Exponential Sums and Newton Polyhedra: Cohomology and Estimates

Abstract

The basic objects of this study are exponential sums on a variety defined over a finite field Fq (q = pa, p = char Fq). As we have remarked in some earlier articles [1], [2], we find it more natural to begin with exponential sums on the torus (Gm)n, extend via the usual toric decomposition of An to exponential sums on affine n-space, and finally proceed via a standard character argument [4] to exponential sums on an affine variety defined over Fq. While this is the natural order of the work, what we do, in fact, in the first part of this article is to combine the first two steps and deal with exponential sums on varieties V of the form V = (Gm)r X As (r + s = n). Let f E Fq[xl,..., xn, (xl ... xr-'] be an arbitrary regular function on V. Then f is a sum of monomials and as such has a well-defined Newton polyhedron A( ff) at infinity. This is the convex closure in Rn of the lattice points which occur as exponents of the terms of f together with the origin. We have indicated in our previous work [1], [2] the description of some of the invariants of the associated L-function in terms of properties of this polyhedron. For example, in [1] we showed how bounds for the degree and total degree of the L-function associated with a general exponential sum on V can be expressed in terms of the volumes of A(f) and the intersections of A(f) and the various coordinate spaces. In the present article, assuming f is nondegenerate and commode with respect to Af f), we show these estimates are sharp. Our methods are p-adic and are based on the work of Dwork [11], [12]. Our main accomplishment, from which our other results follow, is the extension of Dwork's cohomology theory from smooth, projective hypersurfaces in characteristic p to a general class of exponential sums. Given f regular on V, we construct a complex of p-adic Banach spaces on which Frobenius acts. The alternating product of characteristic polynomials of Frobenius describes the associated L-function. In fact, when f is nondegenerate and commode with respect to A( f), the complex is acyclic in dimensions other than 0 and the characteristic