Asymptotic variation of L-functions of exponential sums

TL;DR

This paper investigates the asymptotic behavior of L-functions associated with exponential sums of Laurent polynomials supported on convex polytopes, establishing conditions under which their Newton polygons approach a universal lower bound as prime p grows.

Contribution

It proves that under certain geometric conditions on the polytope, the Newton polygons of L-functions of exponential sums become generic and approach a lower bound as p increases, extending to affine toric hypersurfaces.

Findings

Newton polygons match the generic polygon for large p.

As p approaches infinity, polygons approach a universal lower bound.

For toric hypersurfaces with unimodular triangulation, Newton polygons coincide with Hodge polygons.

Abstract

Let Delta be an integral convex polytope containing the origin of dimension n in the n-dim real space and it is simplicial at all origin-less facets. Let A(Delta) be the space of all Laurent polynomials f parametered by its coefficients supported on the interior of Delta with with prescribed vertices. In this paper we prove that, if the interior points of Delta and its vertices generate almost all integral points in the cone of Delta, then there is a Zariski dense open subset U in A(Delta) defined over the rationals such that for every f in U(bar{Q}) and for p large enough the Newton polygon of the L function of exponential sum of f is equal to the generic Newton polygon for A(Delta)(bar{F_p}), and as p approaches infinity they both approach an absolute lower bound depending only on Delta. This paper also proves the following result for affine toric hypersurfaces: Let T_Delta be the space of all regular affine toric hypersurfaces given by f=0 for all Laurent polynomials f with given Newton polytope Delta. If the set of all integral points in Delta has a unimodular triangulation, then for p large enough the generic Newton polygon of T_Delta over bar{F_p} coincides with the Hodge polygon defined in terms of Hodge numbers of the toric family T_Delta determined solely by Delta.