We study the standard nonuniform arithmetic quotient of the affine Bruhat--Tits building attached to $\operatorname{PGL}_d(\mathbb F_q(\!(t^{-1})\!))$, 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 $\alpha$. We determine its exact integrability threshold, proving that $\alpha$ belongs to $L^r$ 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 $\operatorname{Re}(s)<d$. It admits a meromorphic continuation as a rational function of $q^{s/d}$ 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.
Let $X_{G_h}(\mathbf{x};q)=\sum_{\mu\vdash n}c_\mu(q)e_\mu(\mathbf{x})$ be the chromatic quasisymmetric function of the natural unit interval graph attached to a Hessenberg function $h$. We establish an infinite class, valid in all orders, for which every nonzero polynomial $c_\mu(q)$ has a nonnegative, log-concave coefficient sequence with interval support. Namely, this holds whenever $h$ is abelian and its complement-Ferrers partition $\lambda$ satisfies $\min\{\lambda_1,\ell(\lambda)\}\leq 3$; equivalently, the diagram has at most three rows or at most three columns. Cubic interpolation reduces the rank-three case to a uniform theorem for a difference of two products of four $q$-integers, proved by positive decomposition, interval methods, and finite-window smoothing. The argument also yields explicit formulas for every supported elementary coefficient in complement-Ferrers rank at most three. We also include a connected 13-vertex natural unit interval graph for which one elementary coefficient is positive, palindromic, and unimodal but not log-concave, thereby recording the failure of the unrestricted conjecture. Thus low complement-Ferrers rank gives a substantial positive regime even though coefficientwise $e$-log-concavity fails in general.
Let $k=\mathbb F_q$. We classify, up to two-sided $k$-M\"obius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered from the cover; we prove a converse and an exact equivalence criterion. The same data determine branch arithmetic, geometric and arithmetic monodromy, the constant field, and behavior over every finite extension. In particular, over each $\mathbb F_{q^r}$, permutation is equivalent to exceptionality. We obtain explicit rank-two support formulas, necessary and sufficient occurrence criteria, and exact two-sided class counts in every tame signature, including the surviving cases in characteristics $2$ and $3$. In characteristic greater than $3$, tameness is automatic. A sharp $3$-adic obstruction arises only for a stronger cubic self-endomorphism realization problem, not for the fixed-field classification itself.
We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Ab\'ert, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.
An equivalent formulation of the Riemann hypothesis recently led to a partition expansion naturally indexed by diagonals of the Young lattice. Segovia isolated the hook families $(r,1^m)$ on these diagonals and computed their limiting contributions $\rho_r$, while observing that non-hook families provide a missing contribution. We introduce a bivariate finite generating function that packages all Young shapes on every fixed-excess diagonal at once. For each fixed $r\geq1$, we obtain a diagonal generating polynomial $D_r(n;z)$ and prove \[ A_r(n)\sim C_r\,n\log\log n, \] where $C_r$ is the $(r-1)$st coefficient of an explicit convergent infinite product. Moreover, \[ C_r=\sum_{\nu\vdash r-1} C_\nu , \] giving a canonical decomposition over the partitions of the excess $r-1$. The one-part contribution is Segovia's hook constant $\rho_r$, while the remaining terms give all non-hook corrections simultaneously. We then refine these constants by introducing coefficients $C_{r,d}$ that record simultaneously the Young-lattice excess $r-1$ and the number $d$ of non-unit rows. Row sums recover the fixed-excess constants $C_r$, while column sums recover the depth decomposition in an Abel-regularized multiple-zeta expansion of $e^\gamma$. More precisely, each partition $\nu\vdash r-1$ is identified with an Abel-regularized multiple-zeta block of depth $\ell(\nu)$. Thus the same array $(C_{r,d})$ organizes the decomposition simultaneously by Young-lattice excess and multiple-zeta depth. Our results concern the combinatorial and asymptotic structure of this decomposition, rather than the Riemann hypothesis itself.
We prove that, for every non-uniform arithmetic lattice in $\mathrm{SU}(2,1)$, its inverse images in the universal cover and in all connected finite covers are residually finite. The key new input is that every commensurability class of such lattices contains a congruence arithmetic lattice $\Gamma$ for which $$H^1_{\mathrm{cusp}}(\Gamma\backslash\mathbb B^2,\mathbb C)\ne 0.$$ In particular, the first inner cohomology of this ball quotient is non-zero. The proof uses Rogawski's endoscopic classification for $\mathrm{U}(3)$. A cohomological criterion proved previously by the author then gives the residual-finiteness result. Residual finiteness also yields multiplier systems of arbitrary denominator on suitable finite-index subgroups.
We study Fulton's universal Schubert polynomials $\mathfrak S_w(c)$ as regular functions on the upper unitriangular group $U_N$. The standard triangular cluster structure associates a Schubert $\sf g$-vector to every permutation. Their convex hull is unimodularly equivalent to $\Delta_1\times\cdots\times\Delta_{N-1}$, and their root-degree fibers are parabolic Bruhat intervals realized by the strata of staircase quiver Grassmannians. The geometric (i.e., generic, canonical, and Mirkovi\'c--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that $\mathfrak S_w(c)$ is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of $\mathfrak S_w(c)$ implies $\Lambda_Q$-rigidity of $Z_{\sf g_w}$. We also classify simultaneously the unit columns of the geometric-to-Schubert transitions, determine support components of the PBW-to-geometric and code-to-Schubert transitions, and exhibit a permutation $w\in S_{10}$ for which the three geometric basis elements are distinct.
Jia-Rui Fei· 0 citations
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