A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over the set of rational numbers. Remarks are given concerning the extension of these results to forms defined over number fields.

The paper gives formulae for a module presentation of the module of identities among relations for a presentation of a group, in terms of information on 0- and 1-combings of the Cayley graph. These formulae are seen as a special case of formulae for extending a partial free crossed resolution of a group, given a partial contracting homotopy of the universal cover of the partial resolution.

Two different numerical methods for solving a non-self-adjoint evolution equation are compared in this paper. If the intial function lies in the domain of the operator, a recently proposed method that combines pseudospectral ideas and semigroup theory is shown to be considerably more accurate than a standard discretization method. One example is worked out in detail, but the methodsa used are of much wider applicability.

The algorithms described in this paper were developed to investigate three problems regarding polynomials with restricted coefficients: (i) determining whether there exist polynomials with {0, 1} coefficients and repeated noncyclotomic factors, (ii) searching for polynomials with {−1, 1} coefficients and small Mahler measure, and (iii) finding polynomials with {−1, 0, 1} coefficients with a root of high multiplicity off the unit circle. The results in the first problem presented here answer a question of Odlyzko and Poonen.

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

Abstract:We give an algorithm for finding the module of linear dependencies of the roots of a monic integral polynomial. Using this, we describe an algorithm for constructing the algebraic hull of a given matrix Lie algebra in characteristic zero.

The authors present an algorithm to construct conjugacy class representatives of the solvable primitive subgroups of Sd for a given degree d. Using this method, they determine the solvable primitive permutation groups of degree at most 6560 (that is, 38 – 1), up to conjugacy.

We determine the character table of the endomorphism ring of the permutation module associated with the multiplicity-free action of the sporadic simple Baby Monster group B on its conjugacy class 2B, where the centraliser of a 2B-element is a maximal subgroup of shape 21+22.Co2. This is one of the first applications of a new general computational technique to enumerate big orbits.

For each group G of order up to 30 we compute a small 3-dimensional CW-space X with π1X≌ G and π2X = 0, and we quantify the ‘efficiency’ of X. Furthermore, we give a theoretical result for treating the case when G is a semi-direct product of two groups for which 3-presentations are known. We also describe the ZG-module structure on the second homotopy group π2X2 of the 2-skeleton of X. This module structure can in principle be used to determine the co-homology groups H2(G, A) and H3(G, A) with coefficients in a ZG-module A. Our computations, which involve the Todd–Coxeter procedure for coset enumeration and the LLL algorithm for finding bases of integer lattices, are rather naive in that the LLL algorithm is applied to matrices of dimension a multiple of |G|. Thus, in their present form, our techniques can be used only on small groups (say of order up to several hundred). They can in principle be used to construct (crossed) ZG-resolutions of Z, but again, only for small G. The paper is accompanied by two attachment files. The first of these is a summary of our computations in HTML format. The second contains various GAP programs used in the computations.

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.

We discuss the optimal Markovian coupling before an exponential time of the Kolmogorov diffusion, and a class of related stochastic control problems in which the aim is to hit the origin before an exponential time. We provide a scaling argument for the optimal control in the near field and use rational WKB approximation to obtain the optimal control in the far field, and compare these analytical results with numerical experiments. In some of these optimal control problems, in which the advection velocity field is bounded, we show that the probability of success agrees exactly with its leading-order asymptotic approximation in some areas of the plane, up to an undetermined multiplicative constant. We conjecture a necessary and sufficient condition for this behaviour, which is strongly supported by numerical experiments.

The Weil descent construction of the GHS attack on the elliptic curve discrete logarithm problem (ECDLP) is generalised in this paper, to arbitrary Artin-Schreier extensions. A formula is given for the characteristic polynomial of Frobenius for the curves thus obtained, as well as a proof that the large cyclic factor of the input elliptic curve is not contained in the kernel of the composition of the conorm and norm maps. As an application, the number of elliptic curves that succumb to the basic GHS attack is considerably increased, thereby further weakening curves over GF2155.

Other possible extensions or variations of the GHS attack are discussed, leading to the conclusion that they are unlikely to yield further improvements.

This paper contains a variety of results about the action of Con way‘s largest simple group upon the crosses in the Leech lattice. These results are tailor-made for use in ‘A Monster Graph, I'(Proc. London Math. Soc. (3) 90 (2005) 42-60), where a graph related to the Monster simple group is studied.

Let E1 and E2 be ordinary elliptic curves over a finite field Fp such that #E1(Fp) = #E2(Fp). Tate's isogeny theorem states that there is an isogeny from E1 to E2 which is defined over Fp. The goal of this paper is to describe a probabilistic algorithm for constructing such an isogeny.

The algorithm proposed in this paper has exponential complexity in the worst case. Nevertheless, it is efficient in certain situations (that is, when the class number of the endomorphism ring is small). The significance of these results to elliptic curve cryptography is discussed.

Suppose that p is 3, 5 or 7. In this paper, faithful permutation representations of maximal p-local subgroups are constructed, and the radical p-chains of the Baby Monster B are classified. Hence, the Alperin weight conjecture and the Uno reductive conjecture can be verified for B, the latter being a refinement of Dade's reductive conjecture and the Isaacs-Navarro conjecture.

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on maximal unipotent subgroups of split reductive groups and show how this improves computation in the reductive group itself.

The bulk of this paper consists of tables giving lower bounds for discriminants of number fields up to 48. The lower bounds are obtained by using two different inequalities for the discriminant, one due to Odlyzko, and the other due to Serre. These inequalities are derived from Weil's explicit formula by choosing suitable weight functions. The bounds are compared with actual values of the discriminants, and the relative errors are computed. The computations show that, at least for values computed, the bounds obtained via Odlyzko's inequality are better than those obtained via Serre's inequality, and are generally within a few percentage points of the true value. This difference can be attributed to a difference in the weighting given to the contribution of low zeros by the two inequalities.

We present a domain-theoretic version of Picard's theorem for solving classical initial value problems in ℝn. For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, which converge, from below and above respectively, exponentially fast, to the unique solution of the initial value problem. We provide a detailed analysis of the speed of convergence and the complexity of computing the iterates. The algorithm uses proper data types based on rational arithmetic, where no rounding of real numbers is required. Thus we obtain a sound implementation framework to solve initial value problems. In particular, the use of rational arithmetic guarantees that implementations of our technique will adhere to the bounds on convergence speed and algebraic complexity.

We study weak convergence of an Euler scheme for nonlinear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

In this paper, efficient algorithms are given to test the intersection property and some of its variations on flag-transitive coset geometries. These algorithms are then applied to geometries of some sporadic groups, namely the Mathieu groups M11, M12, M22 and M23, the Janko groups J1, J2 and J3 and the Higman-Sims group HS.