Number Theory A Seminar held at the Graduate School and University C
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Number Theory A Seminar held at the Graduate School and University Center of the City University of New York 1983-84
Edited by D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn and M. B. Nathanson
Springer-Verlag Berlin Heidelberg New York Tokyo
Editors
David V.Chudnovsky Gregory V.Chudnovsky Department of Mathematics, Columbia University New York, NY 10027, USA Harvey Cohn Department of Mathematics, C. U. N. Y., City College New York, NY 10031, USA Melvyn B. Nathanson Department of Mathematics, Rutgers - The State University Newark, NJ 07102, USA
Mathematics Subject Classification: 10-06 ISBN 3-540-15649-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15649-6 Springer-Verlag New York Heidelberg Berlin Tokyo ThIS work IS subject to copyright. All rights are reserved. whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by SpringerNeriag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
INTRODUCTION
This is the second volume of proceedings of the New York Number Theory Seminar.
The Seminar meets every Tuesday
afternoon during the academic year at the Graduate School and University Center of the City University of New York.
The
eleven lectures in this volume are expanded versions of lectures presented to the seminar in 1983-1984.
The
organizers of the Seminar (David and Gregory Chudnovsky. Harvey Cohn and Melvyn Nathanson) hope that the New York Number Theory Seminar will continue to serve the mathematical community as a forum for the presentation of the latest results in higher arithmetic.
TABLE OF CONTENTS
R.T. BUMBY, Hausdorff dimension of sets arising in number theory ••.•••••••••....•.•••••....••••••••••••••.••.•..••..•••• D.V. CHUDNOVSKY, G.V. CHUDNOVSKY, Applications of Padt:! approximations to diophantine Inequalities In values of Gfunct Ions.....................................................
9
D.V. CHUDNOVSKY, G.V. CHUDNOVSKY, Applications of Padt:! approximations to the Grothendleck conjecture on linear differential equations........................................
52
H. COHN, Klein's paradox, the Icosahedron, and ring class fields........................................................
101
L. EHRENPREIS, Transcendental numbers and partial differential equat ions. • • • . . • • • • • • • • • • • • • • • • . • . • . • • . • . • . • • . . • • • • • • • • • • • • • . •
112
D.A. HEJHAL, A classical approach to a well-known spectral correspondence on quatern I on groups...........................
127
W. HOYT, On surfaces associated with an indefinite ternary Iatt I ce. • • • . • . . . . • • . . . • . . • • • . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . • . . .
197
M.I. KNOPP
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