Related papers: The dlt motivic zeta function is not well-defined

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…

We study Brown's definition of the probabilistic zeta function of a finite lattice as a generalization of that of a finite group. We propose a natural alternative or extension that may be better suited for non-atomistic lattices. The…

Simple unsmoothed formulas to compute the Riemann zeta function, and Dirichlet $L$-functions to a power-full modulus, are derived by elementary means (Taylor expansions and the geometric series). The formulas enable square-root of the…

Motivic Serre invariants defined by Loeser and Sebag are elements of the Grothendieck ring of varities modulo $\mathbb{L}-1$. In this paper, we show that we can lift these invariants to modulo the square of $\mathbb{L}-1$ after tensoring…

We define a dynamical zeta function for nondegenerate Liouville domains, in terms of Reeb dynamics on the boundary. We use filtered equivariant symplectic homology to (i) extend the definition of the zeta function to a more general class of…

This article is the continuation of [LS12]. We use categories of matrix factorizations to define a morphism of rings (= a Landau-Ginzburg motivic measure) from the (motivic) Grothendieck ring of varieties over $\mathbb{A}^1$ to the…

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…

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's…

A dynamical zeta function $\zeta$ and a transfer operator $\scr L$ are associated with a piecewise monotone map $f$ of the interval $[0,1]$ and a weight function $g$. The analytic properties of $\zeta$ and the spectral properties of $\scr…

A functional equation for the motivic integral corresponding to the Milnor number of an arc is derived using the Denef-Loeser formula for the change of variables. Its solution is a function of five auxiliary parameters, it is unique up to…

By restricting the variables running over various (possibly different) subfields, we introduce the notion of a partial zeta function. We prove that the partial zeta function is rational in an interesting case, generalizing Dwork's well…

We use the connective formal group law to define a one-parameter ($\beta$-)deformation of the motivic Segre classes of Schubert cells in the $d$-step flag variety. This $\beta$-deformation specializes to the motivic Segre classes of…

Following and generalizing a construction by Kontsevich, we associate a zeta function to any matrix with entries in a ring of noncommutative Laurent polynomials with integer coefficients. We show that such a zeta function is an algebraic…

The main objective of this paper is to prove the monodromy conjecture for the local Igusa zeta function of a quasi-ordinary polynomial of arbitrary dimension defined over a number field. In order to do it, we compute the local Denef-Loeser…

In this work, we begin to uncover the architecture of the general family of zeta functions and multiple zeta values as they appear in the theory of integrable systems and conformal field theory. One of the key steps in this process is to…

This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The…

Assuming the positive solution to the Cut-and-paste problem we prove that the motivic zeta function remains irrational after inverting L.

We introduce the etale framework to study Igusa zeta functions in several variables, generalizing the machinery of vanishing cycles in the univariate case. We define the etale Alexander modules, associated to a morphism of varieties F from…

In this paper we prove motivic versions of the Langlands-Shelstad Fundamental Lemma and Ng\^o's Geometric Stabilization. To achieve this, we follow the strategy from the recent proof by Groechenig, Wyss and Ziegler which avoided the use of…

A new, seemingly useful presentation of zeta functions on complex tori is derived by using contour integration. It is shown to agree with the one obtained by using the Chowla-Selberg series formula, for which an alternative proof is thereby…