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Field name	Details
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 0072-7830 ; ; 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 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.:
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 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.
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/10.1007/978-1-4757-1741-9
Links to Related Works	Subject References: Analysis . Analysis (Mathematics) . Mathematical analysis . Mathematics . Authors: Kubert, Daniel S. . Lang, Serge . Corporate Authors: SpringerLink (Online service) . Series: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics . Classification: 515 .

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