On the estimation of certain exponential sums
On the estimation of certain exponential sums
复制标题
关于某些指数和的估计
DOI：
10.4064/aa-69-4-329-358
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发表时间：
1995
期刊：
影响因子：
0.7
通讯作者：
S. Sperber
中科院分区：
文献类型：
--
作者：
E. Bombieri;S. Sperber
I.设k = Fq是特征为p的有限域，kn = Fqn是k的唯一n次扩张，V是定义在k上的拟投射簇，f ∈ k（V）是V上的有理函数，也定义在k上.通常，V（kn）将表示定义在kn上的V的点的集合。我们还用K表示k的代数闭包，对于k上的方案X，我们用XK表示方案XK = X <$K，即从k到K的基变换后的X。下面我们将假设f在V上处处定义，并且在V上没有极点，所以f：V → Ak是一个态射。设Fp上的一个非平凡的加法特征标为Fp，kn上的一个相应的特征标为<$n =<$0 <$Trkn/ p; kn上的每一个加法特征标都可以表示为<$n（lx），其中l ∈ kn。此外，我们将k上的诱导字符写为而不是1。根据定义，与V、f、kn和字符n相关的指数和为
I. Let k = Fq be a finite field of characteristic p, let kn = Fqn be the unique extension of k of degree n, let V be a quasi-projective variety defined over k and let f ∈ k(V ) be a rational function on V , also defined over k. As usual, V (kn) will denote the set of points of V defined over kn. We also denote by K an algebraic closure of k and for a scheme X over k we denote by XK the scheme XK = X ⊗K, i.e. X after base change from k to K. In what follows we shall assume that f is defined everywhere on V and has no poles on V , so that f : V → Ak is a morphism. Let ψ0 be a non-trivial additive character on Fp and let ψn = ψ0 ◦ Trkn/ p be the corresponding character on kn; every additive character on kn can be written as ψn(lx) for some l ∈ kn. Also, we shall write ψ instead of ψ1 for the character induced on k. The exponential sum associated with V, f, kn and the character ψn is by definition