Mark Sheingorn

A Tribute to Emil Grosswald: Number Theory and Related Analysis

Emil Grosswald was a mathematician of great accomplishment and remarkable breadth of vision. This volume pays tribute to the span of his mathematical interests, which is reflected in the wide range of papers collected here. With contributions by leading contemporary researchers in number theory, modular functions, combinatorics, and related analysis, this book will interest graduate students and specialists in these fields. The high quality of the articles and their close connection to current research trends make this volume a must for any mathematics library.

Parametrizing simple closed geodesy on Γ 3

We exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, 0 3 n , with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for0 3 with Markoff triples. The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.

Hyperbolic reflections on pell's equation

Abstract Let p be a prime э 1 mod(4). This paper shows that arithmetic properties of the fundamental solution ( x 0 , y 0 ) of the Pell equation x 2 − py 2 = −1 are given by geometric properties of the surface H Γ(p) ⌢ Γ 0 (p 2 ) and hyperbolic reflection lines thereon. In particular the length of one such line determines the magnitude of y 0 and the center of the surface either has a cylinder running around it or is vacant, being surrounded by such cylinders, according as p ∤ y 0 or p | y 0 .

Covering the Hecke Triangle Surfaces

In the mid-1980s an equivalence was established between the simple closed geodesics on the Riemann surfaces obtained as quotients of the upper half plane H by any of the following subgroups of the modular group Γ(1) : Γ′, Γ(3), and Γ3. An axis of a hyperbolic element of Γ(1) projects to a simple closed geodesic on one of these surfaces if and only if it does so on the other two.This equivalence was used to obtain a variety of Diophantine and geometric results. In subsequent related investigations, the role of Γ(1) was assumed by the Hecke triangle group Gq for q ≥ 3. (For q = 3, we have Γ(1) = G3.) These works employed the analog of Γ3, denoted Γq.In the context of the Gq, the present paper gives the analog of Γ′, which we denote Θq. As in the case q = 3, we have [Γq:Θq] = 2. A rather full discussion of geometry of Θq\ H is given. In particular, we demonstrate that the equivalence of simple closed geodesics on Γq\ H and Θq\ H does not hold for q ≥ 7.As of this writing, we have not been able to obtain an appropriate analog of Γ(3).