Related papers: A MacDonald formula for zeta functions of varietie…

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over $\mathbb Z$. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products…

Let $\mathfrak{Var}_k^G$ denote the category of pairs $(X,\sigma)$, where $X$ is a variety over $k$ and $\sigma$ is a group action on $X$. We define the Grothendieck ring for varieties with group actions as the free abelian group of…

The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function,…

The motivic zeta function of a smooth and proper $\mathbb{C}((t))$-variety $X$ with trivial canonical bundle is a rational function with coefficients in an appropriate Grothendieck ring of complex varieties, which measures how $X$…

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements.…

We consider zeta functions with values in the Grothendieck ring of Chow motives. Investigating the lambda-structure of this ring, we deduce a functional equation for the zeta function of abelian varieties. Furthermore, we show that the…

In this short note we establish some properties of all those motivic measures which can be exponentiated. As a first application, we show that the rationality of Kapranov's zeta function is stable under products. As a second application, we…

We prove 2-out-of-3 property for rationality of motivic zeta function in distinguished triangles in Voevodsky's category DM. As an application, we show rationality of motivic zeta functions for all varieties whose motives are in the thick…

It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually…

We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. As an illustration, we explain how…

The zeta-function of a complex variety is a power series whose nth coefficient is the nth symmetric power of the variety, viewed as an element in the Grothendieck ring of complex varieties. We prove that the zeta-function of a surface is…

Let K be a field of characteristic 0 and A be a rigid tensor K-linear category. Let M be a finite-dimensional object of A in the sense of Kimura-O'Sullivan. We prove that the "motivic" zeta function of M with coefficients in K\_0(A) has a…

Let $C$ be a projective smooth connected curve over an algebraically closed field of characteristic zero, let $F$ be its field of functions, let $C_0$ be a dense open subset of $C$. Let $X$ be a projective flat morphism to $C$ whose generic…

We introduce the Hadamard topology on the Witt ring of rational functions, giving a simultaneous refinement of the weight and point-counting topologies. Zeta functions of algebraic varieties over finite fields are elements of the rational…

Let $K_0(\mathcal{V}_{K})$ be the Grothendieck ring of varieties over a field $K$ of characteristic zero, and let $\mathbb{L} = [\mathbb{A}^1_{K}]$ denote the Lefschetz class. We prove that if a $K$-variety has $\mathbb{L}$-rational…

In this paper, we study the arithmetic zeta function $$\mathscr{Z}_{\mathcal{X}}(s) = \prod_p \prod_{\substack{x \in \mathcal{X}_p \\ \text{closed}}} \Big( \frac{1}{1-|\kappa(x)|^{-s}} \Big)^{\mathfrak{m}_{p}(x)}$$ associated to a scheme…

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck--Witt group. We show that this enrichment is related to the topology of the…

Combining the idea of motivic zeta function, due to Kapranov, and Pellikaan's definition of a two- variable zeta function for curves over finite fields in the present note we introduce a motivic two- variable zeta function for curves over…

Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\mathbb{F}_{p}$ for primes $p \geq 5$. Recent works by Kaplan and Petrow generalized these results…

In the present paper, we prove an identity for the generating function of the quadruple zeta values. Taking homogeneous parts on both sides of the identity and substituting appropriate values for the variables, we obtain the sum formula for…