Lectures On Modular Forms - Softcover

Lehner, Joseph J.

Softcover

ISBN 10: 0486812421 ISBN 13: 9780486812427

Verlag: Dover Publications Inc., 2017

Inhaltsangabe

This concise volume presents an expository account of the theory of modular forms and its application to number theory and analysis. Suitable for advanced undergraduates and graduate students in mathematics, the treatment starts with classical material and leads gradually to modern developments. Prerequisites include a grasp of the elements of complex variable theory, group theory, and number theory.
The opening chapters define modular forms, develop their most important properties, and introduce the Hecke modular forms. Subsequent chapters explore the automorphisms of a compact Riemann surface, develop congruences and other arithmetic properties for the Fourier coefficients of Klein's absolute modular invariant, and discuss analogies with the Hecke theory as well as with the Ramanujan congruences for the partition function. Substantial notes at the end of each chapter provide detailed explanations of the text's more difficult points.

Über diesen Titel

Über die Autorin bzw. den Autor

Joseph Lehner (1912–2013) was on the staff of the Los Alamos Scientific Laboratory in the 1950s and later taught at Michigan State University, the University of Maryland, and the University of Pittsburgh. He is the author of Dover's A Short Course in Automorphic Functions.

Von der hinteren Coverseite

This concise volume presents an expository account of the theory of modular forms and its application to number theory and analysis. Suitable for advanced undergraduates and graduate students in mathematics, the treatment starts with classical material and leads gradually to modern developments. Prerequisites include a grasp of the elements of complex variable theory, group theory, and number theory.
The opening chapters define modular forms, develop their most important properties, and introduce the Hecke modular forms. Subsequent chapters explore the automorphisms of a compact Riemann surface, develop congruences and other arithmetic properties for the Fourier coefficients of Klein's absolute modular invariant, and discuss analogies with the Hecke theory as well as with the Ramanujan congruences for the partition function. Substantial notes at the end of each chapter provide detailed explanations of the text's more difficult points.
Dover republication of the edition originally published by the National Bureau of Standards, Washington, D.C., 1969.
www.doverpublications.com

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Lectures on Modular Forms

By Joseph Lehner

Dover Publications, Inc.

Copyright © 2017 Joseph Lehner
All rights reserved.

ISBN: 978-0-486-81242-7

Contents

Foreword, 
Introduction, 
I Modular Forms, 
II Modular Forms with Multiplicative Fourier Coefficients. I., 
III Modular Forms with Multiplicative Fourier Coefficients. II., 
IV Automorphisms of Compact Riemann Surfaces, 
V The Fourier Coefficients of j (t). I., 
VI The Fourier Coefficients of j (t). II., 

CHAPTER 1

Modular Forms

1 The Modular Group and Some Subgroups

Throughout these lectures we shall use the letter G to denote the modular group, which is the group of linear-fractional transformations

t' = at + b/ct + d; a, b, c, d, e Z, ad - bc = 1 (1)

Here Z is the set of rational integers. In group theory this group is referred to as LF (2, Z). Though the transformations (1) are the object of our study, it is much more convenient to use matrices. The group

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is not quite isomorphic to G, but we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

We regard G as the matrix group SL (2, Z) in which each matrix is identified with its negative.

We shall be concerned with certain subgroups of G, all of finite index:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and some others. Here it is necessary to be careful about the relation of the matrix to the transformation. Clearly, if -I is in the matrix group, (2) holds; otherwise the two are isomorphic. The first case occurs in the first 3 groups listed above, but -I]ITY [not member of] G'. In general, we shall make no distinction between the matrix and transformation groups, trusting to the context to keep things straight. For applications to number theory the group G0 (n) is especially important.

We shall denote by G an arbitrary subgroup of finite index in G. The group G is discrete, that is, it contains no infinite sequence of distinct matrices that converges to the identity matrix. Almost all of what we do goes over with minor changes to discrete groups of linear-fractional transformations with real coefficients provided they are finitely generated, but we shall not be concerned with the more general case except in Chapter IV.

The geometric theory of linear-fractional transformations is essential in our work. We shall deal only with linear-fractional transformations with real coefficients and determinant 1; call the group of such transformations O. An element of O maps the upper half-plane H on itself and the real axis on itself, and conversely any transformation with these properties can be written with real coefficients and determinant 1 and so belongs to O. We also have the classification into 3 types (disregarding the identity):

elliptic if |a + d| < 2

parabolic if |a + d| = 2

hyperbolic if |a + d| > 2. (3)

The interpretation as noneuclidean motions is familiar. (cf. ch. I notes, No. 1, p. 16). An elliptic transformation has two complex-conjugate fixed points, one lying in H; a hyperbolic transformation, two real (distinct) fixed points; a parabolic transformation, a single real fixed point. In particular, elliptic elements of G can only have traces 0 or ±1; the former are of order 2, the latter of order 3, and these are the only orders possible for elements of G. We shall consider that all groups of linear-fractional transformations act on H or on the real axis.

The points a, b, [member of] H are said to be G-equivalent (or simply equivalent, when G is understood) if Va = b for some element V [member of] G. By this equivalence relation H is partitioned into mutually disjoint equivalence classes or orbits

Gz = {Vz |V [member of] G}.

The concept of orbit leads to the fundamental region, on the one hand, and to the Riemann surface, on the other. These are both realizations of the orbit space H/G, defined as the set of distinct orbits of G.H/G is the space obtained by identifying points in H that are G-equivalent.

To realize the orbit space in H, select one point from each orbit and call the union of these points a fundamental set for G (relative to H). Since we wish to deal with nice topological sets, we modify this concept slightly and define a fundamental region RG to be an open subset of H which contains no distinct G-equivalent points and whose closure contains a point equivalent to every point of H. That fundamental regions exist for the groups of interest to us admits a simple proof (cf. Gunning, ch. I or Short Course, p. 57). Fundamental regions for G and G(2) are shown in the figures; notice that they are actually regions (i.e., connected), which is not required by the definition. There are, of course, many fundamental regions; in particular, V (R) is one if R is, where V [member of] G. The collection of regions {V (R)|V [member of] G} form a network of nonoverlapping regions which, with their boundary points, fill up H. Examples of these striking geometric configurations may be found in many books.

Let A, B be sets in which a multiplication of elements is defined. Write

C = AB

if the set of products {ab|a [member of] A, b [member of] B} is the set C. Write

C = A · B (4)

if for each c [member of] C we have uniquely c = ab, a [member of] A, b [member of] B. Thus, with C a group, A a subgroup, we have (4), where B is a system of right representatives

Theorem 1.Let G = H · A, where H is a subgroup of finite index in G. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a fundamental region for H. ([bar.R] is the closure of R.)

The almost evident proof can be found in Short Course, Theorem 6D, p. 61f. This fundamental region may not be connected; however, connected fundamental regions do exist for all subgroups.

It is possible to select the fundamental region so that it has other desirable properties. The sides of such a fundamental region are arranged in conjugate pairs, the two sides of a pair being equivalent by a group element. These conjugating transformations generate the group.

For example, in R,G we regard the arc of the circle as consisting of two sides separated by the point i. Then the vertical sides are mapped into each other by S, the curved sides by T, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

are standard notations. (Short Course, p. 37.) The vertices are defined to be points of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where two sides meet; they are arranged in cycles, each cycle being a complete G-equivalence class of points on the boundary of RG. Thus RG has the following cycles: {8}, {i}, {e,]ITL2pi/3, eITL2pi/6}; while RG has the cycles: {8}, {0}, {-1, 1} – see figure on p. 3. If one vertex of a cycle is fixed by a parabolic element P, the other vertices are also fixed points of parabolic elements, for V P V-1 fixes V a if...

Details zur Ausgabe

• Verlag: Dover Publications Inc.
• Erscheinungsdatum: 2017
• Sprache: Englisch
• ISBN 10: 0486812421
• ISBN 13: 9780486812427
• Einband: Taschenbuch
• Anzahl der Seiten: 80
• Kontakt zum Hersteller: Dover; info@galileo-mc.de; ; Dachauer Str. 5-7; F�rstenfeldbruck; 82256; Deutschland