The 1-2-3 of modular forms: lectures at a summer school in Nordfjordeid, Norway
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| Beschreibung: | X, 266 Seiten graph. Darst. |
| ISBN: | 9783540741176 9783540741190 3540741178 |
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| 020 | |a 3540741178 |c Pb. : EUR 53.45 (freier Pr.), sfr 87.00 (freier Pr.) |9 3-540-74117-8 | ||
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| 100 | 1 | |a Bruinier, Jan Hendrik |d 1971- |0 (DE-588)12084432X |4 aut | |
| 245 | 1 | 0 | |a The 1-2-3 of modular forms |b lectures at a summer school in Nordfjordeid, Norway |c Jan Hendrik Bruinier ... |
| 246 | 1 | 3 | |a The one-two-three of modular forms |
| 264 | 1 | |a Berlin ; ... |b Springer |c 2008 | |
| 300 | |a X, 266 Seiten |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Universitext | |
| 650 | 4 | |a Forms, Modular | |
| 650 | 4 | |a Hilbert modular surfaces | |
| 650 | 0 | 7 | |a Modulform |0 (DE-588)4128299-1 |2 gnd |9 rswk-swf |
| 655 | 7 | |0 (DE-588)1071861417 |a Konferenzschrift |y 2004 |z Nordfjordeid |2 gnd-content | |
| 689 | 0 | 0 | |a Modulform |0 (DE-588)4128299-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Geer, Gerard van der |d 1950- |0 (DE-588)130489336 |4 aut | |
| 700 | 1 | |a Harder, Günter |d 1938- |0 (DE-588)1011622483 |4 aut | |
| 700 | 1 | |a Zagier, Don |d 1951- |0 (DE-588)120415569 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016444953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016444953 | ||
Datensatz im Suchindex
| _version_ | 1804137567713492992 |
|---|---|
| adam_text | Contents
Elliptic Modular Forms and Their Applications
Don Zagier
...................................................... 1
Foreword
....................................................... 1
1
Basic Definitions
............................................. 3
1.1
Modular Groups, Modular Functions
and Modular Forms
...................................... 3
1.2
The Fundamental Domain of the Full Modular Group
........ 5
4k Finiteness of Class Numbers
.......................... 7
1.3
The Finite Dimensionality of
Мк(Г)
....................... 8
2
First Examples:
Eisenstein
Series and the Discriminant Function
... 12
2.1 Eisenstein
Series and the Ring Structure of Af*(/ )
.......... 12
2.2
Fourier Expansions of
Eisenstein
Series
..................... 15
4k Identities Involving Sums of Powers of Divisors
.......... 18
2.3
The
Eisenstein
Series of Weight
2.......................... 18
2.4
The Discriminant Function and Cusp Forms
................. 20
4k Congruences for
τ(η)
................................ 23
3
Theta Series
................................................. 24
3.1
Jacobi s Theta Series
..................................... 25
4k Sums of Two and Four Squares
........................ 26
4k The Kac-Wakimoto Conjecture
....................... 31
3.2
Theta Series in Many Variables
............................ 31
4k Invariants of Even Unimodular Lattices
................ 33
4k Drums Whose Shape One Cannot Hear
................. 36
4 Hecke
Eigenforms and L-series
................................. 37
4.1 Hecke
Theory
........................................... 37
4.2
L-series of Eigenforms
.................................... 39
4.3
Modular Forms and Algebraic Number Theory
.............. 41
4k Binary Quadratic Forms of Discriminant
—23........... 42
4.4
Modular Forms Associated to Elliptic Curves
and Other Varieties
...................................... 44
VIII Contents
φ
Fermaťs
Last Theorem............................... 46
5
Modular
Forms and Differential Operators
....................... 48
5.1
Derivatives of Modular Forms
............................. 48
φ
Modular Forms Satisfy Non-Linear Differential Equations
. 49
φ
Moments of Periodic Functions
........................ 50
5.2
Rankin-Cohen Brackets and Cohen-Kuznetsov Series
........ 53
φ
Further Identities for Sums of Powers of Divisors
........ 56
φ
Exotic Multiplications of Modular Forms
............... 56
5.3
Quasimodular Forms
..................................... 58
φ
Counting Ramified Coverings of the Toms
.............. 60
5.4
Linear Differential Equations and Modular Forms
............ 61
φ
The Irrationality of
ζ(3)
.............................. 64
4k An Example Coming from Percolation Theory
.......... 66
6
Singular Moduli and Complex Multiplication
.................... 66
6.1
Algebraicity of Singular Moduli
............................ 67
φ
Strange Approximations to
тг
......................... 73
φ
Computing Class Numbers
........................... 74
φ
Explicit Class Field Theory for Imaginary Quadratic Fields
75
φ
Solutions of Diophantine Equations
.................... 76
6.2
Norms and Traces of Singular Moduli
...................... 77
φ
Heights of Heegner Points
............................ 79
φ
The Borcherds Product Formula
....................... 83
6.3
Periods and Taylor Expansions of Modular Forms
............ 83
φ
Two Transcendence Results
........................... 85
φ
Hurwitz
Numbers
................................... 85
φ
Generalized
Hurwitz
Numbers
......................... 89
6.4
CM Elliptic Curves and CM Modular Forms
................ 90
φ
Factorization, Primality Testing, and Cryptography
...... 92
φ
Central Values of
Hecke
L-Series
...................... 95
φ
Which Primes are Sums of Two Cubes?
................ 97
References and Further Reading
................................... 99
Hubert
Modular Forms and Their Applications
Jan Hendrik
Bruinier.............................................
105
Introduction
.....................................................105
1
Hubert Modular Surfaces
......................................106
1.1
The Hubert Modular Group
...............................106
1.2
The Baily-Borel Compactification
.........................109
Siegel
Domains
......................................
Ill
1.3
Hubert Modular Forms
...................................113
1.4
Мк{Г)
is Finite Dimensional
..............................118
1.5 Eisenstein
Series
.........................................119
Restriction to the Diagonal
............................122
The Example Q( /5)
..................................123
Contents
IX
1.6
The ¿-function of a Hubert Modular Form
..................125
2
The Orthogonal Group
0(2,
n)
.................................127
2.1
Quadratic Forms
........................................128
2.2
The Clifford Algebra
.....................................129
2.3
The Spin Group
.........................................133
Quadratic Spaces in Dimension Four
....................135
2.4
Rational Quadratic Spaces of Type
(2,
n)
...................136
The Grassmannian Model
.............................136
The
Projective
Model
.................................137
The Tube Domain Model
..............................137
Lattices
.............................................138
Heegner Divisors
.....................................140
2.5
Modular Forms for
0(2,
n)
................................140
2.6
The
Siegel Theta
Function
................................141
2.7
The Hubert Modular Group as an Orthogonal Group
.........143
Hirzebruch- Zagier Divisors
............................145
3
Additive and Multiplicative Liftings
............................146
3.1
The Doi-Naganuma Lift
..................................146
3.2
Borcherds Products
......................................150
Local Borcherds Products
.............................150
The Borcherds Lift
...................................154
Obstructions
........................................158
Examples
...........................................160
3.3
Automorphic Green Functions
.............................162
A Second Approach
..................................167
3.4
CM Values of Hubert Modular Functions
...................168
Singular Moduli
......................................168
CM Extensions
......................................171
CM Cycles
..........................................172
CM Values of Borcherds Products
......................173
Examples
...........................................175
References
......................................................176
Siegel
Modular Forms and Their Applications
Gerard van
der Geer.............................................
181
1
Introduction
.................................................181
2
The
Siegel
Modular Group
....................................183
3
Modular Forms
..............................................187
4
The Fourier Expansion of a Modular Form
......................189
5
The
Siegel
Operator and
Eisenstein
Series
.......................192
6
Singular Forms
...............................................194
7
Theta Series
.................................................195
8
The Fourier- Jacobi Development of
a
Siegel
Modular Form
........196
9
The Ring of Classical
Siegel
Modular Forms for Genus Two
........198
X
Contents
10
Moduli of Principally Polarized Complex Abelian Varieties
........201
11
Compactifications
............................................204
12
Intermezzo: Roots and Representations
..........................207
13
Vector Bundles Defined by Representations
......................209
14
Holomorphic Differential Forms
................................210
15
Cusp Forms and Geometry
....................................212
16
The Classical
Hecke
Algebra
.............,.....................213
17
The Satake Isomorphism
......................................215
18
Relations in the
Hecke
Algebra
.................................218
19
Satake Parameters
............................................219
20
i-functions
..................................................220
21
Liftings
.....................................................221
22
The Moduli Space of Principally Polarized Abelian Varieties
.......226
23
Elliptic Curves over Finite Fields
...............................226
24
Counting Points on Curves of Genus
2..........................230
25
The Ring of Vector-Valued
Siegel
Modular Forms for Genus
2......232
26
Harder s Conjecture
..........................................235
27
Evidence for Harder s Conjecture
...............................237
References
......................................................241
A Congruence Between
a
Siegel
and an Elliptic Modular Form
Günter
Harder
...................................................247
1
Elliptic and
Siegel
Modular Forms
..............................247
2
The
Hecke
Algebra and a Congruence
...........................250
3
The Special Values of the L-function
...........................252
4
Cohomology with Coefficients
..................................253
5
Why the Denominator?
.......................................257
6
Arithmetic Implications
.......................................258
References
......................................................259
Appendix
.......................................................260
Index
..........................................................263
|
| adam_txt |
Contents
Elliptic Modular Forms and Their Applications
Don Zagier
. 1
Foreword
. 1
1
Basic Definitions
. 3
1.1
Modular Groups, Modular Functions
and Modular Forms
. 3
1.2
The Fundamental Domain of the Full Modular Group
. 5
4k Finiteness of Class Numbers
. 7
1.3
The Finite Dimensionality of
Мк(Г)
. 8
2
First Examples:
Eisenstein
Series and the Discriminant Function
. 12
2.1 Eisenstein
Series and the Ring Structure of Af*(/\)
. 12
2.2
Fourier Expansions of
Eisenstein
Series
. 15
4k Identities Involving Sums of Powers of Divisors
. 18
2.3
The
Eisenstein
Series of Weight
2. 18
2.4
The Discriminant Function and Cusp Forms
. 20
4k Congruences for
τ(η)
. 23
3
Theta Series
. 24
3.1
Jacobi's Theta Series
. 25
4k Sums of Two and Four Squares
. 26
4k The Kac-Wakimoto Conjecture
. 31
3.2
Theta Series in Many Variables
. 31
4k Invariants of Even Unimodular Lattices
. 33
4k Drums Whose Shape One Cannot Hear
. 36
4 Hecke
Eigenforms and L-series
. 37
4.1 Hecke
Theory
. 37
4.2
L-series of Eigenforms
. 39
4.3
Modular Forms and Algebraic Number Theory
. 41
4k Binary Quadratic Forms of Discriminant
—23. 42
4.4
Modular Forms Associated to Elliptic Curves
and Other Varieties
. 44
VIII Contents
φ
Fermaťs
Last Theorem. 46
5
Modular
Forms and Differential Operators
. 48
5.1
Derivatives of Modular Forms
. 48
φ
Modular Forms Satisfy Non-Linear Differential Equations
. 49
φ
Moments of Periodic Functions
. 50
5.2
Rankin-Cohen Brackets and Cohen-Kuznetsov Series
. 53
φ
Further Identities for Sums of Powers of Divisors
. 56
φ
Exotic Multiplications of Modular Forms
. 56
5.3
Quasimodular Forms
. 58
φ
Counting Ramified Coverings of the Toms
. 60
5.4
Linear Differential Equations and Modular Forms
. 61
φ
The Irrationality of
ζ(3)
. 64
4k An Example Coming from Percolation Theory
. 66
6
Singular Moduli and Complex Multiplication
. 66
6.1
Algebraicity of Singular Moduli
. 67
φ
Strange Approximations to
тг
. 73
φ
Computing Class Numbers
. 74
φ
Explicit Class Field Theory for Imaginary Quadratic Fields
75
φ
Solutions of Diophantine Equations
. 76
6.2
Norms and Traces of Singular Moduli
. 77
φ
Heights of Heegner Points
. 79
φ
The Borcherds Product Formula
. 83
6.3
Periods and Taylor Expansions of Modular Forms
. 83
φ
Two Transcendence Results
. 85
φ
Hurwitz
Numbers
. 85
φ
Generalized
Hurwitz
Numbers
. 89
6.4
CM Elliptic Curves and CM Modular Forms
. 90
φ
Factorization, Primality Testing, and Cryptography
. 92
φ
Central Values of
Hecke
L-Series
. 95
φ
Which Primes are Sums of Two Cubes?
. 97
References and Further Reading
. 99
Hubert
Modular Forms and Their Applications
Jan Hendrik
Bruinier.
105
Introduction
.105
1
Hubert Modular Surfaces
.106
1.1
The Hubert Modular Group
.106
1.2
The Baily-Borel Compactification
.109
Siegel
Domains
.
Ill
1.3
Hubert Modular Forms
.113
1.4
Мк{Г)
is Finite Dimensional
.118
1.5 Eisenstein
Series
.119
Restriction to the Diagonal
.122
The Example Q(\/5)
.123
Contents
IX
1.6
The ¿-function of a Hubert Modular Form
.125
2
The Orthogonal Group
0(2,
n)
.127
2.1
Quadratic Forms
.128
2.2
The Clifford Algebra
.129
2.3
The Spin Group
.133
Quadratic Spaces in Dimension Four
.135
2.4
Rational Quadratic Spaces of Type
(2,
n)
.136
The Grassmannian Model
.136
The
Projective
Model
.137
The Tube Domain Model
.137
Lattices
.138
Heegner Divisors
.140
2.5
Modular Forms for
0(2,
n)
.140
2.6
The
Siegel Theta
Function
.141
2.7
The Hubert Modular Group as an Orthogonal Group
.143
Hirzebruch- Zagier Divisors
.145
3
Additive and Multiplicative Liftings
.146
3.1
The Doi-Naganuma Lift
.146
3.2
Borcherds Products
.150
Local Borcherds Products
.150
The Borcherds Lift
.154
Obstructions
.158
Examples
.160
3.3
Automorphic Green Functions
.162
A Second Approach
.167
3.4
CM Values of Hubert Modular Functions
.168
Singular Moduli
.168
CM Extensions
.171
CM Cycles
.172
CM Values of Borcherds Products
.173
Examples
.175
References
.176
Siegel
Modular Forms and Their Applications
Gerard van
der Geer.
181
1
Introduction
.181
2
The
Siegel
Modular Group
.183
3
Modular Forms
.187
4
The Fourier Expansion of a Modular Form
.189
5
The
Siegel
Operator and
Eisenstein
Series
.192
6
Singular Forms
.194
7
Theta Series
.195
8
The Fourier- Jacobi Development of
a
Siegel
Modular Form
.196
9
The Ring of Classical
Siegel
Modular Forms for Genus Two
.198
X
Contents
10
Moduli of Principally Polarized Complex Abelian Varieties
.201
11
Compactifications
.204
12
Intermezzo: Roots and Representations
.207
13
Vector Bundles Defined by Representations
.209
14
Holomorphic Differential Forms
.210
15
Cusp Forms and Geometry
.212
16
The Classical
Hecke
Algebra
.,.213
17
The Satake Isomorphism
.215
18
Relations in the
Hecke
Algebra
.218
19
Satake Parameters
.219
20
i-functions
.220
21
Liftings
.221
22
The Moduli Space of Principally Polarized Abelian Varieties
.226
23
Elliptic Curves over Finite Fields
.226
24
Counting Points on Curves of Genus
2.230
25
The Ring of Vector-Valued
Siegel
Modular Forms for Genus
2.232
26
Harder's Conjecture
.235
27
Evidence for Harder's Conjecture
.237
References
.241
A Congruence Between
a
Siegel
and an Elliptic Modular Form
Günter
Harder
.247
1
Elliptic and
Siegel
Modular Forms
.247
2
The
Hecke
Algebra and a Congruence
.250
3
The Special Values of the L-function
.252
4
Cohomology with Coefficients
.253
5
Why the Denominator?
.257
6
Arithmetic Implications
.258
References
.259
Appendix
.260
Index
.263 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Bruinier, Jan Hendrik 1971- Geer, Gerard van der 1950- Harder, Günter 1938- Zagier, Don 1951- |
| author_GND | (DE-588)12084432X (DE-588)130489336 (DE-588)1011622483 (DE-588)120415569 |
| author_facet | Bruinier, Jan Hendrik 1971- Geer, Gerard van der 1950- Harder, Günter 1938- Zagier, Don 1951- |
| author_role | aut aut aut aut |
| author_sort | Bruinier, Jan Hendrik 1971- |
| author_variant | j h b jh jhb g v d g gvd gvdg g h gh d z dz |
| building | Verbundindex |
| bvnumber | BV023259735 |
| callnumber-first | Q - Science |
| callnumber-label | QA243 |
| callnumber-raw | QA243 |
| callnumber-search | QA243 |
| callnumber-sort | QA 3243 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SD 2004 SK 180 SK 240 |
| classification_tum | MAT 103f |
| ctrlnum | (OCoLC)173239471 (DE-599)DNB985542098 |
| dewey-full | 512.73 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.73 |
| dewey-search | 512.73 |
| dewey-sort | 3512.73 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| genre | (DE-588)1071861417 Konferenzschrift 2004 Nordfjordeid gnd-content |
| genre_facet | Konferenzschrift 2004 Nordfjordeid |
| id | DE-604.BV023259735 |
| illustrated | Illustrated |
| index_date | 2024-07-02T20:31:29Z |
| indexdate | 2024-07-09T21:14:21Z |
| institution | BVB |
| isbn | 9783540741176 9783540741190 3540741178 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016444953 |
| oclc_num | 173239471 |
| open_access_boolean | |
| owner | DE-20 DE-384 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-703 DE-91G DE-BY-TUM DE-188 DE-83 DE-706 |
| owner_facet | DE-20 DE-384 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-11 DE-703 DE-91G DE-BY-TUM DE-188 DE-83 DE-706 |
| physical | X, 266 Seiten graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Bruinier, Jan Hendrik 1971- (DE-588)12084432X aut The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway Jan Hendrik Bruinier ... The one-two-three of modular forms Berlin ; ... Springer 2008 X, 266 Seiten graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Forms, Modular Hilbert modular surfaces Modulform (DE-588)4128299-1 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2004 Nordfjordeid gnd-content Modulform (DE-588)4128299-1 s DE-604 Geer, Gerard van der 1950- (DE-588)130489336 aut Harder, Günter 1938- (DE-588)1011622483 aut Zagier, Don 1951- (DE-588)120415569 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016444953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bruinier, Jan Hendrik 1971- Geer, Gerard van der 1950- Harder, Günter 1938- Zagier, Don 1951- The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway Forms, Modular Hilbert modular surfaces Modulform (DE-588)4128299-1 gnd |
| subject_GND | (DE-588)4128299-1 (DE-588)1071861417 |
| title | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway |
| title_alt | The one-two-three of modular forms |
| title_auth | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway |
| title_exact_search | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway |
| title_exact_search_txtP | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway |
| title_full | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway Jan Hendrik Bruinier ... |
| title_fullStr | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway Jan Hendrik Bruinier ... |
| title_full_unstemmed | The 1-2-3 of modular forms lectures at a summer school in Nordfjordeid, Norway Jan Hendrik Bruinier ... |
| title_short | The 1-2-3 of modular forms |
| title_sort | the 1 2 3 of modular forms lectures at a summer school in nordfjordeid norway |
| title_sub | lectures at a summer school in Nordfjordeid, Norway |
| topic | Forms, Modular Hilbert modular surfaces Modulform (DE-588)4128299-1 gnd |
| topic_facet | Forms, Modular Hilbert modular surfaces Modulform Konferenzschrift 2004 Nordfjordeid |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016444953&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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