John Voight

Zeta functions of varieties over finite fields

Abstract: Let X be a variety defined by a set of polynomial equations in several
variables over a finite field F_q with q elements. In many applications,
one is interested in the number of solutions of these equations over finite
extensions F_{q^r} of F_q, with r >= 1. One can package these positive
integers into a generating series in a suitable way to obtain the zeta
function of X. The zeta function possesses a marvelous structure and is one
of the fundamental objects in algebraic geometry when working over a finite
field. In this talk, we will introduce the zeta function from scratch,
provide several examples, and discuss its properties and applications.

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