Dewey Class 
515 
Title 
Modular Units ([EBook] /) / by Daniel S. Kubert, Serge Lang. 
Author 
Kubert, Daniel S. 
Added Personal Name 
Lang, Serge author. 
Other name(s) 
SpringerLink (Online service) 
Publication 
New York, NY : : Springer New York : : Imprint: Springer, , 1981. 
Physical Details 
XIV, 360 p. : online resource. 
Series 
Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics 00727830 ; ; 244 
ISBN 
9781475717419 
Summary Note 
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 Nth 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.: 
Contents note 
1 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 SiegelRobert 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. 
System details note 
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users) 
Internet Site 
http://dx.doi.org/10.1007/9781475717419 
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