Let K denote a number field, and G a finite abelian group. The ring of algebraic integers in K is denoted in this paper by $/cal{O}_K$, and $/cal{A}$ denotes any $/cal{O}_K$-order in K[G]. The paper describes an algorithm that explicitly computes the Picard group Pic($/cal{A}$), and solves the corresponding (refined) discrete logarithm problem. A tamely ramified extension L/K of prime degree l of an imaginary quadratic number field K is used as an example; the class of $/cal{O}_L$ in Pic($/cal{O}_K[G]$) can be numerically determined.

This paper describes how to obtain bounds on the spectrum of a non-self-adjoint operator by means of what are referred to here as ‘its higher-order numerical ranges’. Proofs of some of their basic properties are given, as well as an explanation of how to compute them. Finally, they are used to obtain new spectral insights into the non-self-adjoint Anderson model in one and two space dimensions.

The maximal subgroups of the finite classical groups are divided by a theorem of Aschbacher into nine classes. In this paper, the authors show how to construct those maximal subgroups of the finite classical groups of linear, symplectic or unitary type that lie in the first eight of these classes. The ninth class consists roughly of absolutely irreducible groups that are almost simple modulo scalars, other than classical groups over the same field in their natural representation. All of these constructions can be carried out in low-degree polynomial time.

The authors present three-point and four-point covers having bad reduction at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 12, 18, 28, and 33. By specializing these covers, they obtain number fields ramified at 2 and 3 only, with Galois group An or Sn for n equal to 9, 10, 11, 12, 17, 18, 25, 28, 30, and 33.

The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus Lg of such genus-g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves Hg. The authors present a birational parameterization of Lg via dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈ Hg with |Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈ Lg such that the Klein 4-group is embedded in the reduced automorphism group of p. Further, for g = 3, they show that for every moduli point p ∈ H3 such that |Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

Several classes of Fermat-type diophantine equations have been successfully resolved using the method of galois representations and modularity. In each case, it is possible to view the proper solutions to the diophantine equation in question as corresponding to suitably defined integral points on a modular curve of level divisible by 2 or 3. Motivated by this point of view, an example of a diophantine equation associated to the modular curve X0(5) is discussed in this paper. The diophantine equation has four terms rather than the usual three terms characteristic of generalized Fermat equations.

The minimal degrees of faithful representations of all sporadic simple groups and their covering groups are determined in this paper.

The use of diagrams in mathematics has traditionally been restricted to guiding intuition and communication. With rare exceptions such as Peirce's alpha and beta systems, purely diagrammatic formal reasoning has not been in the mathematician's or logician's toolkit. This paper develops a purely diagrammatic reasoning system of “spider diagrams” that builds on Euler, Venn and Peirce diagrams. The system is known to be expressively equivalent to first-order monadic logic with equality. Two levels of diagrammatic syntax have been developed: an ‘abstract’ syntax that captures the structure of diagrams, and a ‘concrete’ syntax that captures topological properties of drawn diagrams. A number of simple diagrammatic transformation rules are given, and the resulting reasoning system is shown to be sound and complete.

This paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.

The paper describes a procedure for determining (up to algebraic conjugacy) the conjugacy class in which any element of the Monster lies, using computer constructions of representations of the Monster in characteristics 2 and 7. This procedure has been used to calculate explicit representatives for each conjugacy class.

This paper presents a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. The new procedure is a Monte Carlo algorithm, and it is presented and analysed in the context of black-box groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. The authors' major objective was that the procedures could be proved to be Monte Carlo algorithms, and the costs computed. In addition, they explicitly determined suitable subset chains for six of the sporadic groups, and then implemented the algorithms involving these chains in the GAP computational algebra system. It turns out that sample imple-mentations perform well in practice. The implementations will be made available publicly in the form of a GAP package.

This paper presents a new geometric algorithm to construct a Ck-smooth spline curve that interpolates a given set of data (points and velocities) on a complete Riemannian manifold. Although based on a modification of the De Casteljau procedure, this new algorithm is implemented in only three steps, independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function, which is defined as soon as the degree of smoothness is fixed.

The authors construct two families of genus-3 curves defined over Q(t) with three independent morphisms to an elliptic curve of rank at most two. They give explicit examples of an application of the Dem'janenko-Manin method that completely determines both the set of the Q(t)-rational points of the curves under consideration, and the set of Q-rational points of some specialisations.

Ѕ(2n,k) be the variety of homogeneous polynomials of degree k in 2n variables. The authors of this paper give a computer-assisted proof that there is an analytic open set Ω of Ѕ(4,3) such that the surface F = 0 is a minimal homogeneous involutive variety of ℂ4 for all F ∈ Ω. As part of the proof, they give an explicit example of a polynomial with rational coefficients that belongs to Ω.