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In [19] it was explained how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, $mathbb{F}_1$) to a so-called loose graph (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and known realizations of objects over $mathbb{F}_1$ such as combinatorial $mathbb{F}_1$-projective and $mathbb{F}_1$-affine spaces exactly depict the loose graph which corresponds to the associated Deitmar scheme. In this paper, we first modify the construction of loc. cit., and show that Deitmar schemes which are defined by finite trees (with possible end points) are defined over $mathbb{F}_1$ in Kurokawas sense; we then derive a precise formula for the Kurokawa zeta function for such schemes (and so also for the counting polynomial of all associated $mathbb{F}_q$-schemes). As a corollary, we find a zeta function for all such trees which contains information such as the number of inner points and the spectrum of degrees, and which is thus very different than Iharas zeta function (which is trivial in this case). Using a process called surgery, we show that one can determine the zeta function of a general loose graph and its associated { Deitmar, Grothendieck }-schemes in a number of steps, eventually reducing the calculation essentially to trees. We study a number of classes of examples of loose graphs, and introduce the Grothendieck ring of $mathbb{F}_1$-schemes along the way in order to perform the calculations. Finally, we compare the new zeta function to Iharas zeta function for graphs in a number of examples, and include a computer program for performing more tedious calculations.
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We provide a coherent overview of a number of recent results obtained by the authors in the theory of schemes defined over the field with one element. Essentially, this theory encompasses the study of a functor which maps certain geometries including graphs to Deitmar schemes with additional structure, as such introducing a new zeta function for graphs. The functor is then used to determine automorphism groups of the Deitmar schemes and base extensions to fields.
In a recent paper [3], the authors introduced a map $mathcal{F}$ which associates a Deitmar scheme (which is defined over the field with one element, denoted by $mathbb{F}_1$) with any given graph $Gamma$. By base extension, a scheme $mathcal{X}_k = mathcal{F}(Gamma) otimes_{mathbb{F}_1} k$ over any field $k$ arises. In the present paper, we will show that all these mappings are functors, and we will use this fact to study automorphism groups of the schemes $mathcal{X}_k$. Several automorphism groups are considered: combinatorial, topological, and scheme-theoretic groups, and also groups induced by automorphisms of the ambient projective space. When $Gamma$ is a finite tree, we will give a precise description of the combinatorial and projective groups, amongst other results.
We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $mathbb{S} to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra.
Let $K$ be a discretely-valued field. Let $Xrightarrow Spec K$ be a surface with trivial canonical bundle. In this paper we construct a weak Neron model of the schemes $Hilb^n(X)$ over the ring of integers $Rsubseteq K$. We exploit this construction in order to compute the Motivic Zeta Function of $Hilb^n(X)$ in terms of $Z_X$. We determine the poles of $Z_{Hilb^n(X)}$ and study its monodromy property, showing that if the monodromy conjecture holds for $X$ then it holds for $Hilb^n(X)$ too. Sit $K$ corpus cum absoluto ualore discreto. Sit $ Xrightarrow Spec K$ leuigata superficies cum canonico fasce congruenti $mathcal{O}_X$. In hoc scripto defecta Neroniensia paradigmata $Hilb^n(X)$ schematum super annulo integrorum in $K$ corpo, $R subset K$, constituimus. Ex hoc, Functionem Zetam Motiuicam $Z_{Hilb^n(X)}$, dato $Z_X$, computamus. Suos polos statuimus et suam monodromicam proprietatem studemus, coniectura monodromica, quae super $X$ ualet, ualere super $Hilb^n(X)$ quoque demostrando.
We consider the Ihara zeta function $zeta(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $Xto X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $zeta(u,X)$ and that $zeta(u,X//G)$ divides $zeta(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of $L$-functions of edge-free quotients of the tetrahedron graph $K_4$.
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