A classification of Newton polygons of L-functions on polynomials

TL;DR

This paper classifies the possible Newton polygons of L-functions associated with polynomials over finite fields, showing finiteness and dependence on prime residue classes for large primes.

Contribution

It improves existing results by proving finiteness of Newton polygon forms for degree d polynomials independent of p, and simplifies their determination for large primes.

Findings

Finitely many Newton polygon forms for large p

p-adic order of roots has a specific rational form

Determination of Newton polygons reduces to two primes in the same residue class

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).