L-functions associated with families of toric exponential sums

TL;DR

This paper develops a p-adic cohomology framework for families of nondegenerate toric exponential sums, enabling precise estimates of associated L-functions' degrees, zeros, poles, and p-divisibility properties.

Contribution

It introduces a relative polytope and coefficient ring to compute cohomology and Frobenius maps, providing new bounds for L-functions of toric exponential sums.

Findings

Established degree and p-divisibility estimates for L-functions

Extended methods to affine and Archimedean weight families

Computed relative cohomology and Frobenius action for families

Abstract

We consider arbitrary algebraic families of lower order deformations of nondegenerate toric exponential sums over a finite field. We construct a relative polytope with the aid of which we define a ring of coefficients consisting of p-adic analytic functions with polyhedral growth prescribed by the relative polytope. Using this we compute relative cohomology for such families and calculate sharp estimates for the relative Frobenius map. In applications one is interested in L-functions associated with linear algebra operations (symmetric powers, tensor powers, exterior powers and combinations thereof) applied to the relative Frobenius. Using methods pioneered by Ax, Katz and Bombieri we prove estimates for the degree and total degree of the associated L-function and p-divisibility of the reciprocal zeros and poles. Similar estimates are then established for affine families and pure Archimedean weight families (in the simplicial case).