Introduction to Algebraic Independence Theory
In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on v
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Yuri V. Nesterenko Patrice Philippon (Eds.)
Introduction to Algebraic Independence Theory With contributions from: F. Amoroso, D. Bertrand, W.D. Brownawell, G. Diaz, M. Laurent, Yu.V. Nesterenko, K. Nishioka, P. Philippon, G. Rémond, D. Roy, M. Waldschmidt
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Editors Yuri V. Nesterenko Faculty of Mechanics and Mathematics Moscow University 119899 Moscow, Russia
Patrice Philippon Institut de Mathématiques de Jussieu UMR 7586 du CNRS 4, place Jussieu 75252 Paris Cedex 05, France
E-mail: [email protected]
E-mail: [email protected]
Cataloging-in-Publication Data applied for
Mathematics Subject Classification (2000): 11J91, 11J85, 11J81, 11J89, 11G05, 12H20, 13F20, 14G40, 14G35, 14L10, 30D15, 33B15, 33E05, 34M35, 34M25 ISSN 0075-8434 ISBN 3-540-41496-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10759897 41/3142-543210 - Printed on acid-free paper
Preface
In the last five years there was an essential progress in the development of the of transcendental numbers. A new approach to the arithmetic properties of
theory
values of modular forms and theta-functions time there
was a
It states that the modular function
point
-r
of the
irrationalities.
was
found. First and up to a recent 1941 by Th. Schneider.
unique result of this type established in
j(-r)
has transcendental values at any algebraic from the imaginary quadratic
complex upper-half plane, distinct
It is well known that the value of the modular function at any -r with Imr > 0, is an algebraic number.
imaginary quadratic argument In
1995, K. Barr6-Sirieix, G. Diaz, F. Gramain and G. Philibert proved that the
J(z), connected to j(-r) by the relation j(T) J(e 2"ir), at any algebraic point z of the unit disk is transcendental. In 1969, this assertion was proposed by K. Mahler as a conjecture. The proof rests heavily on the functional modular equations connecting the functions J(z) and J(z') over C for any natural number n. Simultaneously the p-adic analogue of this theorem (conjecture formulated by Yu. Manin, 1971) was proved and it has important consequences in the theory of algebraic numbers. value of the function
=
non-zero
The solution
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