A MacDonald formula for zeta functions of varieties over finite fields

TL;DR

This paper introduces a MacDonald-type formula for the zeta functions of symmetric powers of varieties over finite fields, connecting motivic measures with Witt rings and providing explicit computations.

Contribution

It presents a new MacDonald formula for zeta functions of symmetric powers, linking motivic measures with Witt rings and demonstrating exponentiability.

Findings

Derived explicit formulas for zeta functions of symmetric powers

Showed all lambda-ring motivic measures have exponentiable zeta functions

Computed zeta functions in several explicit cases

Abstract

We provide a formula for the generating series of the zeta function $Z(X,t)$ of symmetric powers $Sym^n X$ of varieties over finite fields. This realizes $Z(X,t)$ as an exponentiable motivic measure whose associated Kapranov motivic zeta function takes values in $W(R)$ the big Witt ring of $R=W(\mathbb{Z})$. We apply our formula to compute $Z(Sym^n X,t)$ in a number of explicit cases. Moreover, we show that all $\lambda$-ring motivic measures have zeta functions which are exponentiable. In this setting, the formula for $Z(X,t)$ takes the form of a MacDonald formula for the zeta function.