Related papers: Zeta functions in triangulated categories

We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k, ZZ[1/e]) (where k is a perfect field of exponential characteristic e). We also establish a motivic version…

In [P] R. Pellikaan introduced a two variable zeta-function for a curve over a finite field and proved that it is a rational function. Here we show that its denominator is absolutely irreducible. This is motivated by work of J. Lagarias and…

A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction…

With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that…

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as…

Let $p$ and $l$ be rational primes such that $l$ is odd and the order of $p$ modulo $l$ is even. For such primes $p$ and $l$, and for $e=l, 2l$, we consider the non-singular projective curves $aY^e = bX^e + cZ^e$ ($abc \neq 0$) defined over…

We propose a construction of a tensor exact category F_X^m of Artin-Tate motivic sheaves with finite coefficients Z/m over an algebraic variety X (over a field K of characteristic prime to m) in terms of etale sheaves of Z/m-modules over X.…

We prove that every profinite group in a certain class with a rational probabilistic zeta function has only finitely many maximal subgroups.

Let $k$ be a field of characteristic zero with a fixed embedding $\sigma:k\hookrightarrow \mathbb{C}$ into the field of complex numbers. Given a $k$-variety $X$, we use the triangulated category of \'etale motives with rational coefficients…

We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to…

Let $X \subset \P^5_{\C}$ be a smooth cubic fourfold.The Kuznetsov component $\sA_X$ is contained in the derived category $D^b(X)$ and the transcendental motive $t(X)$ is contained in the category of Chow motives $\sM_{rat}(\C))$. If $X$…

We analyze the spectrum of the tensor-triangulated category of Artin-Tate motives over the base field R of real numbers, with integral coefficients. Away from 2, we obtain the same spectrum as for complex Tate motives, previously studied by…

We construct and study a candidate for the standard motivic t-structure on the triangulated category of relative cohomological 1-motives with rational coefficients over a noetherian finite dimensional scheme S. This t-structure is defined…

Let $k$ be a field, let $R$ be a commutative ring, and assume the exponential characteristic of $k$ is invertible in $R$. In this note, we prove that isomorphisms in Voevodsky's triangulated category of motives $\mathcal{DM}(k;R)$ are…

Though Joyal's species are known to categorify generating functions in enumerative combinatorics, they also categorify zeta functions in algebraic geometry. The reason is that any scheme $X$ of finite type over the integers gives a "zeta…

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…

Given a finite abelian $p$-group $F$, we prove an efficient recursive formula for $\sigma_a(F)=\sum_{\substack{H\leq F}}|H|^a$ where $H$ ranges over the subgroups of $F$. We infer from this formula that the $p$-component of the…

We introduce a zeta function counting imaginary quadratic number fields by their class numbers. It is proved that such a function is rational depending only on the eight roots of unity of degrees $1$ and $2$. As a corollary, one gets a…

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds…

In this paper, we give an overview of the various general methods in computing the zeta function of an algebraic variety defined over a finite field, with an emphasis on computing the reduction modulo $p^m$ of the zeta function of a…