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arXiv · 1707.03479

A MacDonald formula for zeta functions of varieties over finite fields

Jonathan Huang
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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.

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Jonathan Huang. 2017-07-11. A MacDonald formula for zeta functions of varieties over finite fields. https://arxiv.org/abs/1707.03479

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