Hilbert modular forms for the fields Q(√5), Q(√13) and Q(√17) [Elektronische Ressource] / vorgelegt von Sebastian Mayer

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• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2007
• Nombre de lectures: 14
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Hilbert Modular Forms
for the fields
p p p
Q( 5), Q( 13) and Q( 17)
Sebastian Mayer
August 2007Hilbert Modular Forms
for the fields
p p p
Q( 5), Q( 13) and Q( 17)
Von der Fakultat¨ fur¨ Mathematik, Informatik und Naturwissenschaften der
Rheinisch- Westfalischen¨ Technischen Hochschule Aachen
zur Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
genehmigte Dissertation
vorgelegt von
Diplom-Mathematiker
Sebastian Mayer
aus Dachau
Berichter: Univ.-Prof. Dr. rer. nat. Aloys Krieg
Univ.-Prof. Dr. rer. nat. Jan Hendrik Bruinier
Tag der mundlichen¨ Prufung:¨ 4. Mai 2007
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar¨ .The image on the cover shows a memorial plaque inside the RWTH Aachen installed to
commemorate the life of Ludwig Otto Blumenthal.To my familyContents
Introduction 9
Prolog about Blumenthal’s life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Blumenthal’s contribution to Hilbert (Blumenthal) modular forms . . . . . . . . . . . 12
Architecture of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1 Definitions of Hilbert Modular Forms 19
1.1 Automorphic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2 Standard Definition of Hilbert Modular Forms . . . . . . . . . . . . . . . . . . . 26
1.2.1 Restriction to Quadratic Number Fields . . . . . . . . . . . . . . . . . . 30
1.2.2 The groups and . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
1.3 Orthogonal Hilbert Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . 33
^1.3.1 The Operation of SL(2;o),G(K) andG(K). . . . . . . . . . . . . . . . 35
1.3.2 The Dual Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
#1.3.3 The quadratic formq on the Dual Lattice and onL =L . . . . . . . . . . 40
2 Some Modular Forms 43
2.1 Hilbert Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Theta Series and Modular Embedding . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 Hilbert Poincare´ Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Elliptic Modular Forms with Character . . . . . . . . . . . . . . . . . . . . . . . 51
2.5 Forms for Congruence Subgroups . . . . . . . . . . . . . . . . 54
2.5.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.3 Eisenstein series of Nebentypus . . . . . . . . . . . . . . . . . . . . . . 65
2.5.4 A basis of the plus space . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.6 Vector Valued Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3 Theory of Borcherds Products 73
3.1 The Theorem of Borcherds, Bruinier and Bundschuh . . . . . . . . . . . . . . . 73
3.2 Integers inK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Weyl Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7Contents
3.4 Weyl Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.5 Hirzebruch-Zagier Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4 Properties of Hilbert Modular Forms 93
4.1 Multiplier Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2 Symmetry and Restriction to the Diagonal . . . . . . . . . . . . . . . . . . . . . 97
4.3 Twisted Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.4 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Calculation of Borcherds Products 111
5.1 A Basis for the Plus Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111p
5.1.1 A Basis in the case Q( 5) . . . . . . . . . . . . . . . . . . . . . . . . . 113p
5.1.2 A Basis in the case Q( 13) . . . . . . . . . . . . . . . . . . . . . . . . 114p
5.1.3 A Basis in the case Q( 17) . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Weight and Multiplier Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.3 Fourier Expansion of Borcherds Products . . . . . . . . . . . . . . . . . . . . . 122
6 Rings of Hilbert Modular Forms 127
6.1 Reduction process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 State of Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128p
6.3 The Ring of Hilbert Modular Forms for Q( 5) . . . . . . . . . . . . . . . . . . 130p
6.4 The Ring of Forms for Q( 13) . . . . . . . . . . . . . . . . . . 132p
6.5 The Ring of Hilbert Modular Forms for Q( 17) . . . . . . . . . . . . . . . . . . 134
7 Perspectives 139
A Tables 145
Bibliography 157
Symbols 163
Index 171
List of Tables 173
8Introduction
Prolog about Blumenthal’s life
It is now over a hundred years that David Hilbert gave his sketches on a new type of modular
functions to his doctoral student Ludwig Otto Blumenthal, who made them the foundation of his
¨Habilitation “Uber Modulfunktionen von mehreren Veranderlic¨ hen” (on modular functions of
several variables). Blumenthal developed the theory of nowadays Hilbert Blumenthal modular
forms in three important directions: he investigated the existence of a fundamental domain,
introduced Poincare´ series and proved two theorems of Weierstraß about the maximal number of
algebraically independent modular functions (cf. [Bl03]). Later on he published a treatment of
theta functions ([Bl04b]) built upon the more detailed part of Hilbert’s notes.
It took some time before further results were obtained, since on the one hand algebraic geometry
and the theory of complex functions had to evolve further (cf. [Ge88, p. 4]), on the other hand
politics was directing almost all scientific efforts towards military purpose. The first world war
was forthcoming and Blumenthal, who was by the time professor at the Aachen University of
Technology (RWTH), became the head of some military weather stations (“Feldwetterwarte”)
and in 1918 worked in the construction of aircrafts, from which arose his paper [Bl18] in 1918
(cf. [BV06, p. 7]). Returning to Aachen he continued mathematical work as well as he started
occasionally to work on some historical topics like, for example, his biography of Hilbert [Bl22]
(cf. [Be58, p. 390] and [BV06, p. 25 et seqq.]).
Blumenthal did not only publish in several mathematical fields, he also was managing editor
of the “Mathematische Annalen” from 1906 to 1938, appointed editor of the “Jahresberichte
der Deutschen Mathematiker-Vereinigung” (DMV) from 1924 to 1933 and he wrote English and
French abstracts for the “Zeitschrift fur¨ Angewandte Mathematik und Mechanik” (ZAMM) from
1933 to 1938 (cf. [BV06, p.14 et seqq.]). Both the resignment from his work at the DMV in
1933 and the end of his work for the “Mathematische Annalen” and the ZAMM were neither
accidently nor voluntary. He had to leave because of his Jewish ancestors, in 1938 the state
banned him from his profession.
thBlumenthal was denunciated by students of being a communist and was arrested on April, 27
in 1933, an error which was corrected 2 weeks later. But he was nevertheless suspended from
ndhis lectures and was removed from office on September, the 22 . The formal reason was his
9INTRODUCTION
Figure 1: Ludwig Otto Blumenthal (1876–1944)
10