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Modular Units

  • Authors
  • Daniel S. Kubert
  • Serge Lang

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 244)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Daniel S. Kubert, Serge Lang
    Pages 1-23
  3. Daniel S. Kubert, Serge Lang
    Pages 24-57
  4. Daniel S. Kubert, Serge Lang
    Pages 58-80
  5. Daniel S. Kubert, Serge Lang
    Pages 81-109
  6. Daniel S. Kubert, Serge Lang
    Pages 110-145
  7. Daniel S. Kubert, Serge Lang
    Pages 146-171
  8. Daniel S. Kubert, Serge Lang
    Pages 172-189
  9. Daniel S. Kubert, Serge Lang
    Pages 190-210
  10. Daniel S. Kubert, Serge Lang
    Pages 211-223
  11. Daniel S. Kubert, Serge Lang
    Pages 224-232
  12. Daniel S. Kubert, Serge Lang
    Pages 233-268
  13. Daniel S. Kubert, Serge Lang
    Pages 269-310
  14. Daniel S. Kubert, Serge Lang
    Pages 311-337
  15. Back Matter
    Pages 339-360

About this book

Introduction

In the present book, we have put together the basic theory of the units and cuspidal divisor class group in the modular function fields, developed over the past few years. Let i) be the upper half plane, and N a positive integer. Let r(N) be the subgroup of SL (Z) consisting of those matrices == 1 mod N. Then r(N)\i) 2 is complex analytic isomorphic to an affine curve YeN), whose compactifi­ cation is called the modular curve X(N). The affine ring of regular functions on yeN) over C is the integral closure of C[j] in the function field of X(N) over C. Here j is the classical modular function. However, for arithmetic applications, one considers the curve as defined over the cyclotomic field Q(JlN) of N-th roots of unity, and one takes the integral closure either of Q[j] or Z[j], depending on how much arithmetic one wants to throw in. The units in these rings consist of those modular functions which have no zeros or poles in the upper half plane. The points of X(N) which lie at infinity, that is which do not correspond to points on the above affine set, are called the cusps, because of the way they look in a fundamental domain in the upper half plane. They generate a subgroup of the divisor class group, which turns out to be finite, and is called the cuspidal divisor class group.

Keywords

Arithmetic Divisorenklassengruppe Einheit (Math.) Finite Modular form Modulfunktion function logarithm modular curve

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-1741-9
  • Copyright Information Springer-Verlag New York 1981
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4419-2813-9
  • Online ISBN 978-1-4757-1741-9
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site
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