David C. Hunt

Rational rigidity and the sporadic groups

The question of which finite groups can or do occur as the Galois group of a Galois extension of the rational numbers is one which has been studied since early in the twentieth century. It is conjectured that all finite groups do occur as Galois groups over the rationals: this is known to be true for all solvable groups. (See Jacobson [73 for brief historical notes.) In a breakthrough paper Thompson [9] has defined the new concept of rigidity of a finite group. He then proves that if a group G, with trivial centre, is rigid then an associated group G* is a Galois group over the rationals. In this paper it is shown that 14 of the 26 sporadic simple groups are Galois groups over the rationals. The method used involves the complex character table and knowledge of maximal subgroups of the group. In some cases where the maximal subgroups are not known the classification of finite simple groups is used. Section 1 contains the results, Section 2 describes the notation used and Section 3 details the proof for each of the 14 groups separately.

Elliptic curves of conductor 11

We determine all elliptic curves defined over Q of conductor 11. Firstly, we reduce the problem to one of solving a diophantine equation, namely a certain ThueMahler equation. Then we apply recent sharp inequalities for linear forms in the logarithms of algebraic numbers to bound solutions of that equation. Finally, some straightforward computations yield all solutions of the diophantine equation. Our results are in accordance with the conjecture of Taniyama-Weil for conductor 11. Taniyama and Weil have asked whether all elliptic curves defined over Q of a given conductor N are parametrized by modular functions for the subgroup ro(N) of the modular group. The assertion that this question has a positive answer has become known as the Taniyama-Weil conjecture. While the general question seems shrouded in mystery and quite inaccessible at present, one can at least try to verify the conjecture for small numerical values of N. A considerable amount of work has already been done in this direction (cf. [4], [5], [19] -[24], [29]). However, the first nontrivial case of the conjecture, namely N = 11, has not previously been settled. The aim of this note is to determine all elliptic curves of conductor 11 defined over Q and so to verify the conjecture of Taniyama-Weil for N = 11. It is well known that the problem of finding all elliptic curves defined over Q of a given conductor N can be reduced to finding S-integral points on certain associated curves of genus 1; here S is the set of primes dividing N. For certain values of N, these diophantine equations can easily be solved by congruence techniques. However, this elementary approach does not work for N = 11, and we are forced to solve these equations by using some recent sharp inequalities for linear forms in the logarithms of algebraic numbers. The body of this paper is, thus, given over to solving a diophantine equation by Bakers method. Whilst our computations are of course specific to the particular equation we solve, our methods are quite general. As regards the elliptic curves, we employ the usual notation and terminology. For background and more detailed explanation we refer the reader to the surveys of Swinnerton-Dyer and Birch [31] and of Gelbart [12]; see also Mazur and SwinnertonDyer [18]. 1. An elliptic curve E over a field K has a nonsingular plane cubic model (1) y2 + a1xy + a3y = X3 + a2x2 + a4x + a6 Received July 29, 1979; revised September 19, 1979. 1980 Mathematics Subject Classification. Primary 1OF10, 1OD12, 1OB10, 1OB16, 14K07, 12A30. i 1980 American Mathematical Society 0025-5718/80/0000-01 23/

Determinants in the study of Thue's method and curves with prescribed singularities.

04.00 991 This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:17:10 UTC All use subject to http://about.jstor.org/terms 992 M. K. AGRAWAL, J. H. COATES, D. C. HUNT AND A. J. VAN DER POORTEN with the ai in K. If the characteristic char K of K is not 2, we can replace 4(2y +a1x + a3) byy and 4x byx to obtain (2) y2 = x3 + b2x2 + 8b4x + 16b6

The number of lattice rules having given invariants

Our investigations in the 1980s of Thues method yielded determinants that we were only able to analyse successfully in part. We explain the context of our work, recount our experiences, mention our conjectures, and allude to a number of open questions.

Life and Mathematics of Alfred Jacobus van der Poorten (1942–2010)

A lattice rule is a quadrature rule used for the approximation of integrals over the s -dimensional unit cube. Every lattice rule may be characterised by an integer r called the rank of the rule and a set of r positive integers called the invariants. By exploiting the group-theoretic structure of lattice rules we determine the number of distinct lattice rules having given invariants. Some numerical results supporting the theoretical results are included. These numerical results are obtained by calculating the Smith normal form of certain integer matrices.

Free subgroups of free topological groups

Alf van der Poorten was born in the Netherlands in 1942 and educated in Sydney after his family’s move to Australia in 1951. He was based in Sydney for the rest of his life but travelled overseas for professional reasons several times a year from 1975 onwards. Alf was famous for his research in number theory and for his extensive contributions to the mathematics profession both in Australia and overseas.