Newton polygons for general hyper-Kloosterman sums. (English) Zbl 0582.14005

Cohomologie p-adique, Astérisque 119/120, 267-330 (1984).
[For the entire collection see Zbl 0542.00006.]
Let X be the algebraic group defined over \({\mathbb{F}}_ q\) by the equation \(x_ 1^{b_ 1} x_ 2^{b_ 2}...x_ n^{b_ n} x_{n+1}=1\) and let \(g(x)=(\alpha_ 1x_ 1+...+\alpha_{n+1}x_{n+1}\in {\mathbb{Z}}[x_ 1,...,x_{n+1}].\) Let \(\psi\) be a non-trivial additive character of \({\mathbb{F}}_ q\), char \({\mathbb{F}}_ q=p\). After Katz and Deligne, it is known that, under suitable hypothesis, the L-function attached to generalized Kloosterman sums \(S_ m(\bar g,X,\psi)\) has the property that \(L(Kloos_{n+1})^{(-1)^{n+1}}\) is a polynomial of degree \(n+1\). The author studies in this paper the variation with p of the Newton polygon of \(L(Kloos_{n+1})^{(-1)^{n+1}}\).
Reviewer: P.Bayer


14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11L10 Jacobsthal and Brewer sums; other complete character sums


Zbl 0542.00006