2026/07/30 by Soonki Hong, Sanghoon Kwon
Mathematics · #math.NT #math.GR #msc:20E42 #msc:20G25 #msc:11M41 #msc:05A15
22 pages, comments welcome!
arxiv created 2026/07/30 · arxiv updated 2026/07/31
We study the standard nonuniform arithmetic quotient of the affine Bruhat--Tits building attached to PGLd(\mathbb Fq( (t-1) )), with Haar measure normalized so that a maximal compact subgroup has volume one. We first compute its vertex volume in closed product form. The proof is entirely building-theoretic: vertices are parametrized by a dominant sector, their stabilizers are counted exactly, and the resulting sum over block compositions is evaluated by a cut-set recursion. On the same quotient, we introduce a homothety-invariant normalized lattice-minima height α. We determine its exact integrability threshold, proving that α belongs to Lr precisely for 0<r<d, and establish a sharp cusp-tail estimate of order T-d. The associated positive-moment height zeta function, equivalently the Mellin transform of the cusp-height distribution, converges exactly in the half-plane Re(s)<d. It admits a meromorphic continuation as a rational function of qs/d and has a simple pole at s=d, with an explicit critical coefficient. We also compute the resulting rational functions explicitly for d=3,4,5. Thus the same dominant-sector coordinates simultaneously control volume, cusp decay, and the analytic structure of the height zeta function.