Diviseur thêta et formes différentielles

This article concerns the geometry of algebraic curves in characteristic p > 0. We study the geometric and arithmetic properties of the theta divisor Q{\Theta} associated to the vector bundle of locally exact differential forms of a curve. Among other things, we prove that, for a generic curve of genus ≥ 2, this theta divisor Q{\Theta} is always geometrically normal. We give also some results in the case where either p or the genus of the curve is small. In the last part, we apply our results on Q{\Theta} to the study of the variation of fundamental group of algebraic curves. In particular, we refine a recent result of Tamagawa on the specialization homomorphism between fundamental groups at least when the special fiber is supersingular.     

Divisorial complete curves

There is a natural restriction on special divisors of a smooth projective curve C the Clifford index of C. In this paper we look for curves of Clifford index c on which all special divisors exist which are not prevented by c, and we construct two non-trivial examples for c = 4 and c = 5, respectively. Received: 2 May 2005     

Pure Weierstrass gaps from a quotient of the Hermitian curve

In this paper, by employing some results on Kummer extensions, we give an arithmetic characterization of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve, including the well-studied Hermitian curve as a special case. The cardinality of these pure gaps is explicitly investigated. In particular, the numbers of gaps and pure gaps at a pair of distinct places are determined precisely, which can be regarded as an extension of the previous work by Matthews (2001) considered Hermitian curves. Additionally, some concrete examples are provided to illustrate our results.     

The arithmetic of certain cubic function fields

In this paper, we discuss the properties of curves of the form over a given field K of characteristic different from 3. If satisfies certain properties, then the Jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field. We seek to make this connection concrete and then use it to develop an explicit arithmetic for the Jacobian of such curves. From a purely mathematical perspective, this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory. At the same time, it provides another source of groups which are suitable for Diffie-Hellman type protocols in cryptographic applications.

     

Oriented matroids and complete-graph embeddings on surfaces

We provide a link between topological graph theory and pseudoline arrangements from the theory of oriented matroids. We investigate and generalize a function f that assigns to each simple pseudoline arrangement with an even number of elements a pair of complete-graph embeddings on a surface. Each element of the pair keeps the information of the oriented matroid we started with. We call a simple pseudoline arrangement triangular, when the cells in the cell decomposition of the projective plane are 2-colorable and when one color class of cells consists of triangles only. Precisely for triangular pseudoline arrangements, one element of the image pair of f is a triangular complete-graph embedding on a surface. We obtain all triangular complete-graph embeddings on surfaces this way, when we extend the definition of triangular complete pseudoline arrangements in a natural way to that of triangular curve arrangements on surfaces in which each pair of curves has a point in common where they cross. Thus Ringel's results on the triangular complete-graph embeddings can be interpreted as results on curve arrangements on surfaces. Furthermore, we establish the relationship between 2-colorable curve arrangements and Petrie dual maps. A data structure, called intersection pattern is provided for the study of curve arrangements on surfaces. Finally we show that an orientable surface of genus g admits a complete curve arrangement with at most 2g+1 curves in contrast to the non-orientable surface where the number of curves is not bounded.     

Galois covers of the open p-adic disc

This paper investigates Galois branched covers of the open p-adic disc and their reductions to characteristic p. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.     

Semisimple Types in GLn

This paper is concerned with the smooth representation theory of the general linear group G=GL(F) of a non-Archimedean local field F. The point is the (explicit) construction of a special series of irreducible representations of compact open subgroups, called semisimple types, and the computation of their Hecke algebras. A given semisimple type determines a Bernstein component of the category of smooth representations of G; that component is then the module category for a tensor product of affine Hecke algebras; every component arises this way. Moreover, all Jacquet functors and parabolic induction functors connecting G with its Levi subgroups are described in terms of standard maps between affine Hecke algebras. These properties of semisimple types depend on their special intertwining properties which in turn imply strong bounds on the support of coefficient functions.     

Discrete Variational Calculus for B-Spline Curves

In this work, we study discrete variational problems, for B-spline curves, which are invariant under translation and rotation. We show this approach has advantages over studying smooth variational problems whose solutions are approximated by B-spline curves. The latter method has been well studied in the literature but leads to high order approximation problems. We are particularly interested in Lagrangians that are invariant under the special Euclidean group for which B-spline approximated curves are well suited. The main application we present here is the curve completion problem in 2D and 3D. Here, the aim is to find various aesthetically pleasing solutions as opposed to a solution of a physical problem. Smooth Lagrangians with special Euclidean symmetries involve curvature, torsion, and arc length. Expressions of these, in the original coordinates, are highly complex. We show that, by contrast, relatively simple discrete Lagrangians offer excellent results for the curve completion problem. The novel methods we develop for the discrete curve completion problem are general, and can be used to solve other discrete variational problems for B-spline curves. Our method completely avoids the difficulties of high order smooth differential invariants.     

Non degenerate projective curves with very degenerate hyperplane section

In this paper, we study the Hilbert scheme of non degenerate locally Cohen- Macaulay projective curves with general hyperplane section spanning a linear space of dimension 2 and minimal Hilbert function. The main result is that those curves are almost always the general element of a generically smooth component H_n,d,g of the corresponding Hilbert scheme. Moreover, we show that the curves with maximal cohomology almost always correspond to smooth points of H_n,d,g.All the authors were partially supported by Acción Integrada Italia-España, HI2000-0091, and by the Italian counterpart of the project.     

G. Martens′ Dimension Theorem for Curves of Odd Gonality

For a smooth projective irreducible algebraic curve C of odd gonality, the maximal possible dimension of the variety of special linear systems W _d ^r (C) is d-3r by a result of M. Coppens et al. Furthermore it is known that if the maximum dimension of W(C) for a curve C of odd gonality is attained then C is of very special type of curves by the recent progress made by G. Martens. The purpose of this paper is to chase an extension of the result of G. Martens; if dim W(C)=d-3r-1 for a curve C of odd gonality for some dg-4 and r1, then C must be either a smooth plane sextic, a pentagonal curve of bounded genus or a smooth plane octic.     

Eventually minimal curves

A curve defined over a finite field is maximal or minimal according to whether the number of rational points attains the upper or the lower bound in Hasse-Weils theorem, respectively. In the study of maximal curves a fundamental role is played by an invariant linear system introduced by Rück and Stichtenoth in [6]. In this paper we define an analogous invariant system for minimal curves, and we compute its orders and its Weierstrass points. In the last section we treat the case of curves having genus three in characteristic two.     

Connections between connected topological spaces on the set of positive integers

In this paper we introduce a connected topology T on the set ? of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ? which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of the topological spaces (?, T) and (?, T′).     

Linear series on a general k-gonal curve

This paper is a contribution towards a Brill-Noether theory for the moduli space of smooth &-gonal curves of genusg. Specifically, we prove the existence of certain special divisors on a generalk-gonal curveC of genusg, and we detect an irreducible component of the “expected” dimension in the varietyW ^r _d (C), (r ≤k — 2) of special divisors ofC. The latter induces a new proof of the existence theorem for special divisors on a smooth curve.     

Simplified isogeny formulas on twisted Jacobi quartic curves

Isogenies between elliptic curves play a very important role in elliptic curve related cryptosystems and cryptanalysis. It is widely known that different models of elliptic curves would induce different computational costs of elliptic curve arithmetic, and several works have been devoted to accelerate the isogeny computation on various curve models. This paper studies the case of the Jacobi quartic model, which is a classic form of elliptic curves. A new w-coordinate system on extended Jacobi quartic curves is introduced for Montgomery-like group arithmetic. Explicit formulas for 2-isogenies and odd ℓ-isogenies on the specific curves are presented, and based on the w-coordinate system, the computation of such isogenies could be further simplified.     

The Hilbert function of a curve lying on a smooth cubic surface

Summary Let C be any reduced and irreducible curve, lying on a smooth cubic surface S P ³. In this paper we determine the Hilbert function of C. Moreover we characterize some kinds of curves on S: the arithmetically Cohen-Macaulay curves, the maximal rank curves and the extremal ones.Work done with financial support of M.P.I., while the author was a member of C.N.R.     

Pre-compactness of isospectral sets for the neumann operator on planar domains

The Neumann operator maps the boundary value of a harmonic function tc its normal derivative. The inverse spectral properties of the Neumann operator associated to smooth, planar, Jordan curves are studied. The Riemann mapping theorem is used tc parametrize the set of planar Jordan curves by positive functions on the unit circle. By studying the zeta function associated to the spectrum, it is shown that isospectral sets of these functions are pre-compact in the topology of the L²-Sobolev space of order 5/2 - [euro]. Spectral criteria are given for the limiting curves of an isospectral set to be Jordan. A spectrally determined lower bound on the area of the interior of the curve is given.     

Maximally tangent complex curves for germs of finite type {\mathcal{C}^\infty} pseudoconvex domains in {\mathbb C^3}

In this paper we discuss D’Angelo finite type pseudoconvex domains Ω in ${\mathbb C^3}$ . We are interested in complex curves tangent to higher order. Our main result is that there are only finitely many curves of maximal type. Maximal type has to be taken in a micro-local sense since the maximal type can be different in different directions. And of course to get finiteness we have to ignore higher order irrelevant terms which can be added without restriction. In the process of describing such a curve we find a singular change of coordinates which reduces the curve to a complex line.     

On Inductive Limits for Systems of <Emphasis Type="Italic">C</Emphasis>*-Algebras

We consider a covariant functor from the category of an arbitrary partially ordered set into the category of C*-algebras and their *-homomorphisms. In this case one has inductive systems of algebras over maximal directed subsets. The article deals with properties of inductive limits for those systems. In particular, for a functor whose values are Toeplitz algebras, we show that each such an inductive limit is isomorphic to a reduced semigroup C*-algebra defined by a semigroup of rationals. We endow an index set for a family of maximal directed subsets with a topology and study its properties. We establish a connection between this topology and properties of inductive limits.