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 known rationality theorem. In general, the partial zeta function is probably not rational. But a theorem of Faltings says that the partial zeta function is always nearly rational.
Hasse-Weil zeta functions of SL_2-character varieties of arithmetic two bridge link groups are determined. Special values of the zeta functions at s=0,1,2 are also investigated.
We present new algorithms for computing zeta functions of algebraic varieties over finite fields. In particular, let X be an arithmetic scheme (scheme of finite type over Z), and for a prime p let zeta_{X_p}(s) be the local factor of its zeta function. We present an algorithm that computes zeta_{X_p}(s) for a single prime p in time p^(1/2+o(1)), and another algorithm that computes zeta_{X_p}(s) for all primes p < N in time N (log N)^(3+o(1)). These generalise previous results of the author from hyperelliptic curves to completely arbitrary varieties.
Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the symmetric sum formula, the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations.
It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.
We consider the values at proper fractions of the arithmetic gamma function and the values at positive integers of the zeta function for F_q[theta] and provide complete algebraic independence results for them.
We apply the theory of height zeta functions to study the asymptotic distribution of rational points of bounded height on projective equivariant compactifications of semi-direct products.
The p-cohomology of an algebraic variety in characteristic p lies naturally in the category $D_{c}^{b}(R)$ of coherent complexes of graded modules over the Raynaud ring (Ekedahl-Illusie-Raynaud). We study homological algebra in this category. When the base field is finite, our results provide relations between the the absolute cohomology groups of algebraic varieties, log varieties, algebraic stacks, etc. and the special values of their zeta functions. These results provide compelling evidence that $D_{c}^{b}(R)$ is the correct target for p-cohomology in characteristic p.
In 2007 Chang and Yu determined all the algebraic relations among Goss's zeta values for the rational function field - these are also known as the Carlitz zeta values. Goss raised the problem about algebraic relations among Goss's zeta values for a general base ring A, but very little is known. In this paper we develop a general method and determine all algebraic relations among Goss's zeta values for the base ring A attached to an elliptic curve over a finite field. To our knowledge, these are the first non-trivial solutions of Goss's problem for a base ring whose class number is strictly gre…
We present a new method for computing the zeta function of an algebraic curve over a finite field. The algorithm relies on a trace formula of Harvey to count points on a plane model of the curve. The zeta function of the curve is then obtained by making corrections at singular points. We report on an implementation and provide some examples in MAGMA which demonstrate an improvement over Tuitman's algorithm.
We study the value-distribution of the Hurwitz zeta-function with algebraic irrational parameter $ζ(s;α)=\sum_{n\geq_0}(n+α)^{-s}$. In particular, we prove effective denseness results of the Hurwitz zeta-function and its derivatives in suitable strips containing the right boundary of the critical strip $1+i\mathbb{R}$. This may be considered as a first "weak" manifestation of universality for those zeta-functions.
We present a novel axiomatic framework for establishing horizontal norm relations in Euler systems that are built from pushforwards of classes in the motivic cohomology of Shimura varieties. This framework is uniformly applicable to the Euler systems of both algebraic cycles and Eisenstein classes. It also applies to non-spherical pairs of groups that fail to satisfy a local multiplicity one hypothesis, and thus lie beyond the reach of existing methods. A key application of this work is the construction of an Euler system for the spinor Galois representations arising in the cohomology of Siege…
Based on the notion of Stark units we present a new approach that obtains refinements of log-algebraic identities for Anderson t-modules. As a consequence, we establish a generalization of Chang's theorem on logarithmic interpretations for special characteristic p multiple zeta values (MZV's) and recover many earlier results in this direction. Further, we devise a direct and conceptual way to get logarithmic interpretations for both MZV's and v-adic MZV's. This generalizes completely the work of Anderson and Thakur for Carlitz zeta values.
We study special values of zeta functions of singular varieties over finite fields. We give a new formula of special values by constructing a morphism of homology theories, which we call regulator, from higher Chow group to weight homology. Our regulator is defined by using the notion of weight complex for varieties over a perfect field, which was introduced by Gillet and Soule. The main idea of the proof of our formula of special values is to use weight spectral sequence of homology theories, whose E1 terms are homology groups for smooth projective schemes. Also, to calculate special values, …
For a function field $k$ over a finite field with $\mathbb{F}_q$ as the field of constant, and a finite abelian group $G$ whose exponent is divisible by $q-1$, we study the distribution of zeta zeroes for a random $G$-extension of $k$, ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution.
We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.
Let $X_{0}=\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }\mathbb{F}_{q}$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p>0$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline pro-unipotent fundamental groupoid of $X_{0}$. In part I, we have computed explicitly the Frobenius, and in particular cyclotomic $p$-adic multiple zeta values. In part II, we use part I to understand the algebraic relations of cyclotomic $p$-adic multiple zeta values via explicit formulas ; this is in particular a study of the harmoni…
Let $\mathbb{F}_q$ be a finite field with $q$ elements, $ψ$ a non-zero element of $\mathbb{F}_q$, and $n$ an integer $\geq 3$ prime to $q$. The aim of this article is to show that the zeta function of the projective variety over $\mathbb{F}_q$ defined by $X_ψ\colon x_1^n+...+x_n^n - n ψx_1... x_n=0$ has, when $n$ is prime and $X_ψ$ is non singular (i.e. when $ψ^n \neq 1$), an explicit decomposition in factors coming from affine varieties of odd dimension $\leq n-4$ which are of hypergeometric type. The method we use consists in counting separately the number of points of $X_ψ$ and of some vari…
The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.
We study motivic zeta functions for $\mathds{Q}$-divisors in a $\mathds{Q}$-Gorenstein variety. By using a toric partial resolution of singularities we reduce this study to the local case of two normal crossing divisors where the ambient space is an abelian quotient singularity. For the latter we provide a closed formula which is worked out directly on the quotient singular variety. As a first application we provide a family of surface singularities where the use of weighted blow-ups reduces the set of candidate poles drastically. We also present an example of a quotient singularity under the …
In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence resu…
A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affine self-map of $\mathbb{P}^1(k)$ for $k$ an algebraically closed field of positive characteristic.
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 $π_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theor…
The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the zeta function $L(B/K;s)$ is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups $Γ_1(N)$. The characterization involves the structure of End(B), isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illu…
In this paper we propose different notions of F_zeta-geometry, for zeta a root of unity, generalizing notions of F_1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate F_zeta-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of F_zeta-structures in examples arising from general linear groups, matrix equations over finite fields, and some qu…
In this survey article, we discuss the algebraic structure of q-analogues of multiple zeta values, which are closely related to derivatives of Eisenstein series. Moreover, we introduce the formal double Eisenstein space, which generalizes the formal double zeta space of Gangl, Kaneko, and Zagier. Using the algebraic structure of q-analogues of multiple zeta values, we will present a realization of this space. As an application, we will obtain purely combinatorial proofs of identities among (quasi-)modular forms.
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
In this paper we formulate some conjectures about algebraic flows on Shimura varieties. In the first part of the paper we prove the `logarithmic Ax-Lindemann theorem'. We then prove a result concerning the topological closure of the images of totally geodesic subvarieties of symmetric spaces uniformising Shimura varieties. This is a special case of our conjectures.
Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, …
It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of L. Guo and B. Xie, some evaluation formulas with even arguments and the restricted sum formulas of M. E. Hoffman and their generalizations.
This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter of the survey explains recent results obtained in collaboration with Yuri Tschinkel concerning asymptotics of volumes of height balls in analytic geometry over local fields, or in adelic spaces.
Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety over a finite field in terms of other invariants of the variety. In this article, we present the ultimate such conjecture, and provide evidence for it. In particular, we enhance Voevodsky's Z[1/p]-category of etale motivic complexes with a p-integral structure, and show that, for this category, our conjecture follows from the Tate and Beilinson conjectures. As the conjecture is stated in terms of mot…
We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman- Vladuts theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P.Serre, using results of R.M.Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for…
In 2005, Kayal suggested that Schoof's algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pila's algorithm for abelian varieties.
We show that the motivic zeta functions of smooth, geometrically connected curves with no rational points are rational functions. This was previously known only for curves whose smooth projective models have a rational point on each connected component. In the course of the proof we study the class of a Severi-Brauer scheme over a general base in the Grothendieck ring of varieties.
