The absolute zeta function for a scheme $X$ of finite type over $\mathbb{Z}$ satisfying a certain condition is defined as the limit as $p\to 1$ of the congruent zeta function for $X\otimes\mathbb{F}_p$. In 2016, after calculating absolute zeta functions for a few specific schemes, Kurokawa suggested that an absolute zeta function for a general scheme of finite type over $\mathbb{Z}$ should have an infinite product structure which he called the absolute Euler product. In this article, formulating his suggestion using a torsion free Noetherian $\mathbb{F}_1$-scheme defined by Connes and Consani, we give a proof of his suggestion. Moreover, we show that each factor of the absolute Euler product is derived from the counting function of the $\mathbb{F}_1$-scheme.
We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus $\bR^{\n}/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.
Motivated by recent work of Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, Connes-Consani, and the author, we define some multivariable deformed zeta functions of Hurwitz-Igusa type for a Noetherian $\F_1$-scheme $X$ in the sense of Connes-Consani. Our zeta functions generalize both the zeta functions studied by Deitmar-Koyama-Kurokawa, Kurokawa-Ochiai, and the log derivative of the modified Soulé type zeta function Connes-Consani. We give an explicit presentation for these zeta functions using the Hurwitz zeta functions, and so, we can derive its meromorphicity. When restricted to the log derivati…
Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the singularities of analytic torsions at Brill-Noether loci, and the asymptotic behaviors of analytic torsions with respect to the degree. These new yet intrinsic zetas, both abelian and non-abelian, are expected to play key roles to understand global analysis and geometry of Riemann surfaces, such as the Tamagawa number conjecture for Riemann surfaces, searched …
In recent years Lichtenbaum has conjectured a description for the special values of Hasse--Weil zeta functions in terms of ``Weil-étale cohomology''. In earlier papers we studied a class of foliated dynamical systems which had some similarities with arithmetic schemes. Assuming that certain zeta regularized determinants exist we now prove an analogue of Lichtenbaum's conjectures for a particularly simple class of such dynamical systems: Using results of Álvarez López and Kordyukov we express the leading coefficient $ζ^*_R (0)$ of the Ruelle zeta function $ζ_R (s)$ at $s = 0$ in terms of analyt…
For every $P \in \mathbb{Z}[x_1^{\pm 1}, \ldots, x_d^{\pm 1}] \setminus \{0\}$, and every $\varepsilon > 0$, we prove that there are a computable function $M = M(d,\varepsilon,°{P},h(P)) < \infty$ and a finite union $Z = Z(d,\varepsilon,°{P},h(P))$ of proper torsion cosets $\boldsymbolμ T \subsetneq \mathbb{G}_m^d$ such that, for every $N \in \mathbb{N}$, $Z$ contains all but at most $M$ of the torsion points $\boldsymbolζ \in μ_N^d$ satisfying $|P(\boldsymbolζ)| < e^{-\varepsilon ϕ(N)}$. This extends a well known structural theorem from torsion points lying exactly on a variety to torsion poi…
We revisit the theory of Ihara $L$-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study if the Artin formalism is satisfied by these $L$-functions. As an application, we give a proof of the analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and $L$-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the …
We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sl_n(Q_p). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions.
We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions - the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa - match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations …
Thakur (2010) showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $ζ_A(r)$ and $ζ_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $ζ_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct a $t$-module $E'$. To determine whether an $(r+s)$-tuple $\mathfrak{C}$ in $\mathbb{F}_q(θ)^{r+s}$ gives a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $\mathbf{v}_\mathfrak{C}\in E'(\mathbb{F}_q[θ])$ constructed with respect to the …
For a Dedekind domain $\mathcal{O}$ and a rank two co-torsion module $M\subseteq \mathcal{O}^2$ with invariant factor ideals $\mathcal{L}\supseteq \mathcal{K}$ in $\mathcal{O}$, that is, $\frac{\mathcal{O}^2}{M}\cong \frac{\mathcal{O}}{\mathcal{L}}\oplus \frac{\mathcal{O}}{\mathcal{K}}$ we associate a new projective space invariant element in $\mathbb{PF}^1_{\mathcal{I}}$ where $\mathcal{I}$ is given by the ideal factorization $\mathcal{K} = \mathcal{L}\mathcal{I}$ in $\mathcal{O}$. This invariant element along with the invariant factor ideals determine the module $M$ completely as a subset of…
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
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 …
We prove a new effective Chebotarev density theorem for Galois extensions $L/\mathbb{Q}$ that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of $L$); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of $L$, without assuming GRH. We g…
Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from $\mathrm{gr}^{n}TC^-(-)$ to $\mathrm{gr}^{n}TP(-)$ by $LΩ^{<n}_{-/\mathbb{S}}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension $d$ proper over $\…
