It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

2000 Mathematical Subject Classification:11D72.

Let $f(x, y)$ be a binary cubic form with integral rational coefficients, and suppose that the polynomial $f(x, y)$ is irreducible in $\mathbb{Q}[x, y]$ and no prime divides all the coefficients of $f$. We prove that the set $f(\mathbb{Z}^{2})$ contains infinitely many primes unless $f(a, b)$ is even for each $(a, b)$ in $\mathbb{Z}^{2}$, in which case the set $\frac{1}{2}f(\mathbb{Z}^{2})$ contains infinitely many primes.

2000 Mathematical Subject Classification: primary 11N32; secondary 11N36, 11R44.

This paper proves that a triangulated graph is an outline triple system of even index if and only if it is an amalgamated triple system of even index.

2000 Mathematical Subject Classification: primary 05B07; secondary 05C99.

Let $\varphi:A\to B$ be a homomorphism of finite-dimensional algebras over an algebraically closed field and $\varphi^{(c)}:{\rm mod}_B^c\to{\rm mod}_A^c$ the induced morphism of the associated module schemes for any integer $c\geq 1$. We prove that if the induced functor ${\rm mod\,} B\to{\rm mod\,} A$ is hom-controlled then the restriction of $\varphi^{(c)}$ to any connected component of ${\rm mod}_B^c$ is a composition of a smooth morphism followed by an immersion. Some new results on the types of singularities in the orbit closures of module schemes are also proved.

2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.

A semilinear space S is ultrahomogeneous if each isomorphism between the semilinear structures induced on two finite subsets can be extended to an automorphism of S. We give a complete classification of all finite ultrahomogeneous semilinear spaces. Our theorem extends a result of A. Gardiner on graphs and a result of A. Devillers and J. Doyen on linear spaces.

2000 Mathematical Subject Classification: 05B25, 51E14, 20B25.

Let $\{d_j(n)\}_{j=1}^{\tau(n)}$ denote the increasing sequence of divisors of an integer $n$, and write $\Lambda_k(d)$ for the natural density of the set of those integers $n$ with $d_k(n)=d$. Answering a question raised by Erd\H os, we show that $\Lambda_k(d)$ attains its maximum for $d=K^{1/2+o(1)}$, where $K:=k^{(\log \log k)/\log 2}$ (which, by a theorem of Ramanujan, is roughly the size of the smallest integer having $k$ divisors). A key step of the proof consists in establishing a sublinearity property of the counting function $\Psi_1(x,y)$ of $y$-friable square-free integers not exceeding $x$. This is achieved via a complete study of $\Psi_1(x,y)$ using the saddle-point method, which, in particular, also enables a precise description of the Gaussian behaviour of this function in certain ranges.

2000 Mathematical Subject Classification: 11N25, 11N37.

We investigate the existence of a weak solution $u$ to the quasilinear two-point boundary value problem

(P) $$ - (|u'|^{p-2} u')' = \lambda_k |u|^{p-2} u + f(x) , \quad 0 < x < a ; \quad u(0) = u(a) = 0.$$

We assume that $1 < p < \infty$, $p \not= 2$, $0 < a < \infty$, and that $f \in L^1(0,a)$ is a given function. The number $\lambda_k$ stands for the $k$-th eigenvalue of the one-dimensional $p$-Laplacian. Let $\sin_p(\pi_p x/a)$ denote the eigenfunction associated with $\lambda_1$; then $\sin_p(k \pi_p x/a)$ is the eigenfunction associated with $\lambda_k$. We show the existence of solutions to (P) in the following cases.

(i) When $k=1$ and $f$ satisfies the orthogonality condition

$$\int_0^a f(x) \sin_p(\pi_p x/a) \, dx = 0,$$

the set of solutions is bounded.

(ii) If $k=1$ and $f_t \in L^1(0,a)$ is a continuous family parametrized by $t \in [0,1]$, with

$$\int_0^a f_0(x) \sin_p(\pi_p x/a) \, dx < 0 < \int_0^a f_1(x) \sin_p(\pi_p x/a) \, dx,$$

then there exists some $t^* \in [0,1]$ such that (P) has a solution for $f = f_{t^*}$. Moreover, an appropriate choice of $t^*$ yields a solution $u$ with an arbitrarily large $L^1(0,a)$-norm which means that such $f$ cannot be orthogonal to $\sin_p(\pi_p x/a)$.

(iii) When $k \geq 2$ and $f$ satisfies a set of orthogonality conditions to $\sin_p(k \pi_p x/a)$ and/or $\cos_p(k \pi_p x/a)$ on the subintervals $[ma/k, (m+1)a/k]$ (with $m = 0, 1, \dots, k-1$) and/or $[(m - \frac12 )a/k, (m + \frac12)a/k]$ (with $m = 1, 2, \dots, k-1$), again, the set of solutions is bounded.

(iv) If $k \geq 2$ and $\{ f_t \in L^1(0,a) : t \in [0,1] \}$ is a continuous family satisfying either

$$\int_0^a f_0(x) \sin_p(k \pi_p x/a) \, dx < 0 < \int_0^a f_1(x) \sin_p(k \pi_p x/a) \, dx$$

or another related condition, then there exists some $t^* \in [0,1]$ such that (P) has a solution for $f = f_{t^*}$.

Prüfer's transformation plays the key role in our proofs.

2000 Mathematical Subject Classification: primary 34B16, 47J10; secondary 34L40, 47H30.

Let V be a pencil of curves in ${\bf P}^2$ with one place at infinity, and $X \longrightarrow {\bf P}^2$ the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariski's unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity.

2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

In this paper we study the diophantine equation of the title, which was first introduced by Nagell and Ljunggren during the first half of the twentieth century. We describe a method which allows us, on the one hand when n is fixed, to obtain an upper bound for q, and on the other hand when n and q are fixed, to obtain upper bounds for x and y which are far sharper than those derived from the theory of linear forms in logarithms. We also show how these bounds can be used even when they seem too large for a straightforward enumeration of the remaining possible values of x. By combining all these techniques, we are able to solve the equation in many cases, including the case when n has a prime divisor less than 13, or the case when n has a prime divisor which is less than or equal to 23 and distinct from q.

2000 Mathematical Subject Classification: primary 11D41; secondary 11J86, 11Y50.

Let $L(V)$ be the free Lie algebra on a finite-dimensional vector space $V$ over a field $K$, with homogeneous components $L^n(V)$ for $n \geq 1$. If $G$ is a group and $V$ is a $KG$-module, the action of $G$ extends naturally to $L(V)$, and the $L^n(V)$ become finite-dimensional $KG$-modules, called the Lie powers of $V$. In the decomposition problem, the aim is to identify the isomorphism types of indecomposable $KG$-modules, with their multiplicities, in unrefinable direct decompositions of the Lie powers. This paper is concerned with the case where $G$ has prime order $p$, and $K$ has characteristic $p$. As is well known, there are $p$ indecomposables, denoted here by $J_1,\dots,J_p$, where $J_r$ has dimension $r$. A theory is developed which provides information about the overall module structure of $L(V)$ and gives a recursive method for finding the multiplicities of $J_1,\dots,J_p$ in the Lie powers $L^n(V)$. For example, the theory yields decompositions of $L(V)$ as a direct sum of modules isomorphic either to $J_1$ or to an infinite sum of the form $J_r \oplus J_{p-1} \oplus J_{p-1} \oplus \ldots $ with $r \geq 2$. Closed formulae are obtained for the multiplicities of $J_1,\dots,J_p$ in $L^n(J_p)$ and $L^n(J_{p-1})$. For $r < p-1$, the indecomposables which occur with non-zero multiplicity in $L^n(J_r)$ are identified for all sufficiently large $n$.

2000 Mathematical Subject Classification: 17B01, 20C20.

We describe the current state of progress on the maximal subgroup problem for the Monster sporadic simple group. Any unknown maximal subgroup is an almost simple group whose socle is in one of 19 specified isomorphism classes.

2000 Mathematical Subject Classification:20D08.

In this paper the authors prove the following result.

Let $\alpha$ be an irrational number. Then for any $\varepsilon > 0$, there are infinitely many prime numbers p such that $\| \alpha p \| < p^{-16/49 +\varepsilon}$.

The exponent $\frac{16}{49}$ improves on $\frac{9}{28}$, which was obtained recently by the second author [Sci. China Ser. A 43 (2000) 703-721]. The result is very close to the exponent $\frac{1}{3}$, which can be obtained under the Generalized Riemann Hypothesis.

Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non-trivial for the exponent $\frac{1}{3}$. This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis.

2000 Mathematical Subject Classification: 11N36, 11J71.

This series of papers presents and rigorously analyzes a probabilistic algorithm for finding small prime factors of an integer. The algorithm uses the Jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves. This second paper in the series is concerned with the order of the group of rational points on the Jacobian of a curve of genus 2 defined over a finite field. We prove a result on the distribution of these orders.

2000 Mathematical Subject Classification: 11Y05, 11G10, 11M20, 11N25.

This paper provides a strengthening of the theorems of small cancellation theory. It is proven that disc diagrams contain 'fans' of consecutive 2-cells along their boundaries. The size of these fans is linked to the strength of the small cancellation conditions satisfied by the diagram. A classification result is proven for disc diagrams satisfying small cancellation conditions. Any disc diagram either contains three fans along its boundary, or it is a ladder, or it is a wheel. Similar statements are proven for annular diagrams.

2000 Mathematical Subject Classification: 20F06.

The purpose of this paper is to give a detailed study of the basic theory of C*-categories. The study includes some examples of C*-categories that occur naturally in geometric applications, such as groupoid C*-categories, and C*-categories associated to structures in coarse geometry. We conclude the paper with a brief survey of Hilbert modules over C*-categories.

2000 Mathematical Subject Classification: 18D99, 46L05, 46L08.

This paper continues the study of quantised function algebras $\mathcal{O}_{\epsilon}[G]$ of a semisimple group $G$ at an $\ell$th root of unity $\epsilon$. These algebras were introduced by De Concini and Lyubashenko in 1994, and studied further by De Concini and Procesi and by Gordon, amongst others. Our main purpose here is to increase understanding of the finite-dimensional factor algebras $\mathcal{O}_{\epsilon}[G](g)$, for $g \in G$. We determine the representation type and block structure of these factors, and (for many $g$) describe them up to isomorphism. A series of parallel results is obtained for the quantised Borel algebras $U_{\epsilon}^{\geq 0}$ and $U_{\epsilon}^{\leq 0}$.

2000 Mathematical Subject Classification: 16W35, 17B37.

By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.

2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.

Dedicated to the memory of Helmut Wielandt

In this paper a classification is obtained of all pairs G, H of finite primitive permutation groups such that G < H, the socles of G and H are different, and G and H share a common non-trivial orbital. An application is given to the study of automorphism groups of edge-transitive graphs.

2000 Mathematical Subject Classification: 20B15, 20B25.

Dedicated to the memory of Professor A. I. Kostrikin

The main problem under discussion is to determine, for quasi-simple groups of Lie type G, irreducible representations $\phi$ of G that remain irreducible under reduction modulo the natural prime p. The method is new. It works only for p >3 and for representations $\phi$ that can be realized over an unramified extension of ${\mathbb Q}_p$, the field of p -adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G=SL(2, p). The paper contains a result of independent interest on infinitesimally irrreducible representations $\rho$ of G over an algebraically closed field of characteristic p. Assuming that G is not of rank 1 and $G\neq G_2(5)$, it is proved that either the Jordan normal form of a root element contains a block of size d with 1<d<p -1 or the highest weight of $\rho$ is equal to p -1 times the sum of the fundamental weights.

2000 Mathematical Subject Classification: 20C33, 20G15.

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.

2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.