Topics in number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2009
|
| Series: | Monographs in number theory
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 278 S. |
| ISBN: | 9789812835185 9812835180 |
Staff View
MARC
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Record in the Search Index
| _version_ | 1804138563692920832 |
|---|---|
| adam_text | Contents
Preface
v
Glossary
vii
I Theory of Modular Forms of One Variable
1
1
Group Actions of the Modular Group
3
1.1
The Upper Half-Plane
..................... 3
1.2
The Modular Group
...................... 5
1.3
Modular Forms
......................... 9
1.4
Exercises
............................ 14
2
The Gamma and
Zeta
Functions
17
2.1
The Riemann
Zeta
Function and the Gamma Function
... 17
2.2
Analytic Continuation
..................... 21
2.3
Bernoulli Numbers
....................... 23
2.4
Functional Equation of
Ç(s)
.................. 29
2.5
Exercises
............................ 31
3
Zeta
Functions of Modular Forms
33
3.1
Theta Series as Examples of Modular Forms
......... 33
3.2
Zeta
Functions Attached to Modular Forms
......... 36
3.3 Hecke
Operators on Modular Forms
............. 40
3.4
Exercises
............................ 44
xiii
xjv Contents
4
Dimension Formulas
47
4.1
Dimension Formulas for Modular Forms
........... 47
4.2
Dedekind Eta Function
.................... 50
4.3
Selberg Trace Formula
..................... 54
4.4
Exercises
............................ 59
5
Bernoulli Identities and Applications
61
5.1
Two Classical Bernoulli Identities
............... 61
5.2
Zeta
Functions Associated with Linear Forms
........ 66
5.3
Zeta
Functions Associated with Rational Functions
..... 70
5.4
Kummer s Congruences
.................... 77
5.5
Exercises
............................ 84
6
Euler
Sums and Recent Development
85
6.1
Two Different Approaches for Evaluations
.......... 85
6.2
Analogue of
Euler
Sums
.................... 93
6.2.1
Extended
Euler
sums
[13]............... 93
6.2.2
Generalized
Euler
sums of even weight
[26] ..... 96
6.2.3
Euler
sums over rationally deformed simplices
[28] . 98
6.3
Euler
Sums on Hurwitz
Zeta
Functions
............ 99
6.4
Multiple
Zeta
Values
...................... 107
6.5
Drinfeld Integrals
........................ 119
6.6
Exercises
............................ 126
II Theory of Modular Forms of Several Variables
129
7
Theory of Modular Forms of Several Variables
131
7.1
The Generalized Upper Half-Plane
.............. 131
7.2
The Real Symplectic Group
.................. 133
7.3
The Action of the Symplectic Group
............. 136
7.4
The Generalized Disc
...................... 140
7.5
Exercises
............................ 142
8
The Full Modular Group
145
8.1
The Full Modular Group and Its Subgroups
......... 145
8.2
Regular Elliptic Elements
................... 148
Contents xv
8.3 Siegel
Modular
Forms
..................... 150
8.4
Exercises
............................ 152
9
The Fourier Coefficients of
Eisenstein
Series
155
9.1
A Transformation Formula
................... 155
9.2
Siegel s Main Formula and Its Applications
......... 160
9.3
Theta Series
........................... 162
9.4
An Example to Siegel s Main Theorem
............ 164
9.5
Exercises
............................ 168
10
Theory of Jacobi Forms
171
10.1
Jacobi Forms over Real Numbers
............... 171
10.2
Cayley Numbers
........................ 174
10.3
Jacobi Forms over Cayley Numbers
.............. 177
10.4
Jacobi Forms as Vector-Valued Modular Forms
....... 179
10.5
Examples of Jacobi Forms
................... 183
10.6
Exercises
............................ 187
11 Hecke
Operators and Jacobi Forms
189
11.1 Hecke
Operators on Jacobi Forms
............... 189
11.2 Maaß
Space
........................... 192
11.3
Selberg Trace Formula
..................... 195
11.4
Exercises
............................ 197
12
Singular Modular Forms on the Exceptional Domain
199
12.1
Modular Forms on the Exceptional Domain
......... 199
12.2
Jacobi Forms of Degree Two over Cayley Numbers
..... 203
12.3
Jacobi-Eisenstein Series
.................... 206
12.4
Applications to Singular Modular Forms
........... 208
12.5
Jacobi Cusp Forms
....................... 212
A The p-Adic Numbers 219
A.I The p-Adic Integers
...................... 219
A.2 The p-Adic Valuation
..................... 222
A.3 The Field of p-Adic Numbers
................. 226
A.4 Kummer s Congruences
.................... 228
xvi Contents
В
Weighted Sum Formulae of Multiple
Zeta
Values
235
B.I Introduction and Notations
.................. 235
B.2 The Proof of Theorem B.I.
1
and Its Consequences
..... 238
B.3 The Proof of Theorem B.
1.3
and Generalizations
...... 242
B.4 Generalizations
......................... 244
С
The Density of a Quadratic Form on Cay ley Numbers
251
C.I Preliminaries and Notations
.................. 251
C.2 Interpretations as Densities
.................. 255
C.3 Densities at Finite Places
................... 259
C.4 Densities at Infinite Places
................... 263
Bibliography
269
Index
275
|
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| id | DE-604.BV035274572 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:30:11Z |
| institution | BVB |
| isbn | 9789812835185 9812835180 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017079888 |
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| physical | XVI, 278 S. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | World Scientific |
| record_format | marc |
| series | Monographs in number theory |
| series2 | Monographs in number theory |
| spelling | Eie, Minking 1952- Verfasser (DE-588)140942785 aut Topics in number theory Minking Eie Singapore [u.a.] World Scientific 2009 XVI, 278 S. txt rdacontent n rdamedia nc rdacarrier Monographs in number theory 2 Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Monographs in number theory 2 (DE-604)BV035341434 2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017079888&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Eie, Minking 1952- Topics in number theory Monographs in number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Topics in number theory |
| title_auth | Topics in number theory |
| title_exact_search | Topics in number theory |
| title_full | Topics in number theory Minking Eie |
| title_fullStr | Topics in number theory Minking Eie |
| title_full_unstemmed | Topics in number theory Minking Eie |
| title_short | Topics in number theory |
| title_sort | topics in number theory |
| topic | Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017079888&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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