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 nonarchimedean local field with a finite residue field. To a two-dimensional finite complex XΓ arising as the quotient of the Bruhat-Tits building X associated to Sp4(F) by a discrete torsion-free co-compact subgroup Γ of PGSp4(F), associate the zeta function Z(XΓ,u) which counts geodesic tailless cycles contained in the 1-skeleton of XΓ. 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 XΓ 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.

Original languageEnglish (US)
Pages (from-to)886-923
Number of pages38
JournalInternational Mathematics Research Notices
Volume2013
Issue number4
DOIs
StatePublished - Mar 4 2013

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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