The Zeta Functions of Complexes from $\PGL(3)$: a Representation-theoretic Approach

Artin L-function on PGL3

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].

$\boldsymbol{\text{PGL}}_d$ 的标准算术商的顶点体积、格极小值尾和高度泽塔函数

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。

On finite arithmetic simplicial complexes

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.

Zeta and L-functions of finite quotients of apartments and buildings

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 …

Motivic complexes and special values of zeta functions

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…

有限CW-复形的KU-局部zeta函数

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。

SL_2 的元循环覆盖的局部 zeta 函数的函数方程与 gamma 因子

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。

Fractions de Bernoulli-Carlitz et opérateurs q-Zeta

We introduce a q-deformation of Dirichlet series : for each s, an operator acting on formal power series in q without constant term. We relate Bernoulli-Carlitz numbers to the q-Riemann Zeta operators for negative integers, evaluated on some polynomials. ----- On introduit une déformation des séries de Dirichlet d'une variable complexe s, sous la forme d'un opérateur pour chaque nombre complexe s, agissant sur les séries formelles en une variable q sans terme constant. On montre que les fractions de Bernoulli-Carlitz sont les images de certains polynômes en q par les opérateurs associés à la f…

The p-cohomology of algebraic varieties and special values of zeta functions

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.

Dedekind Zeta motives for totally real fields

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…

The Weil-Etale Topology for Number Rings

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.

图积分、费曼周期与单值多重zeta值

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。

关于复数上贝塞尔函数的汉克尔变换及高斯域上的显式谱公式

On the Hankel Transform of Bessel Functions on Complex Numbers and Explicit Spectral Formulae over the Gaussian Field

本文在复数域上证明贝塞尔函数的汉克尔-梅林与双重傅里叶-梅林变换积分公式，并应用于高斯数域上戴德金zeta函数四次矩及皮卡群中心L值二阶矩的显式谱公式，同时建立分布汉克尔变换理论。