## 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

### MSC:

 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