Exponential sums and Newton polyhedra: Cohomology and estimates
Exponential sums and Newton polyhedra: Cohomology and estimates
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DOI：
10.2307/1971424
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发表时间：
1989-09
影响因子：
4.9
通讯作者：
A. Adolphson;S. Sperber;B. Dwork
中科院分区：
文献类型：
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作者：
A. Adolphson;S. Sperber;B. Dwork
本文研究有限域Fq（q = pa，p = char Fq）上的簇的指数和.正如我们在以前的一些文章[1]，[2]中所指出的，我们发现开始于环面（Gm）n上的指数和，通过An的通常的环面分解推广到仿射n-空间上的指数和，最后通过标准特征引数[4]继续到定义在Fq上的仿射簇上的指数和，这是更自然的。虽然这是工作的自然顺序，但实际上，我们在本文的第一部分中所做的是将前两个步骤联合收割机结合起来，处理形式为V =（Gm）r X As（r + s = n）的簇V上的指数和。令fEFq [xl，...，xn，（xl.是V上的任意正则函数。则f是单项式的和，并且因此在无穷远处具有良好定义的牛顿多面体A（ff）。这是在Rn中的凸封闭的格点出现的指数的条款f连同原点。我们在以前的工作[1]、[2]中已经指出了用这个多面体的性质来描述相关L-函数的一些不变量。例如，在[1]中，我们展示了与V上的一般指数和相关的L-函数的次数和总次数的界限如何用A（f）的体积以及A（f）与各种坐标空间的交集来表示。在本文中，假设f是非退化的，并且关于Af f）是commode的，我们证明了这些估计是尖锐的。我们的方法是p-adic的，基于Dwork [11]，[12]的工作。我们的主要成就，从我们的其他结果如下，是Dwork的上同调理论的延伸，从光滑，射影超曲面的特征p的一般类的指数和。给定V上的f正则，我们构造了一个复形的p-adic Banach空间上的Frobenius行为。弗罗贝纽斯特征多项式的交错积描述了相关的L函数。事实上，当f关于A（f）是非退化的和commode时，复形在除0以外的维度上是非循环的，并且特征
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