Valuation of exponential sums and the generic first slope for Artin–Schreier curves

Abstract

We define the p -density of a finite subset D ⊂ N r , and show that it gives a sharp lower bound for the p -adic valuations of the reciprocal roots and poles of zeta functions and L -functions associated to exponential sums over finite fields of characteristic p . When r = 1 , the p -density of the set D is the first slope of the generic Newton polygon of the family of Artin–Schreier curves associated to polynomials with their exponents in D .