Modular forms: a computational approach
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Math. Soc.
2007
|
| Series: | Graduate studies in mathematics
79 |
| Subjects: | |
| Online Access: | Table of contents only Inhaltsverzeichnis |
| Physical Description: | XV, 268 S. graph. Darst. |
| ISBN: | 0821839608 9780821839607 |
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MARC
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| 100 | 1 | |a Stein, William |d 1974- |e Verfasser |0 (DE-588)137841973 |4 aut | |
| 245 | 1 | 0 | |a Modular forms |b a computational approach |c William Stein ; with an appendix by Paul E. Gunnells |
| 264 | 1 | |a Providence, R.I. |b American Math. Soc. |c 2007 | |
| 300 | |a XV, 268 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 79 | |
| 650 | 4 | |a Espaces algébriques - Informatique - Manuels d'enseignement supérieur | |
| 650 | 4 | |a Formes modulaires - Informatique - Manuels d'enseignement supérieur | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Forms, Modular |x Data processing |v Textbooks | |
| 650 | 4 | |a Algebraic spaces |x Data processing |v Textbooks | |
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Record in the Search Index
| _version_ | 1804136562939658240 |
|---|---|
| adam_text | Contents
Preface
Xl
Chapter
1.
Modular Forms
1
§1.1.
Basic Definitions
1
§1.2.
Modular Forms of Level
1 3
§1.3.
Modular Forms of Any Level
4
§1.4.
Remarks on Congruence Subgroups
7
§1.5.
Applications of Modular Forms
9
§1.6.
Exercises
11
Chapter
2.
Modular Forms of Level
1 13
§2.1.
Examples of Modular Forms of Level
1 13
§2.2.
Structure Theorem for Level
1
Modular Forms
17
§2.3.
The Miller Basis
20
§2.4. Hecke
Operators
22
§2.5.
Computing
Hecke
Operators
26
§2.6.
Fast Computation of Fourier Coefficients
29
§2.7.
Fast Computation of Bernoulli Numbers
29
§2.8.
Exercises
33
Chapter
3.
Modular Forms of Weight
2 35
§3.1. Hecke
Operators
36
§3.2.
Modular Symbols
39
§3.3.
Computing with Modular Symbols
41
vn
viii Contents
§3.4. Hecke Operators 47
§3.5. Computing
the Boundary Map
51
§3.6. Computing
a
Basis
for S2(T0(N))
53
§3.7. Computing 5i2(r0(iV))
Using Eigenvectors
58
§3.8.
Exercises
60
Chapter
4.
Dirichlet Characters
63
§4.1.
The Definition
64
§4.2.
Representing Dirichlet Characters
64
§4.3.
Evaluation of Dirichlet Characters
67
§4.4.
Conductors of Dirichlet Characters
70
§4.5.
The
Kronecker
Symbol
72
§4.6.
Restriction, Extension, and Galois Orbits
75
§4.7.
Alternative Representations of Characters
77
§4.8.
Dirichlet Characters in SAGE
78
§4.9.
Exercises
81
Chapter
5. Eisenstein
Series and Bernoulli Numbers
83
§5.1.
The
Eisenstein Subspace 83
§5.2.
Generalized Bernoulli Numbers
83
§5.3.
Explicit Basis for the
Eisenstein
Subspace
88
§5.4.
Exercises
90
Chapter
6.
Dimension Formulas
91
§6.1.
Modular Forms for T0(N)
92
§6.2.
Modular Forms for Ti(iV)
95
§6.3.
Modular Forms with Character
98
§6.4.
Exercises
102
Chapter
7.
Linear Algebra
103
§7.1.
Echelon Forms of Matrices
103
§7.2.
Rational Reconstruction
105
§7.3.
Echelon Forms over
Q
107
§7.4.
Echelon Forms via Matrix Multiplication
110
§7.5.
Decomposing Spaces under the Action of Matrix
114
§7.6.
Exercises
119
Chapter
8.
General Modular Symbols
121
Contents
ix
§8.1.
Modular Symbols
122
§8.2.
Manin
Symbols
124
§8.3. Hecke
Operators
128
§8.4.
Cuspidal Modular Symbols
133
§8.5.
Pairing Modular Symbols and Modular Forms
137
§8.6.
Degeneracy Maps
142
§8.7.
Explicitly Computing Mk(r0(N))
144
§8.8.
Explicit Examples
147
§8.9.
Refined Algorithm for the Presentation
154
§8.10.
Applications
155
§8.11.
Exercises
156
Chapter
9.
Computing with Newforms
159
§9.1.
Dirichlet Character Decomposition
159
§9.2.
Atkin-Lehner-Li Theory
161
§9.3.
Computing Cusp Forms
165
§9.4.
Congruences between Newforms
170
§9.5.
Exercises
176
Chapter
10.
Computing Periods
177
§10.1.
The Period Map
178
§10.2.
Abelian Varieties Attached to Newforms
178
§10.3.
Extended Modular Symbols
179
§10.4.
Approximating Period Integrals
180
§10.5.
Speeding Convergence Using Atkin-Lehner
183
§10.6.
Computing the Period Mapping
185
§10.7.
All Elliptic Curves of Given Conductor
187
190
191
191
193
194
196
197
197
198
§10.8.
Exercises
Chapter
11.
Solutions to Selected Exercises
§11.1.
Chapter
1
§11.2.
Chapter
2
§11.3.
Chapter
3
§11.4.
Chapter
4
§11.5.
Chapter
5
§11.6.
Chapter
6
§11.7.
Chapter
7
χ
Contents
§11.8.
Chapter
8
199
§11.9.
Chapter
9
201
§11.10.
Chapter
10
201
Appendix A. Computing in Higher Rank
203
§
A
. 1.
Introduction
203
§A.2. Automorphic Forms and Arithmetic Groups
205
ŞA.3.
Combinatorial Models for Group Cohomology
213
§A.4.
Hecke
Operators and Modular Symbols
225
§A.5.
Other Cohomology Groups
232
§A.6. Complements and Open Problems
244
Bibliography
253
Index
265
|
| adam_txt |
Contents
Preface
Xl
Chapter
1.
Modular Forms
1
§1.1.
Basic Definitions
1
§1.2.
Modular Forms of Level
1 3
§1.3.
Modular Forms of Any Level
4
§1.4.
Remarks on Congruence Subgroups
7
§1.5.
Applications of Modular Forms
9
§1.6.
Exercises
11
Chapter
2.
Modular Forms of Level
1 13
§2.1.
Examples of Modular Forms of Level
1 13
§2.2.
Structure Theorem for Level
1
Modular Forms
17
§2.3.
The Miller Basis
20
§2.4. Hecke
Operators
22
§2.5.
Computing
Hecke
Operators
26
§2.6.
Fast Computation of Fourier Coefficients
29
§2.7.
Fast Computation of Bernoulli Numbers
29
§2.8.
Exercises
33
Chapter
3.
Modular Forms of Weight
2 35
§3.1. Hecke
Operators
36
§3.2.
Modular Symbols
39
§3.3.
Computing with Modular Symbols
41
vn
viii Contents
§3.4. Hecke Operators 47
§3.5. Computing
the Boundary Map
51
§3.6. Computing
a
Basis
for S2(T0(N))
53
§3.7. Computing 5i2(r0(iV))
Using Eigenvectors
58
§3.8.
Exercises
60
Chapter
4.
Dirichlet Characters
63
§4.1.
The Definition
64
§4.2.
Representing Dirichlet Characters
64
§4.3.
Evaluation of Dirichlet Characters
67
§4.4.
Conductors of Dirichlet Characters
70
§4.5.
The
Kronecker
Symbol
72
§4.6.
Restriction, Extension, and Galois Orbits
75
§4.7.
Alternative Representations of Characters
77
§4.8.
Dirichlet Characters in SAGE
78
§4.9.
Exercises
81
Chapter
5. Eisenstein
Series and Bernoulli Numbers
83
§5.1.
The
Eisenstein Subspace 83
§5.2.
Generalized Bernoulli Numbers
83
§5.3.
Explicit Basis for the
Eisenstein
Subspace
88
§5.4.
Exercises
90
Chapter
6.
Dimension Formulas
91
§6.1.
Modular Forms for T0(N)
92
§6.2.
Modular Forms for Ti(iV)
95
§6.3.
Modular Forms with Character
98
§6.4.
Exercises
102
Chapter
7.
Linear Algebra
103
§7.1.
Echelon Forms of Matrices
103
§7.2.
Rational Reconstruction
105
§7.3.
Echelon Forms over
Q
107
§7.4.
Echelon Forms via Matrix Multiplication
110
§7.5.
Decomposing Spaces under the Action of Matrix
114
§7.6.
Exercises
119
Chapter
8.
General Modular Symbols
121
Contents
ix
§8.1.
Modular Symbols
122
§8.2.
Manin
Symbols
124
§8.3. Hecke
Operators
128
§8.4.
Cuspidal Modular Symbols
133
§8.5.
Pairing Modular Symbols and Modular Forms
137
§8.6.
Degeneracy Maps
142
§8.7.
Explicitly Computing Mk(r0(N))
144
§8.8.
Explicit Examples
147
§8.9.
Refined Algorithm for the Presentation
154
§8.10.
Applications
155
§8.11.
Exercises
156
Chapter
9.
Computing with Newforms
159
§9.1.
Dirichlet Character Decomposition
159
§9.2.
Atkin-Lehner-Li Theory
161
§9.3.
Computing Cusp Forms
165
§9.4.
Congruences between Newforms
170
§9.5.
Exercises
176
Chapter
10.
Computing Periods
177
§10.1.
The Period Map
178
§10.2.
Abelian Varieties Attached to Newforms
178
§10.3.
Extended Modular Symbols
179
§10.4.
Approximating Period Integrals
180
§10.5.
Speeding Convergence Using Atkin-Lehner
183
§10.6.
Computing the Period Mapping
185
§10.7.
All Elliptic Curves of Given Conductor
187
190
191
191
193
194
196
197
197
198
§10.8.
Exercises
Chapter
11.
Solutions to Selected Exercises
§11.1.
Chapter
1
§11.2.
Chapter
2
§11.3.
Chapter
3
§11.4.
Chapter
4
§11.5.
Chapter
5
§11.6.
Chapter
6
§11.7.
Chapter
7
χ
Contents
§11.8.
Chapter
8
199
§11.9.
Chapter
9
201
§11.10.
Chapter
10
201
Appendix A. Computing in Higher Rank
203
§
A
. 1.
Introduction
203
§A.2. Automorphic Forms and Arithmetic Groups
205
ŞA.3.
Combinatorial Models for Group Cohomology
213
§A.4.
Hecke
Operators and Modular Symbols
225
§A.5.
Other Cohomology Groups
232
§A.6. Complements and Open Problems
244
Bibliography
253
Index
265 |
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| illustrated | Illustrated |
| index_date | 2024-07-02T17:45:52Z |
| indexdate | 2024-07-09T20:58:23Z |
| institution | BVB |
| isbn | 0821839608 9780821839607 |
| language | English |
| lccn | 2006047950 |
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| spelling | Stein, William 1974- Verfasser (DE-588)137841973 aut Modular forms a computational approach William Stein ; with an appendix by Paul E. Gunnells Providence, R.I. American Math. Soc. 2007 XV, 268 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 79 Espaces algébriques - Informatique - Manuels d'enseignement supérieur Formes modulaires - Informatique - Manuels d'enseignement supérieur Datenverarbeitung Forms, Modular Data processing Textbooks Algebraic spaces Data processing Textbooks Modulform (DE-588)4128299-1 gnd rswk-swf Modulform (DE-588)4128299-1 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1803-8 Graduate studies in mathematics 79 (DE-604)BV009739289 79 http://www.loc.gov/catdir/toc/fy0710/2006047950.html Table of contents only Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015681804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stein, William 1974- Modular forms a computational approach Graduate studies in mathematics Espaces algébriques - Informatique - Manuels d'enseignement supérieur Formes modulaires - Informatique - Manuels d'enseignement supérieur Datenverarbeitung Forms, Modular Data processing Textbooks Algebraic spaces Data processing Textbooks Modulform (DE-588)4128299-1 gnd |
| subject_GND | (DE-588)4128299-1 |
| title | Modular forms a computational approach |
| title_auth | Modular forms a computational approach |
| title_exact_search | Modular forms a computational approach |
| title_exact_search_txtP | Modular forms a computational approach |
| title_full | Modular forms a computational approach William Stein ; with an appendix by Paul E. Gunnells |
| title_fullStr | Modular forms a computational approach William Stein ; with an appendix by Paul E. Gunnells |
| title_full_unstemmed | Modular forms a computational approach William Stein ; with an appendix by Paul E. Gunnells |
| title_short | Modular forms |
| title_sort | modular forms a computational approach |
| title_sub | a computational approach |
| topic | Espaces algébriques - Informatique - Manuels d'enseignement supérieur Formes modulaires - Informatique - Manuels d'enseignement supérieur Datenverarbeitung Forms, Modular Data processing Textbooks Algebraic spaces Data processing Textbooks Modulform (DE-588)4128299-1 gnd |
| topic_facet | Espaces algébriques - Informatique - Manuels d'enseignement supérieur Formes modulaires - Informatique - Manuels d'enseignement supérieur Datenverarbeitung Forms, Modular Data processing Textbooks Algebraic spaces Data processing Textbooks Modulform |
| url | http://www.loc.gov/catdir/toc/fy0710/2006047950.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015681804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT steinwilliam modularformsacomputationalapproach |