Paula Beazley Cohen

On the coefficients of the transformation polynomials for the elliptic modular function

In this article an estimate is obtained for the coefficients of the transformation polynomials for the elliptic modular function. This enables us to answer some questions raised by Mahler in [ 5 ] and [ 6 ].

Generators of quantum stochastic flows and cyclic cohomology

This paper relates cyclic cohomology, as obtained from traces with suitable domains on free product algebras, to the construction, by using the quantum stochastic calculus of Hudson and Parthasarathy, of quantum (stochastic) flows. As an application of the methods of the present article, we show how the characters of finitely summable Fredholm modules, as constructed by Connes, can be recovered from quantum flows in both the specialisation to the pure gauge and to the Brownian case. We also relate the regularized trace of Connes, used in the construction of these characters, to Lindblad generators.

On the Modular Function and Its Importance for Arithmetic

The modular function

On the Non-commutative Torus of Real Dimension Two

j(\tau ) = \exp ( - 2i\pi \tau ) + 744 + \sum\limits_{n = 1}^\infty {a_n \exp (2i\pi n\tau )} , a_n \in Z,

Humbert Surfaces and Transcendence Properties of Automorphic Functions

automorphic with respect to the action of SL(2,Z) on the Poincare upper half plane of those τ ∈ C with positive imaginary part, is very important for the theory of elliptic curves and of modular forms. Indeed, the values of j parametrise the isomorphism classes over C of elliptic curves. In this lecture, we give an introduction to the modular function, and explain in particular a celebrated result of Th. Schneider (1937) which says that the j function takes an algebraic value at an algebraic point τ if and only if τ is imaginary quadratic, that is the associated class of elliptic curves has complex multiplication. We also discuss some more recent results.

A

It is well known that Hall’s transformation factorizes into a composition of two isometric maps to and from a certain completion of the dual of the universal enveloping algebra of the Lie algebra of the initial Lie group. In this paper this fact will be demonstrated by exhibiting each of the maps in turn as the composition of two isometries. For the first map we use classical stochastic calculus, and in particular a stochastic analogue of the Dyson perturbation expansion. For the second map we make use of quantum stochastic calculus, in which the circumambient space is the complexification of the Lie algebra equipped with the ad-invariant inner product. 1991 Mathematics Subject Classification: Primary 81S25; Secondary 44A15. Work supported by EPSRC Grant GR/LG1811 and Alliance Franco-British Joint Research Programme PN96.104. The paper is in final form and no version of it will be published elsewhere.

C^*

The aim of the following notes is to recall how, according to the theory of Alain Connes, one may introduce a differentiable structure on a dense sub-algebra of the irrational rotation C*-algebra introduced in the lectures of Jean Bellissard. This sub-algebra may be viewed as a non-commutative generalisation of the classical torus of real dimension two. We compute the variation of the value at the origin of the meromorphic continuation of the zeta-function of the Laplacian on this algebra as the metric on the non-commutative torus varies within a given conformal class. This manifests the involved intervention into pseudo-differential computations of the non-commutativity, even for the simplest of non-commutative differential objects.