The Zeta Functions of Complexes from \Sp(4)

2011/09/18 by Yang Fang, Wen-Ching Winnie Li, Fang, Yang +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1109.3854

openalex publication_date 2011/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \Sp₄(F) by a discrete torsion-free cocompact subgroup Γ of \PGSp₄(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.

Citations

• Expanders and the Affine Building of \rm Sp_n

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• A fixed point theorem for the action of SL_n over local fields on symmetric spaces of infinite dimension and finite rank

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