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Partial Zeta Functions of Algebraic Varieties over Finite Fields

✍ Scribed by Daqing Wan

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 147 KB
Volume: 7
Category: Article
ISSN: 1071-5797
DOI: 10.1006/ffta.2000.0288

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✦ Synopsis

DEDICATED TO PROFESSOR CHAO KO ON THE OCCASION OF HIS 90TH BIRTHDAY

Motivated by arithmetic applications, we introduce the notion of a partial zeta function which generalizes the classical zeta function of an algebraic variety de"ned over a "nite "eld. We then explain two approaches to the general structural properties of the partial zeta function in the direction of the Weil-type conjectures. The "rst approach, using an inductive "bred variety point of view, shows that the partial zeta function is rational in an interesting case, generalizing Dwork's rationality theorem. The second approach, due to Faltings, shows that the partial zeta function is always nearly rational.

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