In response to the article by Ivan Kausz: A discriminant and an upper bound for w 2 for hyperelliptic arithmetic surfaces

We define a natural discriminant for a hyperelliptic curve X of genus g over a field K as a canonical element of the (8g+4)th tensor power of the maximal exterior product of the vectorspace of global differential forms on X. If v is a discrete valuation on K and X has semistable reduction at v, we compute the order of vanishing of the discriminant at v in terms of the geometry of the reduction of X over v. As an application, we find an upper bound for the Arakelov self-intersection of the relative dualizing sheaf on a semistable hyperelliptic arithmetic surface.     

An -module structure of the cokernel of the second Johnson homomorphism

The Johnson homomorphisms τ_k (k1) give Abelian quotients of a series of certain subgroups of the mapping class group of a surface. Morita determined the rational image of the second Johnson homomorphism τ₂. In this paper, we study the structure of the torsion part of the cokernel of τ₂. First, we determine the rank of the cokernel over . Although we do it first by computing explicitly, later we improve the proof, using the Birman–Craggs homomorphism, obtained by the classical Rohlin invariant of homology 3-spheres. Since τ₂ is equivariant with respect to the action of the mapping class group, Im τ₂ is -invariant and hence acts on the cokernel. Moreover, computing this action explicitly, we show that the action reduces to that of the finite symplectic group .     

Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

An Oblivious Routing Algorithm for Two-Dimensional Meshes of Constant Queue-Size

This paper shows an important exception to the common perception that three-dimensional meshes are more powerful than two-dimensional ones. Let N be the total number of processors. Then permutation routing over three-dimensional mesh computers needs Θ(N^2/3) steps while it takes Θ(N^1/2) steps over two-dimensional ones under the following conditions: (1) The path of each packet must be determined solely by its initial position and destination, i.e., the algorithm must be oblivious. (2) Each path must be “elementary,” i.e., it must be shortest and as straight as possible. Thus the conditions, under which, somewhat surprisingly, three-dimensional meshes are significantly less powerful than two-dimensional ones for the fundamental network operation, are quite reasonable in practice.     

String matching in quantum time

We show how to determine whether a given pattern p of length m occurs in a given text t of length n in time (where allows for logarithmic factors in m and n/m) with inverse polynomial failure probability. This algorithm combines quantum searching algorithms with a technique from parallel string matching, called Deterministic Sampling.     

Existence of <Emphasis Type="Italic">g</Emphasis><Superscript>1</Superscript><Subscript><Emphasis Type="Italic">g</Emphasis>–2</Subscript>’s on a double covering of a hyperelliptic curve

Abstract Let X be a non–hyperelliptic curve of genus g which is a double covering of a hyperelliptic curve C of genus h. In this paper, we prove that, if h≥ 3 and g≥ 4h+5, then X admits a complete, base point free g¹_g–2. Moreover, if h=3, this result holds under the mild condition g≥ 4h+3=15. Keywords: Double covering of hyperelliptic curves, Pencil of degree g–2 Mathematics Subject Classification (2000:) 14H30, 14H45     

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.      

On a family of sequences defined recursively in (II)

This paper is a second part to previous work (see Finite Fields Appl. 9 (2003) 211). Different conjectures stated there are proven here. We are concerned with sequences (x_i)_i1 in such that the continued fraction expansion [x₁T,x₂T,…,x_nT,…] in is algebraic over . These algebraic elements correspond in some way to quadratic real numbers for which the continued fraction expansion is well known.     

p -SOLVABILITY AND A GENERALIZATION OF PRIME GRAPHS OF FINITE GROUPS

Abstract

The real Torelli mapping, from the moduli space of real curves of genus g to the moduli space of g-dimensional real principally polarized abelian varieties, sends a real curve into its real Jacobian. The real Schottky problem is to describe its image. The results contained in the present paper concern hyperelliptic real curves and in particular real curves of genus 2. We exhibit also some counterexamples for the non-hyperelliptic case.     

Curves of genus 2 with group of automorphisms isomorphic to or

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.