A first course in modular forms:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
2010
|
| Edition: | [Nachdr.] |
| Series: | Graduate texts in mathematics
228 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 436 S. graph. Darst. |
| ISBN: | 9781441920058 |
Staff View
MARC
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| 245 | 1 | 0 | |a A first course in modular forms |c Fred Diamond ; Jerry Shurman |
| 250 | |a [Nachdr.] | ||
| 264 | 1 | |a New York |b Springer |c 2010 | |
| 300 | |a XV, 436 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
........................................................
vii
1
Modular
Forms, Elliptic Curves, and Modular Curves
..... 1
L.
1
First definitions and examples
............................. 1
1.2
Congruence subgroups
................................... 11
1.3
Complex tori
........................................... 24
1.4
Complex tori as elliptic curves
............................ 31
1.5
Modular curves and moduli spaces
......................... 37
2
Modular Curves as Riemann Surfaces
..................... 45
2.1
Topology
............................................... 45
2.2
Charts
................................................. 48
2.3
Elliptic points
........................................... 52
2.4
Cusps
.................................................. 57
2.5
Modular curves and Modularity
........................... 63
3
Dimension Formulas
....................................... 65
3.1
The genus
.............................................. 65
3.2
Automorphic forms
...................................... 71
3.3
Meromorphic differentials
................................. 77
3.4
Divisors and the Riemann-Roch Theorem
.................. 83
3.-5 Dimension formulas for even
к
............................ 85
3.Ö
Dimension formulas for odd A
............................. 89
3.7
More on elliptic points
................................... 92
3.5 More on cusps
.......................................... 98
3.9
More dimension formulas
.................................106
4 Eisenstein
Series
...........................................109
4.1 Eisenstein
series for SL2(Z)
...............................109
4.2 Eisenstein
series for
Г(Х)
when
k> ¿
......................
Ul
4.3
Dirichlet characters. Gauss sums, and eigenspaces
...........116
xiv Contents
4.4
Gamma, zeta,
and
L-functions
............................120
4.5 Eisenstein
series for the eigenspaces when
к
> 3.............126
4.6 Eisenstein
series of weight
2...............................130
4.7
Bernoulli numbers and the Hurwitz
zeta
function
............133
4.8 Eisenstein
series of weight
1...............................138
4.9
The Fourier transform and the Mellin transform
.............143
4.10
Nonholomorphic
Eisenstein
series
..........................147
4.11
Modular forms via theta functions
.........................155
5 Hecke
Operators
...........................................163
5.1
The double coset operator
................................163
5.2
The (d) and Tp operators
.................................168
5.3
The (n) and Tn operators
................................178
5.4
The
Petersson
inner product
..............................181
5.5
Adjoints
of the
Hecke
Operators
...........................183
5.6
Oldforms and Newforms
..................................187
5.7
The Main Lemma
.......................................189
5.8
Eigenforms
.............................................195
5.9
The connection with L-functions
..........................200
5.10
Functional equations
.....................................204
5.11 Eisenstein
series again
...................................205
6
Jacobians and Abelian Varieties
...........................211
6.1
Integration, homology, the Jacobian, and Modularity
.........212
6.2
Maps between Jacobians
.................................217
6.3
Modular Jacobians and
Hecke
operators
....................226
6.4
Algebraic numbers and algebraic integers
...................230
6.5
Algebraic eigenvalues
....................................233
6.6
Eigenforms, Abelian varieties, and Modularity
..............240
7
Modular Curves as Algebraic Curves
......................249
7.1
Elliptic curves as algebraic curves
.........................250
7.2
Algebraic curves and their function fields
...................257
7.3
Divisors on curves
.......................................268
7.4
The Weil pairing algebraically
.............................275
7.5
Function fields over
С
....................................279
7.6
Function fields over
Q
....................................287
7.7
Modular curves as algebraic curves and Modularity
..........290
7.8 Isogenies
algebraically
....................................295
7.9 Hecke
operators algebraically
.............................300
8
The Eichler-Shimura Relation and L-functions
............309
8.1
Elliptic curves in arbitrary characteristic
...................310
8.2
Algebraic curves in arbitrary characteristic
.................317
8.3
Elliptic curves over
Q
and their reductions
.................322
Contents xv
8.4
Elliptic curves over
Q
and their reductions
.................329
8.5
Reduction of algebraic curves and maps
....................336
8.6
Modular curves in characteristic p: Igusa s Theorem
.........347
8.7
The Eichler-Shimura Relation
............................349
8.8
Fourier coefficients, L-functions, and Modularity
.............356
9
Galois Representations
....................................365
9.1
Galois number fields
.....................................366
9.2
The
¿-adic
integers and the
¿-adic
numbers
.................372
9.3
Galois representations
....................................376
9.4
Galois representations and elliptic curves
...................382
9.5
Galois representations and modular forms
..................386
9.6
Galois representations and Modularity
.....................391
Hints and Answers to the Exercises
............................401
List of Symbols
................................................421
Index
..........................................................427
References
.....................................................433
|
| any_adam_object | 1 |
| author | Diamond, Fred 1964- Shurman, Jerry Michael 1959- |
| author_GND | (DE-588)110035459X (DE-588)173095909 |
| author_facet | Diamond, Fred 1964- Shurman, Jerry Michael 1959- |
| author_role | aut aut |
| author_sort | Diamond, Fred 1964- |
| author_variant | f d fd j m s jm jms |
| building | Verbundindex |
| bvnumber | BV036808281 |
| classification_rvk | SK 180 SK 240 |
| ctrlnum | (OCoLC)706020971 (DE-599)BVBBV036808281 |
| discipline | Mathematik |
| edition | [Nachdr.] |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV036808281 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:48:42Z |
| institution | BVB |
| isbn | 9781441920058 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020724317 |
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| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XV, 436 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Diamond, Fred 1964- Verfasser (DE-588)110035459X aut A first course in modular forms Fred Diamond ; Jerry Shurman [Nachdr.] New York Springer 2010 XV, 436 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 228 Algebraische Geometrie (DE-588)4001161-6 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Modulform (DE-588)4128299-1 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Modulform (DE-588)4128299-1 s DE-604 Zahlentheorie (DE-588)4067277-3 s Algebraische Geometrie (DE-588)4001161-6 s Shurman, Jerry Michael 1959- Verfasser (DE-588)173095909 aut Graduate texts in mathematics 228 (DE-604)BV000000067 228 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020724317&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Diamond, Fred 1964- Shurman, Jerry Michael 1959- A first course in modular forms Graduate texts in mathematics Algebraische Geometrie (DE-588)4001161-6 gnd Zahlentheorie (DE-588)4067277-3 gnd Modulform (DE-588)4128299-1 gnd |
| subject_GND | (DE-588)4001161-6 (DE-588)4067277-3 (DE-588)4128299-1 (DE-588)4123623-3 |
| title | A first course in modular forms |
| title_auth | A first course in modular forms |
| title_exact_search | A first course in modular forms |
| title_full | A first course in modular forms Fred Diamond ; Jerry Shurman |
| title_fullStr | A first course in modular forms Fred Diamond ; Jerry Shurman |
| title_full_unstemmed | A first course in modular forms Fred Diamond ; Jerry Shurman |
| title_short | A first course in modular forms |
| title_sort | a first course in modular forms |
| topic | Algebraische Geometrie (DE-588)4001161-6 gnd Zahlentheorie (DE-588)4067277-3 gnd Modulform (DE-588)4128299-1 gnd |
| topic_facet | Algebraische Geometrie Zahlentheorie Modulform Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020724317&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT diamondfred afirstcourseinmodularforms AT shurmanjerrymichael afirstcourseinmodularforms |