Modular Units

Daniel S. Kubert, Serge Lang (auth.)0387905170, 9780387905174

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.

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Table of contents :
Front Matter….Pages i-xiii
Distributions on Toroidal Groups….Pages 1-23
Modular Units….Pages 24-57
Quadratic Relations….Pages 58-80
The Siegel Units Are Generators….Pages 81-109
The Cuspidal Divisor Class Group on X ( N )….Pages 110-145
The Cuspidal Divisor Class Group on X 1 ( N )….Pages 146-171
Modular Units on Tate Curves….Pages 172-189
Diophantine Applications….Pages 190-210
Unramified Units….Pages 211-223
More Units in the Modular Function Field….Pages 224-232
Siegel-Robert Units in Arbitrary Class Fields….Pages 233-268
Klein Units in Arbitrary Class Fields….Pages 269-310
Computation of a Unit Index….Pages 311-337
Back Matter….Pages 339-360