A classification of Newton polygons of L-functions on polynomials

Abstract

Considering the L-function of exponential sums associated to a polynomial over a finite field F_q, Deligne proved that a reciprocal root's p-adic order is a rational number in the interval [0, 1]. Based on hypergeometric theory, in this paper we improve this result that there are only finitely many possible forms of Newton polygons for the L-function of degree d polynomials independent of p, when p is larger than a constant D^*(theorem 4.3), i.e., a reciprocal root's p-adic order has form (up-v)/(D^*(p-1)) in which u, v have finitely many possible values. Furthermore, when p>D^*, to determine the Newton polygon is only to determine it for any two specified primes p_1, p_2>D^* in the same residue class of D^*(theorem 4.5).