Schottky Groups and Mumford Curves
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817 Lothar Gerritzen Marius van der Put
Schottky Groups and Mumford Curves
Springer-Verlag Berlin Heidelberg New York 1980
Authors Lothar Gerritzen Ruhr-Universit~t Bochum, Institut for Mathematik, Geb~ude NA 2/33 Postfach 102148 4 6 3 0 Bochum 1 Federal Republik of Germany Marius van der Put University of Groningen, Department of Mathematics, W S N - g e b o u w Paddepoel Groningen The Netherlands
A M S Subject Classifications (1980): 10 D30, 14 G 20, 14 H30, 14 H40, 14Kxx, 3 0 G 0 5 , 32 Gxx, 3 2 K 1 0 ISBN 3-540-10229-9 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-10229-9 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data. Gerritzen, Lothar, 1941Schottky groups and Mumford curves. (Lecture notes in mathematics; 817) Bibliography: p. Includes index. 1. Curves, Algebraic. 2. Fields, Algebraic. 3. Discontinuous groups. 4. Automorphic forms. 5. Analytic spaces. I. Put, Marius van der, 1941-joint author. II. Title. III.Series: Lecture notes in mathematics (Berlin); 817. QA3.L28. no. 817. [QA567]. 510s [512'.33] 80-20755 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 the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Introduction The idea of i n v e s t i g a t i n g zations of curves
the p-adic version of classical uniformi-
is due to John Tate who showed that an elliptic
curve over a p-adic
field K whose
j-invariant has absolute value
greater than I can be a n a l y t i c a l l y uniformized.
While Tate's original
paper has never been p u b l i s h e d there are good accounts available,
of his work
see [34].
The g e n e r a l i z a t i o n of the above result of Tare to curves of higher genus has been given by David Mumford "Analytic
rings".
in ~972 in a work called
c o n s t r u c t i o n of d e g e n e r a t i n g curves over complete
The main result of the paper states
that there is a o n e - t o - o n e
correspondence between
a) conjugacy classes of Schottky groups b)
local
F c PGL2(K )
i s o m o r p h i s m classes of curves C over K which are the generic
fibers of normal
schemes over the v a l u a t i o n ring K of K whose
closed fiber is a split degenerate In these Notes we call the curves
curve.
that M u m f o r d has a s s o c i a t e d
to
p-adic Schottky groups Mumford curves.
Manin has called them
When M u m f o r d received the Fields medal
in 1974 his discovery was
S c h o t t k y - M u m f o r d curves
in [26].
praised by Tate when he described
the work of Mumford.
I want to mention briefly p-adic un
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