Certain relative quadric hypersurfaces in 3-dimensional projective space bundles over a projective curve

We classify the certain type of relative quadric hypersurfaces of 3-dimensional projective space bundles over a projective line or an elliptic curve whose fiber is the direct product of 2 projective lines.     

Construction of rational surfaces yielding good codes

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound à la Weil of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over F₇ and a [91,18,53] code over F₉ are discovered, these codes beat the best known codes up to now.     

Canonical geometrically ruled surfaces

We prove the existence of canonical scrolls; that is, scrolls playing the role of canonical curves. First of all, they provide the geometrical version of Riemann Roch Theorem: any special scroll is the projection of a canonical scroll and they allow to understand the classification of special scrolls in P^N. Canonical scrolls correspond to the projective model of canonical geometrically ruled surfaces over a smooth curve. We also prove that the generic canonical scroll is projectively normal except in the hyperelliptic case and for very particular cases in the nonhyperelliptic situation. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Integral Models of Extremal Rational Elliptic Surfaces

Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson's classification has an integral model with good reduction everywhere (except for those of type X ₁₁(j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reducing one of these integral models mod p.     

Legendre Elliptic Curves over Finite Fields

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.     

Elliptic CR-manifolds and shear invariant ordinary differential equations with additional symmetries

We classify the ordinary differential equations that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can only be 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable isotropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.     

Ovoids and monomial ovals

This paper is a contribution to the classification of ovoids. We show, under some rather technical assumptions, that if an ovoid of PG(3, q) has a pencil of monomial ovals, then it is either an elliptic quadric or a Tits ovoid. Further, we show that if an ovoid of PG(3, q) has a bundle of translation ovals, again under some extra assumptions, then the ovoid is an elliptic quadric or a Tits ovoid.     

Einstein-Weyl spaces and (1 , n) -curves in the quadric surface

Following the ideas of Hitchin on the twistoral approach to 3-dimensional Einstein-Weyl geometry we construct a series of complex surfaces containing rational curves with self-intersection number 2. These mini twistor spaces are obtained by taking an n-fold covering of a neighbourhood of a (1 , n)- curve in the quadric CP₁ x CP₁ branched along the curve. We describe the corresponding Einstein-Weyl geometry on the parameter space of curves.     

Enriques surfaces and Jacobian elliptic K3 surfaces

This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.     

Lifting the j-invariant: Questions of Mazur and Tate

In this paper we analyze the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We show that its coordinates are rational functions on the j-invariant of the elliptic curve in characteristic p. In particular, we prove that the second coordinate is always regular at j=0 and j=1728, even when those correspond to supersingular values. A proof is given which yields a new proof for some results of Kaneko and Zagier about the modular polynomial.     

二次C-曲线研究

给出具有相同控制顶点的二次C-曲线与二次有理Bézier曲线表示同一参数曲线段的充要条件,由此得到了二次C-曲线不能精确表示双曲线段的结论;另外,还给出了二次C-曲线在任意一点的细分公式.     

On a special case of the intersection of quadric and cubic surfaces

The intersection curve between two surfaces in three-dimensional real projective space RP³ is important in the study of computer graphics and solid modelling. However, much of the past work has been directed towards the intersection of two quadric surfaces. In this paper we study the intersection curve between a quadric and a cubic surface and its projection onto the plane at infinity. Formulas for the plane and space curves are given for the intersection of a quadric and a cubic surface. A family of cubic surfaces that give the same space curve when we intersect them with a quadric surface is found. By generalizing the methods in Wang et al. (2002) [6] that are used to parametrize the space curve between two quadric surfaces, we give a parametrization for the intersection curve between a quadric and a cubic surface when the intersection has a singularity of order 3.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Ovoids ofPG(3, 16) are elliptic quadrics,II

In a previous paper [8] the authors have shown that every ovoid ofPG(3, 16) is an elliptic quadric. The arguments used a computer and also depended on the computer-aided classification of hyperovals ofPG(2, 16) (see [3]). Recently (see [9]) the classification of hyperovals ofPG(2,16) has been obtained without the use of a computer. The present paper completes a computer-free proof that every ovoid ofPG(3,16) is an elliptic quadric.     

Affine Gelfand-Dickey brackets and holomorphic vector bundles

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem leads to the problem of classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly.     

Sur la dynamique du groupe d'automorphismes des surfaces K3

We study the dynamics of the automorphisms group of K3 surfaces. Assuming that the surface contains two elliptic fibrations that are invariant by non-periodic automorphisms, we give the classification of invariant probability measures. We also describe the closure of orbits and then give applications to the repartition of rational points on K3 surfaces.     

The modularity of some Q-curves

A Q-curve is an elliptic curve, defined over a number field, that is isogenous to each of its Galois conjugates. Ribet showed that Serre's conjectures imply that such curves should be modular. Let E be an elliptic curve defined over a quadratic field such that E is 3-isogenous to its Galois conjugate. We give an algorithm for proving any such E is modular and give an explicit example involving a quotient of J_o (169). As a by-product, we obtain a pair of 19-isogenous elliptic curves, and relate this to the existence of a rational point of order 19 on J₁ (13).     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

On the modularity of elliptic curves over -adic exercises

We complete the proof that every elliptic curve over the rational numbers is modular.