TY - JOUR

T1 - The zeta functions of complexes from Sp(4)

AU - Fang, Yang

AU - Li, Wen Ching Winnie

AU - Wang, Chian Jen

N1 - Funding Information:
This research was partially supported by the NSF grant DMS-0801096 (to Y.F.), supported by the NSF grants DMS-0801096 and DMS-1101368 (to W.-C.W.L.), and also was partially supported by the NSC grant 99-2115-M-008-001 (to C.-J.W.).

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874387868&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874387868&partnerID=8YFLogxK

U2 - 10.1093/imrn/rns007

DO - 10.1093/imrn/rns007

M3 - Article

AN - SCOPUS:84874387868

SN - 1073-7928

VL - 2013

SP - 886

EP - 923

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 4

ER -