This article is cited in 26 papers

Finiteness of the extension of $\mathbb Q$ generated by Frobenius traces, in finite characteristic

Pierre Deligne

Institute for Advanced Study, School of Mathematics, 1 Einstein Drive, Princeton, NJ 08540 USA

Abstract: Let $\mathscr{F}_0$ be a $\bar{\mathbb{Q}}_l$-sheaf on a scheme $Z_0$ of finite type over $\mathbb{F}_q$. We show the existence of a finite type extension $E\subset\bar{\mathbb{Q}}_l$ of $\mathbb{Q}$ such that all local factors of the $L$-function of $\mathscr{F}_0$ have coefficients in $E$. When $Z_0$ is normal and connected, and $\mathscr{F}_0$ is an irreducible $l$-adic local system whose determinant is of finite order, $E$ can be taken to be a finite extension of $\mathbb{Q}$.

Key words and phrases: $l$-adic sheaves, Frobenius traces.

MSC: 14F20, 14G15

Received: June 21, 2011

Language: French

DOI: 10.17323/1609-4514-2012-12-3-497-514

Bibliographic databases:

• 
• 
•