Showing papers in "Bulletin of The London Mathematical Society in 1993"

Lectures on Mechanics

Abstract: lectures on mechanics What to say and what to do when mostly your friends love reading? Are you the one that don't have such hobby? So, it's important for you to start having that hobby. You know, reading is not the force. We're sure that reading will lead you to join in better concept of life. Reading will be a positive activity to do every time. And do you know our friends become fans of lectures on mechanics as the best book to read? Yeah, it's neither an obligation nor order. It is the referred book that will not make you feel disappointed.

Galois Properties of Division Fields of Elliptic Curves

Abstract: Let k be a number field, let k be an algebraic closure, and write G = Gal (k/k) for the relative Galois group. If E is an elliptic curve defined over k, then G acts on the group E(k) of points of E defined over k. In particular, for any positive integer n it acts on the group En of points of E with finite order dividing n. From now on, suppose n — I is prime. We can regard Et as a vector space (of dimension 2) over the finite field F, with / elements, and so there is a natural homomorphism (j>t from G to the corresponding general linear group GL(.Ej). A fundamental result of Serre ([16,17]) says that if E has no complex multiplication over k, then (fr^G) = GL (EJ for all sufficiently large /. In other words, there exists l0, depending only on k and E, such that t{G) = GL(£Z) for all / > /0. Up to now it seems that no general estimate for /0 has been written down. Serre gives a number of results for special classes of elliptic curves. For example, Corollaire 1 of [17, p. 308] yields a simple estimate when k = Q and E is semistable. In his later paper [18] he was able to eliminate the semistability condition by assuming the Generalized Riemann Hypothesis (see Theoreme 22 and Lemme 15, p. 196). But in a talk at the D.-P.-P. Seminaire in April 1988, he did announce an effective estimate in the general case. In this note we give a general upper bound for /„. As Serre himself pointed out during a conference at Schloss Ringberg in July 1988, such a result is a relatively simple deduction from some isogeny estimates proved by us. Indeed, our exposition in Sections 3 and 4 follows closely a talk he gave there on this subject. After recording the necessary isogeny estimates in Section 2, we apply these in Section 3 to rule out some particular possibilities for (p^G). Then in Section 4 we prove our main result by appealing to the group-theoretical analysis of [17]. We also prove two further results of a similar type. In Section 5 we generalize to several elliptic curves, and in Section 6 we consider the corresponding problem for several points of infinite order on a single elliptic curve. To state our main result we define the Weil height of the elliptic curve E as the (absolute logarithmic) Weil height of its y-invariant.

Line-transitive automorphism groups of linear spaces

Abstract: In this paper we prove the following theorem Let S be a linear space Assume that S has an automorphism group G which is line-transitive and point-imprimitive with k < 9 Then S is one of the following:(a) A projective plane of order 4 or 7, (a) One of 2 linear spaces with v = 91 and k = 6, (b) One of 467 linear spaces with v = 729 and k = 8 In all cases the full automorphism group Aut(S) is known

Balanced Presentations of the Trivial Group

Abstract: The group Fn is not the normal closure of fewer than n of its elements (since, to put it briefly, its abelianization FJ[Fn,Fn], where [Fn,Fn] denotes the commutator subgroup, cannot be generated by fewer than n elements). However, there are many w-element subsets {rl9...,rn} of Fn whose normal closure is the whole of Fn: (r r Y = F V i> ••• > n/ nWe shall call such (unordered) n-tuples annihilating for Fn. There would appear to be a great variety of annihilating n-tuples for Fn, ranging from the totally obvious (such as {x1,...,xn} or any other free basis) to the arbitrarily recondite. (We give two moderately subtle examples below.) Note that each annihilating w-tuple {rx,...,rn} yields a finite presentation of zero deficiency of the trivial group:

A New Construction of the Baby Monster and Its Applications

Abstract: In this paper we show how to construct 4370 x 4370 matrices over GF(2), generating Fischer's {3,4}transposition group B, popularly known as the Baby Monster. This for the first time gives a construction which permits effective calculations to be performed in this very large simple group. We use this to give a new existence proof, and describe briefly applications to subgroup structure, geometry, 2-modular characters, and genus actions.