Extremal Riemann Surfaces From the Proceedings of the AMS Special Session with Related Papers January 4-5, 1995 San Francisco, California J. R. Quine Peter Sarnak Editors http://dx.doi.org/10.1090/conm/201 Selected Titles in This Series 201 J. R. Quine and Peter Sarnak, Editors, Extremal Riemann surfaces, 1997 200 F. Dias, J.-M. Ghidaglia, and J.-C. Saut, Editors, Mathematical problems in the theory of water waves, 1996 199 G. Banaszak, W. Gajda, and P. KrasOli, Editors, Algebraic K-theory, 1996 198 Donald G. Saari and Zhihong Xia, Editors, Hamiltonian dynamics and celestial mechanics, 1996 197 J. E. Bonin, J. G. Oxley, and B. Servatius, Editors, Matroid theory, 1996 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry, 1996 195 Warren Dicks and Enric Ventnra, The group fixed by a family of injective endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, "Lie algebras and their representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and related topics, 1996 192 Tomek Bartoszytiski and Marion Scheepers, Editors, Set theory, 1995 191 Thong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally, Jr., Editors, Representation theory and harmonic analysis, 1995 190 Monrad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F. Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A. M. Marcantognini, G. A. Mendoza, M. D. Moran, A. Octavio, and W. O. Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors, Homotopy theory and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups, 1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara, and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995 185 RaUl E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995 184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on Algebra, 1995 183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995 182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The eech Centennial, 1995 180 David E. Keyes and Jinchao Xu, Editors, Domain decomposition methods in scientific and engineering computing, 1994 179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic geometry and quantization, 1994 178 Helime Barcelo and Gil Kalai, Editors, Jerusalem Combinatorics '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr., Editors, Representation theory and analysis on homogeneous spaces, 1994 176 David Ballard, Foundational aspects of "non" standard mathematics, 1994 175 Paul J. Sally, Jr., Moshe Flato, James Lepowsky, Nicolai Reshetikhin, and Gregg J. Zuckerman, Editors, Mathematical aspects of conformal and topological field theories and quantum groups, 1994 174 Nancy Childress and John W. Jones, Editors, Arithmetic geometry, 1994 173 Robert Brooks, Carolyn Gordon, and Peter Perry, Editors, Geometry of the spectrum, 1994 (Continued in the back of this publication) Extremal Riemann Surfaces CoNTEMPORARY MATHEMATICS 201 Extremal Riemann Surfaces From the Proceedings of the AMS Special Session with Related Papers January 4-5, 1995 San Francisco, California J. R. Quine Peter Sarnak Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor Andy Magid Michael Vogelius Clark Robinson Peter M. Winkler This volume is an outgrowth of the AMS Special Session on Extremal Riemann Sur- faces held at the Annual Meeting of the American Mathematical Society, San Francisco, California, from January 4-5, 1995. 1991 Mathematics Subject Classification. Primary 30Fxxj Secondary 14Hxx, 58Gxx. Library of Congress Cataloging-in-Publication Data Extremal Riemann surfaces / J. R. Quine, Peter Sarnak, editors. p. cm.-(Contemporary mathematics, ISSN 0271-4132 ; 201) "From the proceedings of the AMS special session, January 4-5, 1995, San Francisco, Califor- nia." Includes bibliographical references. ISBN 0-8218-0514-2 (alk. paper) 1. Riemann surfaces-Congresses. 2. Extremal problems (Mathematics)-Congresses. I. Quine, J. R. (John R.), 1943-. II. Sarnak, Peter. III. Series: Contemporary mathemat- ics (American Mathematical Society) ; v. 201. QA333.E96 1996 515'.223-dc20 96-45737 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made bye-mail to reprint-permission~ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) © 1997 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @; The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. 10 9 8 7 6 5 4 3 2 1 0201 00 99 98 97 Contents Preface IX Extremal geometries PETER SARNAK 1 Extremal Riemann surfaces with a large number of systoles PAUL SCHMUTZ SCHALLER 9 On arithmetic genus 2 subgroups of triangle groups M. NAATANEN AND T. KUUSALO 21 Some lattices obtained from Riemann surfaces M. BERNSTEIN AND N. J. A. SLOANE 29 Jacobian of the Picard curve J. R. QUINE 33 Fermat's quartic curve, Klein's curve, and the tetrahedron RUBi E. RODRiGUEZ AND ViCTOR GONZALES-AGUILERA 43 Riemann surfaces admitting large automorphism groups RAVI S. KULKARNI 63 The splitting of some Jacobi varieties using their automorphism groups JOHN F. X. RIES 81 Number theory, theta identities, and modular curves ROBERT BROOKS, HERSHEL M. FARKAS, AND IRWIN KRA 125 Uniformization of some quotients of modular curves ROBERT BROOKS AND YAACOV KOPELIOVICH 155 Teichmiiller disks and Veech's F-structures CLIFFORD J. EARLE AND FREDERICK P. GARDINER 165 On the geometry and spectral asymptotics of degenerating hyperbolic three manifolds JOZEF DODZIUK AND JAY JORGENSON 191 Differential equations for the quadratic periods of abelian differentials DEBRA CURTIS AND MARVIN TRETKOFF 207 vii viii CONTENTS Zeta functions on 8 2 CARLO MORPURGO 213 A minimum problem for heat kernels of flat tori ALBERT BAERNSTEIN II 227 Preface This volume is an outgrowth of the AMS Special Session on Extremal Riemann Surfaces held at the Annual Meeting of the American Mathematical Society in January of 1995 in San Francisco, California. Not all the people invited to talk were able to attend, but each was also asked to contribute a paper for the volume, and we are pleased at this result. The idea for the Special Session first came in 1994 when Peter Sarnak delivered a colloquium lecture at Florida State University. I had become interested in the computation of Jacobians of Riemann surfaces because of a lecture that Cliff Earle had given in Tallahassee in a previous year, and also because of a paper Buser and Sarnak had recently written on period matrices. We discussed questions related to Jacobians and noted connections with recent work of Paul Schmutz Schaller on systoles. At dinner at Angelo's restaurant on the Gulf of Mexico the plan was put in motion to have a Special Session in San Francisco and invite Paul to come to the U.S. and invite all the experts we could think of on extremal problems related to Riemann surfaces, Jacobians, and lattices to attend. Perhaps the Gulf sunset inspired us to make such an ambitious plan, but it was fortunate that most of it worked thanks to the enthusiastic cooperation of everyone we asked to be involved. What is an extremal Riemann surface? This depends on your point of view. The survey paper of Sarnak, first in this volume, gives his perspective. Clearly some function has to be maximized or minimized, and one can choose the function. Next one chooses a domain of surfaces to look at, usually some Teichmiiller space or family of curves, and then local or global extrema are sought. Some of the oldest questions about Riemann surfaces fit this scheme. A Hurwitz surface maximizes the order of the automorphism group for a given genus. A Wiman curve maximizes the order of a cyclic group of automorphisms. The papers of Kulkarni and Ries concern these subjects. One can restrict the domain of investigation to a subset of Teichmiiller space. Natural candidates are the curve families in lower genus. The definitions of these families are based on subgroups of the automorphism group. Varying the param- eters for them, one can study the variation of hyperbolic geometry of the surface and the geometry of the Jacobian. The paper of Ries and that of Rodriguez and GonzaJes-Aguilera give this kind of information for some families of genus 3 curves. In the paper of Curtis and Tretkoff the variation of quadratic periods is studied for a one parameter family of Riemann surfaces. Some more recent extremal questions are analogs of finding the Hermite con- stant in the theory of lattices.