynomial(s) such that the valuation of the associated exponential sum meets the bound. In the 90s, Moreno and Moreno took into account the prime p. Using a reduction to the ground fied method, they replaced the degrees of the polynomials by their p-weights (the sums of their base p digits) and found a new bound, improving the existing ones in many cases [6]. Recently, in a joint work with Castro, Kumar and Shum, O. Moreno reduced the problem of estimating the p-adic valuation of an exponential sum to the one of estimating the minimal p-weight of the set of solutions a system of modular equations [7]. They shown that their bound is tight. Note that it must be finer than the preceding ones. It is the first one really taking into account only the monomials appearing: when we use a Newton polyhedron, 1991 Mathematics Subject Classification. 11M38,14F30,52B20.