Counting curves over finite fields

This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the Euler characteristic of the cohomology of local systems on moduli spaces of curves of low genus and its implications for modular forms.Comment: 25 pages, to appear in Finite Fields and their Application

On the cyclicity of the rational points group of abelian varieties over finite fields

We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among certain families of them, when $q$ tends to infinity. Some of these bounds require an additional hypothesis. In the case of surfaces, we prove that this hypothesis is achieved and, over all $\mathbb{F}_q$-isogeny classes with endomorphism algebra being a field and where $q$ is an even power of a prime, we prove that the one with maximal number of rational points is cyclic and ordinary.Comment: 13 pages, this is a preliminary version, comments are welcom

Counting points on curves over families in polynomial time

This note concerns the theoretical algorithmic problem of counting rational points on curves over finite fields. It explicates how the algorithmic scheme introduced by Schoof and generalized by the author yields an algorithm whose running time is uniformly polynomial time for curves in families.Comment: 7 page

Maps between curves and arithmetic obstructions

Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.Comment: 8 page

Counting abelian varieties over finite fields via Frobenius densities

Let $[X,\lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $\nu_v([X,\lambda])$ for each place $v$ of $\mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,\lambda]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.Comment: Main text by Achter, Altug and Gordon; appendix by Li and Ru

Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.Comment: 44 page

Two lectures on the arithmetic of K3 surfaces

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update

A p-adic quasi-quadratic point counting algorithm

In this article we give an algorithm for the computation of the number of rational points on the Jacobian variety of a generic ordinary hyperelliptic curve defined over a finite field of cardinality $q$ with time complexity $O(n^{2+o(1)})$ and space complexity $O(n^2)$, where $n=\log(q)$. In the latter complexity estimate the genus and the characteristic are assumed as fixed. Our algorithm forms a generalization of both, the AGM algorithm of J.-F. Mestre and the canonical lifting method of T. Satoh. We canonically lift a certain arithmetic invariant of the Jacobian of the hyperelliptic curve in terms of theta constants. The theta null values are computed with respect to a semi-canonical theta structure of level $2^\nu p$ where $\nu >0$ is an integer and p=\mathrm{char}(\F_q)>2. The results of this paper suggest a global positive answer to the question whether there exists a quasi-quadratic time algorithm for the computation of the number of rational points on a generic ordinary abelian variety defined over a finite field.Comment: 32 page