On the number of curves of genus 2 over a finite field

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujolàs, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in , the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in .     

Hyperelliptic curves of genus three over finite fields of even characteristic

In this paper we classify hyperelliptic curves of genus 3 defined over a finite field k of even characteristic. We consider rational models representing all k-isomorphy classes of curves with a given arithmetic structure for the ramification divisor and we find necessary and sufficient conditions for two models of the same type to be k-isomorphic. Also, we compute the automorphism group of each curve and an explicit formula for the total number of curves.     

Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus of the moduli space M₃ of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385-389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198-200] on the hyperelliptic locus we obtain a complete picture of these strata for M₃.     

On Lang's Conjecture for Surfaces of General Type

Abstract

Let m be a nonnegative integer. We explicitly bound the number of rational curves with arithmetic genus no greater than m on a surface of general type by geometric invariants of the surface. We also briefly discuss the possibility of bounding all rational curves lying on a surface of general type in ?³.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

On the rational equivalence of full decomposable forms

Let k be any finite normal extension of the rational field Q and fix an order D of k invariant under the galois group $\text{G(}\text{k}\text{Q}\text{)}$. Consider the set F of the full decomposable forms which correspond to the invertible fractional ideals of D. In a recent paper the author has given arithmetic criteria to determine which classes of improperly equivalent forms in F integrally represent a given positive rational integer m. These criteria are formulated in terms of certain integer sequences which satisfy a linear recursion and need only be considered modulo the primes dividing m. Here, for the most part, we consider partitioning F under rational equivalence. It is a found that the set of rationally equivalent classes in F is a group under composition of forms analogous to Gauss' and Dirichlet's classical results for binary quadratic forms. This leads us to given criteria as before to determine which classes of rationally equivalent forms in F rationally represent m. Moreover, by applying the genus theory of number fields, we find arithmetic criteria to determine when everywhere local norms are global norms if the Hasse norm principle fails to hold in $\text{k}\text{Q}$.     

Galois sections for abelianized fundamental groups

Given a smooth projective curve X of genus at least 2 over a number field k, Grothendieck’s Section Conjecture predicts that the canonical projection from the étale fundamental group of X onto the absolute Galois group of k has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of k but not over k. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundamental group ‘with abelianized geometric part’ onto the Galois group. We also point out the relation to the elementary obstruction of Colliot-Thélène and Sansuc. This article has an appendix by E. V. Flynn.     

The arithmetic of Jacobian groups of superelliptic cubics

We present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data.

     

Genus Two Curves with Many Elliptic Subcovers

We determine all genus 2 curves, defined over ?, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in ?₂. For each component, we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to ?-isomorphism) defined over ?, which have degree 2 and 3 elliptic subcovers also defined over ?.     

Hyperbolic uniformization of Fermat curves

The ground-breaking research on the uniformization of curves was conducted at the beginning of the last century. Nevertheless, there are few examples in the literature of algebraic curves for which an explicit uniformization is known. In this article we obtain an explicit uniformization of the Fermat curves F _N , for each . The results presented here are based in part on an earlier study of the second author [6] in which each Riemann surface F _N (ℂ) was described as a quotient of the complex disk by a Fuchsian group Γ. 2000 Mathematics Subject Classification Primary—11F03, 11F06; Secondary—11F30 This work was partially supported by MCYT BFM2000-0627 and BMF2003-01898.     

Rational classes and divisors on curves of genus 2

We describe a method of looking for rational divisor classes on a curve of genus 2. We have an algorithm to decide if a given class of divisors of degree 3 contains a rational divisor. It is known that the shape of the kernel of Cassel’s morphism (X − T) is related to the existence of rational classes of degree 1. Our key tool is the dual Kummer surface.V. G. L. Neumann supported by CNPq, Brazil     

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

We first build the moduli spaces of real rational pseudo-holomorphic curves in a given real symplectic 4-manifold. Then, following the approach of Gromov and Witten [3, 19, 11], we define invariants under deformation of real symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points. Mathematics Subject Classification (2000) 14N10, 14P25, 53D05, 53D45     

Deformation and rigidity results for the 2k‐Ricci tensor and the 2k‐Gauss‐Bonnet curvature

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are 2k‐Einstein (in the sense that their 2k‐Ricci tensor is constant) or have constant 2k‐Gauss‐Bonnet curvature. The results hold for a family of manifolds containing all non‐flat space forms and the main ingredients in the proofs are explicit formulae for the linearizations of the above invariants obtained by means of the formalism of double forms.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

Curves of genus g on an abelian variety of dimension g

In this paper we prove a general theorem concerning the number of translation classes of curves of genus g belonging to a fixed cohomology class in a polarized abelian variety of dimension g. For g = 2 we recover results of Göttsche and Bryan-Leung. For g = 3 we deduce explicit numbers for these classes.     

Local points of twisted Mumford quotients and Shimura curves

We determine over which fields twisted Mumford quotients have rational points. Using the $p$-adic uniformization, we apply these results to Shimura curves, and show some new cases for which the jacobians are even in the sense of [PS]. Mathematics Subject Classification (2000):14G20, 14G35The first author was partially supported by grants from the NSF and PSC-CUNYThe first two authors were partially supported by a joint Binational Israel-USA Foundation grant     

The Hasse principle and the Brauer-Manin obstruction for curves

We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the conjecture that the Tate-Shafarevich group of the Jacobian is finite.Mathematics Subject Classification (2000): Primary 11G30; Secondary 11G10, 14H40The author was supported by EPSRC grant GR/R82975/01.     

Extremal elliptic surfaces in characteristic 2 and 3

We show that extremal elliptic surfaces in characteristic 2 and 3 are unirational surfaces. Our strategy of proof is to determine explicit equations for the generic fibers. The method also applies to the classification of elliptic curves over k(T) with given places of bad reduction where k is any perfect field of characteristic 2 or 3. Received: 17 September 1999 / Revised version: 19 April 2000     

Witt’s Theorem for Noncommutative Conic Curves

Let k be a field. We extend the main result in Nyman (J. Algebra 434, 90–114, 2015) to show that all homogeneous noncommutative curves of genus zero over k are noncommutative \(\mathbb {P}^{1}\)-bundles over a (possibly) noncommutative base. Using this result, we compute complete isomorphism invariants of homogeneous noncommutative curves of genus zero, allowing us to generalize a theorem of Witt.