# Exponentiation of Motivic Zeta Functions

 dc.contributor.advisor Ramachandran, Niranjan en_US dc.contributor.author Huang, Jonathan Andrew en_US dc.date.accessioned 2017-06-22T06:02:56Z dc.date.available 2017-06-22T06:02:56Z dc.date.issued 2017 en_US dc.identifier https://doi.org/10.13016/M2RZ9J dc.identifier.uri http://hdl.handle.net/1903/19393 dc.description.abstract We provide a formula for the generating series of the Weil zeta function $Z(X,t)$ of symmetric powers $\Sym^n X$ of varieties $X$ over finite fields. This realizes the zeta function $Z(X,t)$ as an exponentiable measure whose associated Kapranov motivic zeta function takes values in $W(R)$ the big Witt ring of $R=W(\Z)$. We apply our formula to compute $Z(\Sym^n X,t)$ in a number of explicit cases. Any motivic zeta function $\zeta_\mu$ of a measure $\mu$ factoring through the Grothendieck ring of Chow motives is itself exponentiable; in fact, this applies to $\zeta_\mu$ as a motivic measure itself. We prove a condition for which any motivic measure taking values in a $\lambda$-ring has an associated motivic zeta function $Z = \zeta_\mu$ that is itself an exponentiable measure $\mu_Z = Z$, and this process is shown to iterate indefinitely. This involves a study of the case of $\lambda$-ring-valued motivic measures. Finally, we provide an understanding of MacDonald's formula in this context. en_US dc.language.iso en en_US dc.title Exponentiation of Motivic Zeta Functions en_US dc.type Dissertation en_US dc.contributor.publisher Digital Repository at the University of Maryland en_US dc.contributor.publisher University of Maryland (College Park, Md.) en_US dc.contributor.department Mathematics en_US dc.subject.pqcontrolled Mathematics en_US dc.subject.pquncontrolled Big Witt ring en_US dc.subject.pquncontrolled Grothendieck ring of varieties en_US dc.subject.pquncontrolled Motivic measures en_US dc.subject.pquncontrolled Motivic zeta functions en_US
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