The Digit Principle

A number of constructions in function field arithmetic involve extensions from linear objects using digit expansions. This technique is described here as a method of constructing orthonormal bases in spaces of continuous functions. We illustrate several examples of orthonormal bases from this viewpoint, and we also obtain a concrete model for the continuous functions on the integers of a local field as a quotient of a Tate algebra in countably many variables.Comment: 20 pages, 0 figures, LaTeX, to appear in Journal of Number Theor

Irregular singularities in Liouville theory

Motivated by problems arising in the study of N=2 supersymmetric gauge theories we introduce and study irregular singularities in two-dimensional conformal field theory, here Liouville theory. Irregular singularities are associated to representations of the Virasoro algebra in which a subset of the annihilation part of the algebra act diagonally. In this paper we define natural bases for the space of conformal blocks in the presence of irregular singularities, describe how to calculate their series expansions, and how such conformal blocks can be constructed by some delicate limiting procedure from ordinary conformal blocks. This leads us to a proposal for the structure functions appearing in the decomposition of physical correlation functions with irregular singularities into conformal blocks. Taken together, we get a precise prediction for the partition functions of some Argyres-Douglas type theories on the four-sphere.Comment: 84 pages, 6 figure

Dynamics of charged fluids and 1/L perturbation expansions

Some features of the calculation of fluid dynamo systems in magnetohydrodynamics are studied. In the coupled set of the ordinary linear differential equations for the spherically symmetric $\alpha^2-$dynamos, the problem represented by the presence of the mixed (Robin) boundary conditions is addressed and a new treatment for it is proposed. The perturbation formalism of large$-\ell$ expansions is shown applicable and its main technical steps are outlined.Comment: 16 p

Instantaneous Bethe-Salpeter equation: utmost analytic approach

The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.Comment: LaTeX, 23 pages, 2 figures, version to appear in Phys. Rev.

Power series expansions of modular forms and their interpolation properties

Let x be a CM point on a modular or Shimura curve and p a prime of good reduction, split in the CM field K. We define an expansion of an holomorphic modular form f in the p-adic neighborhood of x and show that the expansion coefficients give information on the p-adic ring of definition of f. Also, we show that letting x vary in its Galois orbit, the expansions coefficients allow to construct a p-adic measure whose moments squared are essentially the values at the centre of symmetry of L-functions of the automorphic representation attached to f based-changed to K and twisted by a suitable family of Grossencharakters for K.Comment: 45 pages. In this new version of the paper the restriction on the weight in the expansion principle in the quaternionic case has been removed. Also, the formula linking the square of the moment to the special value of the L-function has been greatly simplified and made much more explici