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OverviewIn 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. Full Product DetailsAuthor: D. Kubert , S. LangPublisher: Springer-Verlag New York Inc. Imprint: Springer-Verlag New York Inc. Edition: Softcover reprint of hardcover 1st ed. 1981 Volume: 244 Dimensions: Width: 15.50cm , Height: 1.90cm , Length: 23.50cm Weight: 1.160kg ISBN: 9781441928139ISBN 10: 1441928138 Pages: 360 Publication Date: 01 December 2010 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of Contents1 Distributions on Toroidal Groups.- 2 Modular Units.- 3 Quadratic Relations.- 4 The Siegel Units Are Generators.- 5 The Cuspidal Divisor Class Group on X(N).- 6 The Cuspidal Divisor Class Group on X1 (N).- 7 Modular Units on Tate Curves.- 8 Diophantine Applications.- 9 Unramified Units.- 10 More Units in the Modular Function Field.- 11 Siegel-Robert Units in Arbitrary Class Fields.- 12 Klein Units in Arbitrary Class Fields.- 13 Computation of a Unit Index.- Appendix: The Logarithm of the Siegel Functions.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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