Morgan Ward pursued the study of elliptic divisibility sequences, originally initiated by Lucas, and Chudnovsky and Chudnovsky subsequently suggested looking at elliptic divisibility sequences for prime appearance. The problem of prime appearance in these sequences is examined here, from both a theoretical and a practical viewpoint. We show calculations, together with a heuristic argument, to suggest that these sequences contain only finitely many primes.

A module over a finite-dimensional algebra is called a ‘diamond’ if it has a simple top and a simple socle. Using covering theory, the classification of all diamonds for algebras of finite representation type over algebraically closed fields can be reduced to representation-directed algebras. The author proves a criterion referring to the positive roots of the corresponding Tits quadratic form, which makes it easy to check whether a representation-directed algebra has a faithful diamond. Using an implementation of this criterion in the CREP program system on representation theory, he is able to classify all exceptional representation-directed algebras having a faithful diamond. He obtains a list of 157 algebras up to isomorphism and duality. The 52 maximal members of this list are presented at the end of this paper.

The authors determine all the absolutely irreducible representations of degree up to 250 of quasi-simple finite groups, excluding groups that are of Lie type in their defining characteristic. Additional information is also given on the Frobenius-Schur indicators and the Brauer character fields of the representations.

The author considers the development of algorithms for deciding whether a finitely generated matrix group over a field of positive characteristic is finite. A deterministic algorithm for deciding the finiteness is presented for the case of a field of transcendence degree one over a finite field.

Corrigendum to Proposition 14, on page 87 of the paper ‘Reduction of binary cubic and quartic forms’, LMS Journal of Computation and Mathematics, Volume 2, pp. 62–92.

The authors study a new method for coset enumeration in finitely presented groups. Their method uses prefix Gröbner basis computation in the monoid ring ${\mathbb{K}}[{\cal M}]$, where ${\mathbb{K}}$ is a computable field and ${\cal M}$ a monoid presented by a convergent string-rewriting system. The method is compared to well-known methods for Todd-Coxeter enumeration, using examples from the literature where studies of these methods are reported. New insights into coset enumeration were gained using three different kinds of orderings, combined with new frameworks and strategies implemented in MRC 1.2.

The author has determined, for all simple simply connected reductive linear algebraic groups defined over a finite field, all the irreducible representations in their defining characteristic of degree below some bound. These also give the small degree projective representations in defining characteristic for the corresponding finite simple groups. For large rank l, this bound is proportional to l3, and for rank less than or equal to 11 much higher. The small rank cases are based on extensive computer calculations.

Modular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} for all α,β in the extended upper half plane and γ ∊ Γ. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.

This paper announces the creation of a database for number fields. It describes the contents and the methods of access, indicates the origin of the polynomials, and formulates the aims of this collection of fields.

This paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of new, simple proofs of two known results. First, it is shown that an irreducible invariant algebraic curve of degree n > 1 without multiple points for a polynomial vector field of degree m satisfies n ≤ m + 1. Second, a proof is given that quadratic polynomial vector fields have no algebraic limit cycles of degree 3.