Ming-Hsuan Kang, Wen-Ching Winnie Li
In this paper we obtain a closed form expression of the zeta function $Z(X_Γ, u)$ of a finite quotient $X_Γ= Γ\backslash PGL_3(F)/PGL_3(O_F)$ of the Bruhat-Tits building of $PGL_3$ over a nonarchimedean local field $F$. Analogous to a graph zeta function, $Z(X_Γ, u)$ is a rational function and it satisfies the Riemann hypothesis if and only if $X_Γ$ is a Ramanujan complex.
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We prove an approximation to an Ihara-formula for the zeta function of an arithmetic quotient of the Bruhat-Tits building of a p-adic PGL(3).
Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$
本文引入了chamber zeta函数，通过欧拉乘积扩展了Ihara-Bass框架，并证明了其与chamber转移算子特征多项式的关系，从而获得闭gallery类的计数结果。
Edge zeta function and closed cycles in the standard non-uniform complex from $\operatorname{PGL}_3$
本文定义加权复形的边 zeta 函数，并利用特定方向截断技术，给出标准非一致 $\operatorname{PGL}_3$ 复形边 zeta 函数的有理函数公式，进而得到建筑中测地线闭圈数量的精确公式。
The zeta function attached to a finite complex $X_Γ$ arising from the Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of $X_Γ$. In this paper we reprove the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed cha…
Zeta functions of $\mathrm{PGL}_n$ over non-Archimedean local fields
研究非阿基米德局部域上\(\mathrm{PGL}_n\)的泽塔函数，通过引入几何与组合\(k -\)测地线并证明其重合，在商空间上定义相关函数，得到泽塔函数交错积与非分歧\(L -\)函数的等同关系，并扩展到\(\mathrm{PGL}_n(D)\)。
We study Artin $L$-functions on a finite $2$-dimensional complex $X_Γ$ arising from PGL$_3$ attached to finite-dimensional representations $ρ$ of its fundamental group. Some key properties, such as rationality, functional equation, and invariance under induction, of these functions are proved. Moreover, using a cohomological argument, we establish a connection between the Artin $L$- functions and the $L$-function of $ρ$, extending the identity on zeta functions of $X_Γ$ obtained in [KL,KLW].
Vertex volumes, lattice-minima tails, and height zeta functions for the standard arithmetic quotient of $\operatorname{PGL}_d$
该研究针对 $\text{PGL}_d$ 的标准算术商，基于建筑理论计算顶点体积，引入同伦不变归一性格极小值高度，确定其可积阈值与尖点尾估计，推导高度泽塔函数的收敛域与解析延拓性质，明确计算了 $d=3,4,5$ 时的有理函数。
We compute the Euler-Poincaré characteristic of quotients of the Bruhat-Tits building of PGL(n) under the action of arithmetic groups arising from central division algebras over rational function fields of positive characteristic. We use this result to determine the structure of the quotient simplicial complex in certain cases.
Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's Z[1/p]-category of etale motivic complexes with a p-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. As the conjecture is stated in terms of mot…
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 $\Sp_4(F)$ by a discrete torsion-free cocompact subgroup $Γ$ of $\PGSp_4(F)$, 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 …
KU-local zeta-functions of finite CW-complexes
本文通过用复K-理论替换平展上同调、用Adams运算替换Frobenius算子，定义了有限CW-复形的$KU$-局部zeta函数，证明其解析延拓与函数方程，并揭示其特殊值可恢复$KU$-局部稳定同伦群。
Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL_2
本文为 SL_2 的元循环二重覆盖上的超尖点表示引入局部 zeta 函数，证明其函数方程并用 Bessel 函数的 Mellin 变换给出 gamma 因子，进而通过局部 theta 对应将其与 PGL_2 上二元对称矩阵预齐性向量空间的球面函数 zeta 积分等同起来。
In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G=PGL3 and PGSp4 over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara's theorem for finite quotients of the Bruhat-Tits trees. This identity is shown to agree with the q=1 version of the analogous identities for finite quotients of the building of G established in (KL1, KLW, FLW), verifying the philosophy of the field with one …
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
Let $k$ be a totally real number field. For every odd $n\geq 3$, we construct a Dedekind zeta motive in the category $\MT(k)$ of mixed Tate motives over $k$. By directly calculating its Hodge realisation, we prove that its period is a rational multiple of $π^{n[k:\Q]}ζ^*_k(1-n)$, where $ζ^*_k(1-n)$ denotes the special value of the Dedekind zeta function of $k$. We deduce that the group $\Ext^1_{\MT(k)} (\Q(0),\Q(n))$ is generated by the cohomology of a quadric relative to hyperplanes. This proves a surjectivity result for certain motivic complexes for $k$ that have been conjectured to calculat…
We would like to construct a new Grothendieck topology for arithmetic schemes, whose cohomology groups associated with motivic complexes of sheaves are finitely generated and whose Euler characteristics are related to special values of zeta-functions. In this paper we construct this topology for rings of algebraic integers and show that the cohomology of the constant sheaf Z is related to the behavior of zeta-functions at s = 0.
Graph integrals, Feynman periods, and single-valued multiple zeta values
研究图积分是什么数的问题，通过证明原始规范积分与变形量子化中的复位置空间积分一致，得出图的规范积分值为单值多重zeta值，且每个单值多重zeta值是特定图费曼周期的有理线性组合，还表明在交换图复形中相关上闭链一致。
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