SELECTED PAPERS OF YU. I. MANIN

Transcription

1 SELECTED PAPERS OF YU. I. MANIN

2 World Scientific Series in 20th Century Mathematics Published Vol. 1 Vol. 2 Vol. 3 The Neumann Compendium edited by F. Brddy and T. Va'mos 40 Years in Mathematical Physics by L. D. Faddeev Selected Papers of Yu I Manin by Y. Manin Forthcoming A Mathematician and His Mathematical Work Selected Papers of S. S. Chem edited by S. Y. Cheng, P. Li and G. Tian

3 World Scientific Series in 20th Century Mathematics Vol. 3 SELECTED PAPERS OF YU. I. MANIN Yu. I. Manin Max-Planck-Institut fur Mathematik Germany World Scientific Singapore* New Jersey London* Hong Kong

4 Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore USA office: Suite IB, 1060 Main Street, River Edge, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Manin, Yu. I. [Selections, English] Selected papers of Yu. I. Manin / Yu. I. Manin. p. cm. - (World Scientific series in 20th century mathematics ; vol. 3) Includes bibliographical references. ISBN Mathematics. I. Title. II. Series. QA3.M dc CIP British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 1996 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore by Uto-Print

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7 vii PREFACE I aim grateful to the World Scientific Publishers for their suggestion to publish my Selected Papers. This volume contains about one fourth of my articles written before The choice was difficult and it was not governed by any systematic criteria. Probably, I simply picked those which I enjoyed most. It was my joy and privilege to collaborate with many friends and colleagues, and I am thankful to all the co-authors of the papers reprinted here: V. V. Batyrev, A. A. Beilinson, V. G. Drinfeld, J. Franke, V. A. Iskovskikh, M. M. Kapranov, D. R. Lebedev, A. A. Panchishkin, A. 0. Radul, V. V. Schechtman, Yu. Tschinkel, M. M. Vishik, Ju. G. Zarkhin. A considerable part of these texts was initially published in Russian. The English translations are not always of the best quality, but I did not try to improve them, restricting myself to correcting a few mistakes. The Publishers suggested that I supply some comments. For the most part, I have chosen to give references for further reading and to sketch the subsequent development of the ideas presented in my works rather than reminisce about their conception. The reason is that most mathematical texts have a transient value, if any. Their survival is secured only if their content becomes a part of oral tradition, even a commonplace. The author may lament that his ideas were misrepresented, misattributed, and misnamed, but who could care less. Yu. I. Manin. Bonn, October 1995.

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9 . ix CONTENTS Preface vii PART I. Algebraic Geometry The Hasse-Witt matrix of an algebraic curve 3 Russian: Izv. AN SSSR, ser. mat. 25:1 (1961), English: Amer. Math. Soc. Transl. Ser. 2, 45 (1965), Rational points of algebraic curves over functional fields 23 Russian: Izv. AN SSSR, ser. mat. 27:6 (1963), English: Amer. Math. Soc. Transl. Ser. 2, 50 (1966), Letter to the editor 69 Russian: Izv. AN SSSR, ser. mat. 53:2 (1989). English: Math. USSR Izvestija 34:2 (1990), Correspondences, motives, and monoidal transformations 71 Russian: Mat. Sbornik77:4 (1968), English: Math. USSR-Sb. 6 (1968), Three-dimensional quartics and counterexamples to the Liiroth problem (with V. A. Iskovskih) 103 Russian: Mat. Sbornik 86:1 (1971), English: Math. USSR-Sb. 15 (1971), New directions in geometry 129 Russian: Uspekhi Mat. Nauk 39:6 (1984), English: Russian Math. Surveys 39:6 (1984), Arrangements of hyperplanes, higher braid groups and higher Bruhat orders (with V. V. Schechtman) 162 Algebraic Number Theory, Advanced Studies in Pure Math. 17 (1989), , Academic Press PART II. Modular Forms and Diophantine Equations The p-torsion of elliptic curves is uniformly bounded 185 Russian: Izv. AN SSSR, ser. mat. 33:3 (1969), English: Math. USSR-Izv. 3 (1969), Le groupe de Brauer-Grothendieck en geom&rie diophantienne 191 Actes Congr. Int. Math. Nice, Vol. 1 (1971), 401-4H, Gauthier- Villars.

10 X, Parabolic points and zeta-functions of modular curves 202 Russian: Izv. AN SSSR, ser. mat. 36:1 (1972), English: Math. USSR-Izv. 6:1 (1972), Height on families of abelian varieties (with Ju. G. Zarhin) 248 Russian: Mat. Sbornik 89:2 (1972), English: Math. USSR-Sb. 18:2 (1972), Periods of p-adic Schottky groups (with V. G. Drinfeld) 259 J. Reine Angew. Math. 262/263 (1973), Periods of parabolic forms and p-adic Hecke series 268 Russian: Mat. Sbornik 92:3 (1973), English: Math. USSR-Sb. 21:3 (1973), The values of p-adic Hecke series at integer points of the critical strip 291 Russian: Mat. Sbornik 93:4 (1974), English: Math. USSR-Sb. 22:4 (1974), p-adic Hecke series of imaginary quadratic fields (with M. M. Vishik) 298 Russian: Mat. Sbornik 95:5 (1974), English: Math. USSR-Sb. 24:3 (1974), Convolutions of Hecke series and their values at lattice points (with A. A. Panchishkin) 325 Russian: Mat. Sbornik 104:4 (1977), English: Math. USSR-Sb. 33:4 (1977), Rational points of bounded height on Fano varieties (with J. Franke, Yu. Tschinkel) 358 Inv. Math. 95 (1989), Sur le nombre des points rationnels de hauteur bornee des varietes algebriques (with V. V. Batyrev) 373 Math. Ann. 286 (1990), Points of bounded height on del Pezzo surfaces (with Yu. Tschinkel) 390 Compositio Math. 85 (1993), Notes on the arithmetic of Fano threefolds 408 Compositio Math. 85 (1993),

11 xi PART III. Differential Equations and Mathematical Physics Conservation laws and Lax representation of Benney's long wave equations (with D. R. Lebedev) 429 Phys. Lett. A 74 (1979), Gelfand-Dikii Hamiltonian operator and the coadjoint representation of the Volterra group (with D. R. Lebedev) 432 Russian: Funkts. Anal, i ego Prilozhen 13:4 (1979), English: Functional Anal. Appl. 13:4 (1979), A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy (with A. O. Radul) 438 Comm. Math. Phys. 98 (1985), The twistor transformation and algebraic-geometric constructions of solutions of the equations of field theory (with M. M. Kapranov) 451 Russian: Uspekhi Mat. Nauk41:5 (1986), English: Russian Math. Surveys 41:5 (1986), Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super)curves 480 Russian: Funkts. Anal, i ego Prilozhen 20:5 (1986), English: Functional Anal. Appl. 20:3 (1986), The partition function of the Polyakov string can be expressed in terms of theta-functions 483 Phys. Lett. B 172:2 (1986), The Mumford form and the Polyakov measure in string theory (with A. A. Beilinson) 485 Comm. Math. Phys. 107:3 (1986), Some remarks on Koszul algebras and quantum groups 503 Ann. Inst. Fourier 37:4 (1987), Reflections on arithmetical physics 518 Conformal Invariance and String Theory (Poiana Brasov, 1987), , Academic Press, Notes on quantum groups and quantum de Rham complexes 529 Teoreticheskaya i Matematicheskaya Fizika 92:3 (1992), 4^5-450.

12 xii PART IV. Miscellany Mathematics as Metaphor 557 Proc. of ICM, Kyoto 1990, vol. 2, , The AMS of Japan and Springer- Verlag. Arakelov's Drawing 565 Related References and Comments 567 List of Publications by Yu. I. Manin 589