False Hyperelliptic Surfaces with Section

Let X be a nonsingular relatively minimal projective surface over an algebraically closed field of characteristic p > 0. We call X a false hyperelliptic surface if X satisfies the following conditions: (1) c₂(X) = 0, c₁(X)² = 0, dim Alb (X) = 1, and (2) All fibres of the Albanese mapping of X are rational curves with only one cusp of type xp^v + yⁿ = 0. In this article, we consider a false hyperelliptic surface whose Albanese mapping has a cross-section. We prove that every false hyperellyptic surface with section arises from an elliptic ruled surface and that every false hyperelliptic surface has an elliptic fibration with multiple fibre. Moreover, we construct an example of false hyperelliptic surface with section, whose elliptic fibration has a multiple fibre of supersingular elliptic curve of multiplicity p^v (v > 1).     

Counting isomorphism classes of pointed hyperelliptic curves of genus 4 over finite fields with even characteristic

This paper is devoted to counting the number of isomorphism classes of pointed hyperelliptic curves over finite fields. We deal with the genus 4 case and the finite fields are of even characteristics. The number of isomorphism classes is computed and the explicit formulae are given. This number can be represented as a polynomial in q of degree 7, where q is the order of the finite field. The result can be used in the classification problems and it is useful for further studies of hyperelliptic curve cryptosystems, e.g. it is of interest for research on implementing the arithmetics of curves of low genus for cryptographic purposes. It could also be of interest for point counting problems; both on moduli spaces of curves, and on finding the maximal number of points that a pointed hyperelliptic curve over a given finite field may have.     

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

We consider families of curves with extra automorphisms in ?₃, the moduli space of smooth hyperelliptic curves of genus g = 3. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. For each family of positive dimension where |Aut (C)|>4, we determine the possible distributions of weights of 2-Weierstrass points.     

Sur certains sous-groupes de torsion de jacobiennes de courbes hyperelliptiques de genreg ≥ 1

We construct a family of hyperelliptic curves of genusg defined over Q whose Jacobians have a rational point of order 2g(2g+1). Forl = 2g ² + 5g + 5, we construct a family of genusg hyperelliptic curves defined over Q, such that their Jacobians have a rational point of orderl orl / 2 orl / 4. We also construct a hyperelliptic curve of genusg defined over Q, which does not belong to the previous family, and whose Jacobian has a rational point of orderl.      

The p-rank strata of the moduli space of hyperelliptic curves

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p?3. This yields a strong technique that allows us to analyze the stratum of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to Z if g?4. Furthermore, we prove that the Z/?-monodromy of every irreducible component of is the symplectic group Sp2g(Z/?) if g?3, and ?≠p is an odd prime (with mild hypotheses on ? when f=0). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.     

Elliptic and hyperelliptic curves over supersimple fields in characteristic 2

In this paper, we extend a previous result of A. Pillay and the author regarding existence of rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple fields to the even characteristic case. We give a detailed exposition of the affine models of these families of curves in characteristic 2 and the transformations between members in the same rational isomorphism class.     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

On the moments of Kloosterman sums and fibre products of Kloosterman curves

Let q=p^r with p=3 and r2. We give a recursion formula for the moments of a Kloosterman sum over the finite field , which utilizes known weight formulae for the ternary Melas code M of length q−1. The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As an application for the formulae, and for their analogues obtained earlier in case p=2, we get the exact number of rational points on fibre products of certain Kloosterman curves. As a corollary we obtain identities between Ramanujan's tau-function, Kronecker class numbers, and Dickson polynomials.     

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

A Geometric Interpretation and a New Proof of a Relation by Cornalba and Harris

Abstract

In the 1980's Cornalba and Harris discovered a relation among the Hodge class and the boundary classes in the Picard group with rational coefficients of the moduli space of stable, hyperelliptic curves. They proved the relation by computing degrees of the classes involved for suitable one-parameter families. In the present article we show that their relation can be obtained as the class of an appropriate, geometrically meaningful empty set, thus conforming with Faber's general philosophy of finding relations among tautological classes in the Chow ring of the moduli space of curves. The empty set we consider is the closure of the locus of smooth, hyperelliptic curves having a special ramification point.     

Manifolds of support sets of chebyshev polynomials

We consider a parameterization of the set of polynomialsT _n(E, x) whose deviation from zero is the least on a systemE consisting of several intervals on the real axis. We point out a new way for obtaining the equations which describe the boundary of the maximum set of the least deviationE ⁺ ⊃E. We describe the geometry of the variety of all possible setsE ⁺; this manifold is embedded in the moduli space of hyperelliptic curves with real branch points. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 828–836, June, 2000.     

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.     

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications.(a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n.(b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.     

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X₀(31) (and X₀(22) as a by-product) and X₀(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X₀(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X₀(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation. The authors thank the FNR (project FNR/04/MA6/11) for their support.     

The Hodge conjecture for certain moduli varieties

For smooth projective varietiesX over ℂ, the Hodge Conjecture states that every rational Cohomology class of type (p, p) comes from an algebraic cycle. In this paper, we prove the Hodge conjecture for some moduli spaces of vector bundles on compact Riemann surfaces of genus 2 and 3.     

Rédei-Funktionen für Zweierpotenzen

L. Rédei has introduced in 1946 a class of rational functions over finite fields of orderp ^t –p being an odd prime — and has proved some interesting results on permutations which are induced by these functions. Recently, similar results have been found for factor rings of the integers of orderp ^t ; moreover, these functions have been applied to construct public key cryptosystems. In this paper we consider functions of Rédei type over finite fields and factor rings of the integers of order 2 ^t . We show that most of the results for oddp also hold in this case.

     

Exponential sum estimates over subgroups and almost subgroups of \mathbb{Z}_{Q}^{*} , where Q is composite with few prime factors

In this paper we extend the exponential sum results from [BK] and [BGK] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q^δ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand, the methods from [BGK] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog–Szemeredi theorem. As a corollary, we get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p^r (p prime) for all r. Only the case r = 2 has been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [KS]. Submitted: October 2004 Revision: June 2005 Accepted: August 2005     

Moduli of Polarised Abelian Surfaces

The moduli variety parametrising abelian surfaces over ? with a polarisation of type (1, p) may be compactified by toroidal methods. We show that if p is a prime and p ≥ 173 then the compactified moduli variety is of general type.     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.