Jacobi-Like Forms, Pseudodifferential Operators, and Quasimodular Forms

307 Pages·2019·1.969 MB·English

by  Young Ju Choie, Min Ho Lee

Springer Monographs in Mathematics
YoungJu Choie
Min Ho Lee
Jacobi-Like Forms,
Pseudodifferential
Operators, and
Quasimodular
Forms
Springer Monographs in Mathematics
Editors-in-Chief
Isabelle Gallagher, Paris, France
Minhyong Kim, Oxford, UK
Series Editors
Sheldon Axler, San Francisco, USA
Mark Braverman, Princeton, USA
Maria Chudnovsky, Princeton, USA
Tadahisa Funaki, Tokyo, Japan
Sinan C. Güntürk, New York, USA
Claude Le Bris, Marne la Vallée, France
Pascal Massart, Orsay, France
Alberto A. Pinto, Porto, Portugal
Gabriella Pinzari, Padova, Italy
Ken Ribet, Berkeley, USA
René Schilling, Dresden, Germany
Panagiotis Souganidis, Chicago, USA
Endre Süli, Oxford, UK
Shmuel Weinberger, Chicago, USA
Boris Zilber, Oxford, UK
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YoungJu Choie Min Ho Lee
(cid:129)
Jacobi-Like Forms,
Pseudodifferential Operators,
and Quasimodular Forms
123
YoungJu Choie Min HoLee
Department ofMathematics Department ofMathematics
POSTECH University of Northern Iowa
Pohang,Korea (Republicof) Cedar Falls,IA, USA
ISSN 1439-7382 ISSN 2196-9922 (electronic)
SpringerMonographs inMathematics
ISBN978-3-030-29122-8 ISBN978-3-030-29123-5 (eBook)
https://doi.org/10.1007/978-3-030-29123-5
MathematicsSubjectClassification(2010): 11F11,11F50
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Preface
This book is concerned with various topics centered around connections of
quasimodular forms with Jacobi-like forms and automorphic pseudodifferen-
tialoperatorsandisintendedforresearchersandgraduatestudentsinnumber
theory.Inparticular,itcontainsadetailedexpositionofsuchconnectionsfor
the first time in book form.
Modular forms for a discrete subgroup Γ of SL(2,R) have been playing
an important role in number theory for a long time. Quasimodular forms
generalizemodularformsandwereintroducedbyKanekoandZagierin1995
and have been studied in connection with various topics in number theory
and other areas of mathematics in recent years.
Jacobi-likeformsforΓ areformalpowerseries,whosecoefficientsareholo-
morphic functions on the Poincar´e upper half plane H, invariant under a
certain right action of Γ. They are also closely linked to pseudodifferential
operators, which are formal Laurent series in ∂−1 having holomorphic func-
tions on H as coefficients. The operation of SL(2,R) on H given by linear
fractional transformations determines a natural right action of SL(2,R) on
the space of pseudodifferential operators, and pseudodifferential operators
that are invariant under the restriction of this action to Γ are automorphic
pseudodifferential operators.
There are mutual correspondences among Jacobi-like forms, automorphic
pseudodifferentialoperators,andsequencesofmodularformsofcertaintype.
Ontheotherhandtheseobjectsarecloselylinkedtoquasimodularformsdue
tothefactthateachcoefficientfunctionofaJacobi-likeformoranautomor-
phic pseudodifferential operator is a quasimodular form. As a result, there is
anaturalprojectionmapcarryingaJacobi-likeformtooneofitscoefficients.
On the other hand, this map has a right inverse which may be regarded as
a lifting map carrying a quasimodular form to a Jacobi-like form and having
thegivenquasimodularformasoneofitscoefficients.Similarprojectionand
lifting maps can also be considered between automorphic pseudodifferential
operators and quasimodular forms.
v
vi Preface
The main goal of this book is to explore relations among quasimodular
forms, Jacobi-like forms, automorphic pseudodifferential operators and se-
quences of modular forms and discuss various applications of such relations.
The first author is partially supported by NRF-2018R1A4A1023590 and
NRF-2017R1A2B2001807.
Pohang, Korea, September 30, 2019 YoungJu Choie
Cedar Falls, Iowa, USA September 30, 2019 Min Ho Lee
Contents
Introduction.................................................. 1
1 Formal Power Series and Pseudodifferential Operators.... 7
1.1 Correspondences of Power Series ......................... 7
1.2 Actions of SL(2,R) on Power Series ...................... 12
1.3 Jacobi-like Forms and Modular Series ..................... 18
1.4 Pseudodifferential Operators ............................. 24
2 Jacobi-like Forms and Pseudodifferential Operators....... 31
2.1 Power Series Bundles ................................... 31
2.2 Automorphic Pseudodifferential Operators of Mixed Weight.. 36
2.3 Bundles of Pseudodifferential Operators ................... 40
2.4 Poincar´e Series......................................... 43
2.5 Linear Maps and Rankin–Cohen Brackets ................. 46
3 Hecke Operators.......................................... 53
3.1 Jacobi-like Forms, Modular Series, and Hecke Operators..... 53
3.2 Hecke Operators on Pseudodifferential Operators ........... 60
3.3 Differential Equations and Modular Forms................. 67
3.4 Hecke Operators and Differential Equations................ 72
4 Lie Algebras.............................................. 77
4.1 Lie Algebras of Power Series ............................. 77
4.2 Lie Algebra Homomorphisms ............................ 80
4.3 Equivariant Splittings................................... 83
4.4 Lie Algebras of Jacobi-like Forms......................... 87
5 Heat Operators........................................... 91
5.1 Radial Heat Operators .................................. 91
5.2 Modular Series......................................... 100
5.3 Pseudodifferential Operators ............................. 102
vii
viii Contents
6 Group Cohomology....................................... 107
6.1 Cohomology of Groups.................................. 107
6.2 Hecke Operators ....................................... 111
6.3 Jacobi-like Forms and Group Cohomology ................. 115
7 Quasimodular Forms ..................................... 121
7.1 Quasimodular and Modular Forms........................ 121
7.2 Polynomials and Formal Power Series ..................... 127
7.3 Quasimodular Polynomials .............................. 130
7.4 Hecke Operators ....................................... 137
7.5 Vector Bundles......................................... 141
8 Quasimodular and Modular Polynomials.................. 149
8.1 Correspondences of Polynomials.......................... 149
8.2 Modular and Quasimodular Polynomials .................. 153
8.3 Poincar´e Series......................................... 157
8.4 Heat Operators ........................................ 161
9 Liftings of Quasimodular Forms........................... 165
9.1 Liftings of Modular Forms to Quasimodular Forms ......... 165
9.2 Liftings of Quasimodular Forms to Jacobi-like Forms........ 168
9.3 Lie Brackets ........................................... 176
9.4 Rankin–Cohen Brackets on Quasimodular Forms ........... 181
10 Quasimodular Forms and Vector-valued Modular Forms .. 185
10.1 Vector-valued Forms .................................... 185
10.2 Symmetric Tensor Representations........................ 189
10.3 Scalar- and Vector-valued Modular Forms ................. 194
10.4 Quasimodular Polynomials and Vector-valued Modular Forms 202
11 Differential Operators on Modular Forms................. 207
11.1 Jacobi-like Forms and Quasimodular Forms with Character .. 207
11.2 Differential Operators on Modular Forms .................. 211
11.3 Theta Functions........................................ 215
11.4 Differential Operators Associated to Theta Functions ....... 218
12 Half-integral Weight Forms ............................... 223
12.1 Quasimodular Forms of Half-integral Weight ............... 223
12.2 Shimura Correspondences ............................... 227
12.3 Shintani Liftings ....................................... 231
13 Projective Structures ..................................... 235
13.1 Projective Structures on Riemann Surfaces ................ 235
13.2 Pseudodifferential Operators ............................. 238
13.3 The Degree Four Case .................................. 244
13.4 The Degree Five Case................................... 246
Contents ix
14 Applications of Quasimodular Forms...................... 251
14.1 Quasimodular Forms for the Full Modular Group........... 251
14.2 The Work of Kaneko–Zagier ............................. 256
14.3 Covers of Elliptic Curves ................................ 263
14.4 Partitions ............................................. 268
14.5 Modular Linear Differential Equations .................... 276
14.6 Holomorphic Anomaly Equations......................... 281
14.7 Other Applications ..................................... 284
14.7.1 Gromov–Witten Invariants ........................ 284
14.7.2 Covers of Pillowcases ............................. 286
14.7.3 Square-tiled Surfaces.............................. 287
14.7.4 Curves on Abelian Surfaces........................ 289
References.................................................... 291
List of Notation .............................................. 297
Index......................................................... 301