L-functions associated with families of tone exponential sums

JOURNAL OF NUMBER THEORY

卷:144

Article

关键词: p-Adic cohomology;    L-functions;    Exponential sums;    Dwork theory;   

DOI  :  10.1016/j.jnt.2014.03.016

来源: Elsevier

【 摘 要 】

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 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). (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free