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arXiv · 1109.3854

The Zeta Functions of Complexes from $\Sp(4)$

Yang Fang·Wen-Ching Winnie Li·Chian-Jen Wang
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Abstract

Let $F$ be a non-archimedean local field with a finite residue field. To a 2-dimensional finite complex $X_Γ$ arising as the quotient of the Bruhat-Tits building $X$ associated to $\Sp_4(F)$ by a discrete torsion-free cocompact subgroup $Γ$ of $\PGSp_4(F)$, associate the zeta function $Z(X_Γ, u)$ which counts geodesic tailless cycles contained in the 1-skeleton of $X_Γ$. Using a representation-theoretic approach, we obtain two closed form expressions for $Z(X_Γ, u)$ as a rational function in $u$. Equivalent statements for $X_Γ$ being a Ramanujan complex are given in terms of vertex, edge, and chamber adjacency operators, respectively. The zeta functions of such Ramanujan complexes are distinguished by satisfying the Riemann Hypothesis.

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Yang Fang, Wen-Ching Winnie Li, Chian-Jen Wang. 2012-03-04. The Zeta Functions of Complexes from $\Sp(4)$. https://arxiv.org/abs/1109.3854

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