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 finite fields $\mathbf{F}_q$ such that $q=p^α\equiv 1(\bmod {e})$. We see that the Fermat curves correspond precisely to those curves among each class (for $e=l,2l$), that are maximal or minimal over $\mathbf{F}_q$. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. For $e=2l$, we explicitly determine the $ζ$-function(s) for this class of curves, over $\mathbf{F}_q$, as rational functions in the variable $t$, for distinct cases of $a,b$, and $c$, in $\mathbf{F}_q^*$. The $ζ$-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.
We reprove some results about the Minakshisundaram-Pleijel zeta functions of spheres and study their values at nonpositive integers. The results are extended to zeta functions of real projective spaces.
We introduce a new approach to study height zeta functions of projective spaces and projective bundles. To study height zeta functions of projective spaces $Z(\mathbb{P}^n, H_{\mathcal{O}(1)}; s)$, we apply the Riemann-Roch theorem of Arakelov vector bundles by van der Geer and Schoof to the integrand of an integral expression of $Z(\mathbb{P}^n, H_{\mathcal{O}(1)}; s)$. We give a proof of the analytic continuation and functional equations of height zeta functions of projective spaces with respect to various height functions. Motivic analogues of these results are also proved. We also study he…
I give a formula for the zeta function of a projective toric hypersurface over a finite field and estimate its Newton polygon. As an application this formula allows us to compute the exact number of rational points on the families of Calabi-Yau manifolds in Mirror Symmetry.
We present here continued fraction for Zeta(3) parametrized by some family of points (F,G) on projective line. This family of points can be obtained if from full projective line would be removed some no more than countable exeptional set of finite points. A countable set, which contains the above exeptional set of finite points, is specified also.
We give improvements of the deformation method for computing the zeta function of a generic projective hypersurface in characteristic~$p$ that either reduce the dependence on~$p$ of the time complexity to $\tilde{O}(p^{1/2})$ or that of the space complexity to $\tilde{O}(\log(p))$ while remaining polynomial in the other input parameters.
Given a hypersurface, $X$, prime $p$, the zeta function is a generating function for the number of $\mathbb{F}_{p}$ rational points of $X$. Until now, there is no algorithm for computing hypersurfaces with ADE singularities. Scott Stetson and Vladimir Baranovsky provided an algorithm with Mathematica for the ordinary double point case. In this paper, I go over a Sage algorithm for computing the zeta function of a hypersurface with ADE singularities over 3-dimensional projective space. To make the algorithm more efficient, I established an equivalence between a polynomial belonging to the Jacob…
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.
We compute a basis for the p-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give p-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.
In this paper we prove that there exists a Zariski dense open subset U defined over the rationals Q in the space of all one-variable rational functions with arbitrary k poles of prescribed orders, such that for every geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a prime p has Newton polygon approaching the Hodge polygon as p approaches infinity. As an application to algebraic geometry, we prove that the p-adic Newton polygon of the zeta function of a p-cover of the projective line totally ramified at arbitrary k points of prescribed orders has an asymptotic gener…
Given a projective variety X defined over a finite field, the zeta function of divisors attempts to count all irreducible, codimension one subvarieties of X, each measured by their projective degree. When the dimension of X is greater than one, this is a purely p-adic function, convergent on the open unit disk. Four conjectures are expected to hold, the first of which is p-adic meromorphic continuation to all of C_p. When the divisor class group (divisors modulo linear equivalence) of X has rank one, then all four conjectures are known to be true. In this paper, we discuss the higher rank case…
In this brief note, we will investigate the number of points of bounded (twisted) height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the $p$-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is $p$-adic meromorphic.
With every path on a projective line minus zero, one and infinity there is associated a measure. We are studying a sum of two such measures associated to paths from the tangential point at zero to roots of one. We show that the obtained measure can be defined very elementary. Integrating agaist this measure we get p-adic Hurwitz zeta functions constructed previously by Shiratani.
We study the distribution of the zeroes of the zeta functions of the family of Artin-Schreier covers of the projective line over $\mathbb{F}_q$ when $q$ is fixed and the genus goes to infinity. We consider both the global and the mesoscopic regimes, proving that when the genus goes to infinity, the number of zeroes with angles in a prescribed non-trivial subinterval of $[-π,π)$ has a standard Gaussian distribution (when properly normalized).
We consider the one-parameter family of hypersurfaces in $\Pj^5$ with projective equation (X_1^5+X_2^5+X_3^5+X_4^5+X_5^5) = 5λX_1 X_2... X_5, (writing $λ$ for the parameter), proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation throughout the complex plane.
Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over the moduli spaces, all these are connected with the zeros of rank $n$ non-abelian zeta functions of the field of rationals, which are known lie on the central line except a finitely many if $n\geq 2$. Certainly, when $n=1$, our current work recovers that of Armitage, which from th…
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.
We present a method for computing the zeta function of a smooth projective variety over a finite field which proceeds by induction on the dimension. We have implemented our approach for some surfaces using the Magma programming language, and present some explicit examples which we have computed.
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…
For a Dedekind domain $\mathcal{O}$ and a rank two co-torsion module $M\subseteq \mathcal{O}^2$ with invariant factor ideals $\mathcal{L}\supseteq \mathcal{K}$ in $\mathcal{O}$, that is, $\frac{\mathcal{O}^2}{M}\cong \frac{\mathcal{O}}{\mathcal{L}}\oplus \frac{\mathcal{O}}{\mathcal{K}}$ we associate a new projective space invariant element in $\mathbb{PF}^1_{\mathcal{I}}$ where $\mathcal{I}$ is given by the ideal factorization $\mathcal{K} = \mathcal{L}\mathcal{I}$ in $\mathcal{O}$. This invariant element along with the invariant factor ideals determine the module $M$ completely as a subset of…
A new method is devised for calculating the Igusa local zeta function $Z_f$ of a polynomial $f(x_1,\dots,x_n)$ over a $p$-adic field. This involves a new kind of generating function $G_f$ that is the projective limit of a family of generating functions, and contains more data than $Z_f$. This $G_f$ resides in an algebra whose structure is naturally compatible with operations on the underlying polynomials, facilitating calculation of local zeta functions. This new technique is used to expand significantly the set of quadratic polynomials whose local zeta functions have been calculated explicitl…
In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for…
We consider certain $K3$-fibered Calabi--Yau threefolds. One class of such Calabi--Yau threefolds are constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these Calabi--Yau threefolds. We also consider deformations of our Calabi--Yau threefolds, and we study the variation of the zeta-functions using $p$-adic rigid cohomology theory.
Let $X_λ$ and $X_λ'$ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249].
For a scheme $X$ whose $\mathbb F_q$-rational points are counted by a polynomial $N(q)=\sum a_iq^i$, the $\mathbb{F}_1$-zeta function is defined as $ζ(s)=\prod(s-i)^{-a_i}$. Define $χ=N(1)$. In this paper we show that if $X$ is a smooth projective scheme, then its $\mathbb{F}_1$-zeta function satisfies the functional equation $ζ(n-s) = (-1)^χζ(s)$. We further show that the $\mathbb{F}_1$-zeta function $ζ(s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $ζ(r+N-s) = (-1)^χ( ζ(s) )^{(-1)^r}$.
We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic $p$ that runs in time $p^{1/2 + o(1)}$. We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples. The algorithm relies on Gonçalves's generalization of Kedlaya's algorithm for cyclic covers, and Harvey's work on Kedlaya's algorithm for large characteristic.
We construct a cohomology theory with compact support H^i_c(X_ar,Z(n))$ for separated schemes of finite type over a finite field, which should play a role analog to Lichtenbaum's Weil-etale cohomology groups for smooth and projective schemes. In particular, if Tate's conjecture holds and rational and numerical equivalence agree up to torsion, then the groups H^i_c(X_ar,Z(n)) are finitely generated, form an integral version of l-adic cohomology with compact support, and admit a formula for the special values of the zeta-function of X.
Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly well-suited to work with polynomials in small characteristic that have few monomials (relative to their dimension). Our method covers toric, affine, and projective hypersurfaces and also can be used to compute the L-function of an exponential sum.
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.
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, …
Let p be an odd prime. We consider the cyclotomic extension T := Z_(p)[zeta_{p^2}] of S := Z_(p), with galois group G := (Z/p^2)^*. Since this extension is wildly ramified, the SG-module T is not projective. We calculate its cohomology ring H^*(G, T; S), carrying the cup product induced by the ring structure of T. Proceeding in a somewhat greater generality, our results also apply to certain Lubin-Tate extensions.
Let X be a regular scheme, projective and flat over the integers. Let A be the constant in the conjectured functional equation for the zeta-function of X. We give a conjecture computing A in terms of Euler characteristics of derived exterior powers of the sheaf of Kahler differentials on X, and prove this conjecture when the dimension of X is 1 or 2. This conjecture essentially says that the formulas for the special values of the zeta-function of X given by the author in a previous preprint are compatible with the functional equation.
In this paper, we construct four different theories of integration, two that are for Voevodsky motives, one for mixed $\ell$-adic sheaves, and a fourth theory of integration for rational mixed Hodge structures. We then show that they circumvent some of the complications of classical motivic integration, leading to new arithmetic and geometric results concerning K-equivalent $k$-varieties. For example, in addition to recovering known results regarding K-equivalent smooth projective complex varieties, we show that K-equivalent smooth projective $\mathbb{F}_q$-varieties have isomorphic rational $…
We discuss several conjectures about derived equivalent varieties, defined over fields of arbitrary characteristics, and implications among them. In particular we show that the (conjectural) derived invariance of the Hasse-Weil Zeta functions of smooth projective varieties over finite fields, implies the derived invariance of the Hodge diamond of complex algebraic varieties.
The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and l-adic) and a number of less obvious (p-adic) constraints. We consider the converse question, in the style of Honda-Tate: given a function Z satisfying all these constraints, does there exist a K3 surface whose zeta-function equals Z? Assuming semi-stable reduction, we show that the answer is yes if we allow a finite extension of the finite field. An important ingredient in the proof is the construction of complex projective K3 surfaces with complex multiplication by a given CM field.
We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends Kedlaya's algorithm to a very general class of curves using a map to the projective line. We develop all the necessary bounds, analyse the complexity of the algorithm and provide some examples computed with our implementation.
Let X be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on X with bounded degree as it goes to the infinity. By this estimate, we can define a certain kind of zeta functions associated with groups of cycles. We also consider an analogue in Arakelov geometry.
We study mirror symmetric pairs of Calabi--Yau manifolds over finite fields. In particular we compute the number of rational points of the manifolds as a function of the complex structure parameters. The data of the number of rational points of a Calabi--Yau $X/\mathbb{F}_q$ can be encoded in a generating function known as the congruent zeta function. The Weil Conjectures (proved in the 1970s) show that for smooth varieties, these functions take a very interesting form in terms of the Betti numbers of the variety. This has interesting implications for mirror symmetry, as mirror symmetry exchan…
Consider a smooth projective curve $\overline{C}$ over a finite field $\mathbb{F}_q$, equipped with a simply branched morphism $\overline{C} \to \mathbb{P}^1$ of degree $d \leq 5$. Assume char$\, \mathbb{F}_q > 2$ if $d \leq 4$, and char$\, \mathbb{F}_q > 3$ if $d=5$. In this paper we describe how to efficiently compute a lift of $\overline{C}$ to characteristic zero, such that it can be fed as input to Tuitman's algorithm for computing the Hasse-Weil zeta function of $\overline{C} / \mathbb{F}_q$. Our method relies on the parametrizations of low rank rings due to Delone-Faddeev and Bhargava.
Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l di…
We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the Betti numbers conjectures for the zeta function of the variety.
We study fixed points of iterates of dynamically affine maps (a generalisation of Lattès maps) over algebraically closed fields of positive characteristic $p$. We present and study certain hypotheses that imply a dichotomy for the Artin-Mazur zeta function of the dynamical system: it is either rational or non-holonomic, depending on specific characteristics of the map. We also study the algebraicity of the so-called tame zeta function, the generating function for periodic points of order coprime to $p$. We then verify these hypotheses for dynamically affine maps on the projective line, general…
We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic.
We calculate the total derived functor for the map from the Weil-etale site introduced by Lichtenbaum to the etale site for varieties over finite fields. In particular, there is a long exact sequence relating Weil-etale cohomology and etale cohomology. In the second half of the paper, we apply this to study the Weil-etale cohomology of the motivic complex for smooth and projective varieties. These groups are expected to be finitely generated, to give an integral model for l-adic cohomology, and to be related to special values of the zeta function. We give necessary and sufficient conditions fo…
If C is a smooth projective curve over a number field k, then, under fair hypotheses, its L-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is X-mean-periodic for some appropriate functional space X. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such L-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of GL_{2g}-automorphic representations…
This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelov-theoretic version of motivic cohomology is the conjecture on special values of $L$-functions and zeta functions formulated by the second author. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, $h$-descent. We round off the picture with a purity result an…
Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of hyperelliptic curves whose genera tend to infinity, these measures tend to the natural measure on the space of rank two bundles. This is a function field analogue of Duke's theorem on the equidistribution of Heegner points, and can be proven similarly: it follows from a manipulation of zeta functions, plus the Riemann Hypothesis for curves. Likewise, a sequence o…
Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field $\mathbb{F}_q$, is the Euler characteristic of the derived de Rham cohomology of $X/\mathbb{Z}$ modulo the Hodge filtration $F^n$. In this note, we extend this result to arbitrary separated schemes of finite type over $\mathbb{F}_q$ of dimension at most $d$, provided resolution of singularities for schemes of dimension at most $d$ holds. More precisely, we show that Geisser's generalization of Milne's factor, whenever it is well defined, is the Euler characteristic of the …
