The mod $p$ representation associated to an elliptic curve is called split or non-split dihedral if its image lies in the normaliser of a split or non-split Cartan subgroup of $\GL_2(\f_p)$, respectively. Let $\xsplit$ and $\xnonsplit$ denote the modular curves which classify elliptic curves with split and non-split dihedral mod $p$ representation, respectively. We call such curves split and non-split{\it Cartan modular curves}. The curve $\xsplit$ is isomorphic to the curve $X_0^+(p^2)$. Using the Selberg trace formula for Hecke operators, we verify that the jacobian of $\xnonsplit$ is isogenous to the new part of the jacobian of $X_0^+(p^2)$.

1991 Mathematics Subject Classification: primary 11G18; secondary 11F72.

Let $S(x,y)$ be the set $S(x,y)=\{ 1 \leq n \leq x : P(n)\leq y\}$, where $P(n)$ denotes the largest prime factor of $n$. We study $E_f(x,y;\theta)=\sum_{n\inS(x,y)}f(n)e^{2\pi in\theta}$, where $f$ is a multiplicative function. When $f=1$ and when $f=\mu$, we widen the domain of uniform approximation using the method of Fouvry and Tenenbaum and making explicit the contribution of the Siegel zero.

Soit $S(x,y)$ l'ensemble $S(x,y)=\{ 1 \leq n \leq x : P(n)\leq y\}$, o\`u$P(n)$ d\'esigne le plus grand facteur premier de $n$. Nous étudions $E_f(x,y;\theta)=\sum_{n\in S(x,y)}f(n)e^{2\pi in\theta}$, lorsque $f$ est une fonction multiplicative. Quand $f=1$ et quand $f=\mu$, nous élargissons le domaine d'approximation uniforme enutilisant la méthode d\'eveloppée par Fouvry et Tenenbaum et en explicitant la contribution du zéro de Siegel.

1991 Mathematics Subject Classification: 11N25, 11N99.

This paper describes parabolic induction, for smooth representations of finite length of the general linear group $\mbox{GL}(N, F)$ of a non-archimedean local field $F$, in terms of a functor between categories of modules over certain affine Hecke algebras, using the categorical equivalences developed in the Bushnell-Kutzko classification of the admissible dual of $\mbox{GL}(N, F)$. This result is used to prove that the Zelevinsky automorphism, which is an involutive automorphism on the representation ring of the category of smooth representations of finite length of $\mbox{GL}(N, F)$, preserves the irreducible representations. The result also implies a simplification of the study of multiplicities of composition factors of induced representations.

1991 Mathematics Subject Classification: 11S37, 22E50.

We describe an algorithm to determine whether a matrix group over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm is based on finding elements with certain properties by independent random selection from the group. Very few matrix groups other than the classical groups contain subsets of elements with the required properties, and a classification of such matrix groups is the principal theoretical basis of the algorithm. The classification, which is proved here, builds on work of Guralnick, Penttila, Saxl and the second author, and depends on the finite simple group classification. Moreover, probability estimates for finding such elements in classical groups are derived, and these are used to give a theoretical analysis of the performance of the algorithm. The algorithm has been implemented in the computer systems {\sf GAP} and{\scMagma}.

1991 Mathematics Subject Classification: primary 20G40, 11E57, 20-04, 60B99; secondary 15A30.

A famous theorem of Donaldson describes a correspondence between $\mbox{SU}(2)$ monopoles over three-dimensional Euclidean space and maps from $\Bbb{CP}^1$ to itself. This paper generalises this to monopoles with arbitrary gauge group $G,$ and new maps from $\Bbb{CP}^1$ to flag manifolds $G/H$ are produced. This is in line with a conjecture of Atiyah and Murray, following a similar result of Atiyah's on hyperbolic monopoles.

Donaldson's approach, also followed by Hurtubise and Murray in previous proofs of many important cases of the current result, depends on a description of monopoles in terms of a system of ordinary differential equations known as Nahm's equations. In contrast, our approach is more direct, and the bulk of the paper is concerned with describing the rational map associated to a particular framed monopole, via solutions to a scattering equation (first introduced by Hitchin) along parallel lines.

A subsidiary section of the paper analyses rational maps into flag manifolds, constructing canonical lifts into larger flag manifolds, and into the (complexified) Lie group. The main result is dependent upon additional work in a companion paper, ‘Construction of Euclidean monopoles’, to appear in the same journal, where a procedure is described for recovering a monopole from its rational map.

1991 Mathematics Subject Classification: 53C80, 58D27, 58E15, 58G11.

This paper describes a procedure for the construction of monopoles on three-dimensional Euclidean space, starting from their rational maps. A companion paper, ‘Euclidean monopoles and rational maps’, to appear in the same journal, describes the assignment to a monopole of a rational map, from $\Bbb{CP}^1$ to a suitable flag manifold. In describing the reverse direction, this paper completes the proof of the main theorem therein.

A construction of monopoles from solutions to Nahm's equations (a system of ordinary differential equations) has been well-known for certain gauge groups for some time. These solutions are hard to construct however, and the equations themselves become increasingly unwieldy when the gauge group is not $\mbox{SU}(2).$

Here, in contrast, a rational map is the only initial data. But whereas one can be reasonably explicit in moving from Nahm data to a monopole, here the monopole is only obtained from the rational map after solving a partial differential equation.

A non-linear flow equation, essentially just the path of steepest descent down the Yang-Mills-Higgs functional, is set up. It is shown that, starting from an ‘approximate monopole’ - constructed explicitly from the rational map - a solution to the flow must exist, and converge to an exact monopole having the desired rational map.

1991 Mathematics Subject Classification: 53C07, 53C80, 58D27, 58E15, 58G11.

Green's function $G(x)$ of a zero mean random walk on the $N$-dimensional integer lattice ($N\geq 2$) is expanded in powers of $1/|x|$ under suitable moment conditions. In particular, we find minimal moment conditions for $G(x)$ to behave like a constant times the Newtonian potential (or logarithmic potential in two dimensions) for large values of $|x|$. Asymptotic estimates of $G(x)$ in dimensions $N\geq 4$, which are valid even when these moment conditions are violated, are computed. Such estimates are applied to determine the Martin boundary of the random walk. If $N= 3$ or $4$ and the random walk has zero mean and finite second moment,the Martin boundary consists of one point, whereas if $N\geq 5$, this is not the case, because non-harmonic functions arise as Martin boundary points for a large class of such random walks. A criterion for when this happens is provided.

1991 Mathematics Subject Classification: 60J15, 60J45, 31C20.