Book contents
- Frontmatter
- Dedication
- Contents
- Organization
- Acknowledgements
- 1 Introduction
- Part I Modular forms and their variants
- 2 Elliptic functions
- 3 Modular forms for SL(2; Z)
- 4 Variants of modular forms
- 5 Quantum fields on a torus
- 6 Congruence subgroups and modular curves
- 7 Modular forms for congruence subgroups
- 8 Modular derivatives and vector-valued modular forms
- 9 Modular graph functions and forms
- Part II Extensions and applications
- Part III Appendix
- References
- Index
8 - Modular derivatives and vector-valued modular forms
from Part I - Modular forms and their variants
Published online by Cambridge University Press: aN Invalid Date NaN
Book contents
- Frontmatter
- Dedication
- Contents
- Organization
- Acknowledgements
- 1 Introduction
- Part I Modular forms and their variants
- 2 Elliptic functions
- 3 Modular forms for SL(2; Z)
- 4 Variants of modular forms
- 5 Quantum fields on a torus
- 6 Congruence subgroups and modular curves
- 7 Modular forms for congruence subgroups
- 8 Modular derivatives and vector-valued modular forms
- 9 Modular graph functions and forms
- Part II Extensions and applications
- Part III Appendix
- References
- Index
Summary
In this chapter, we construct differential equations in the modular parameter and find solutions to these equations in simple cases. The solutions can generically be assembled into vector-valued modular forms, which have proven fruitful in recent works in mathematics and physics. We will establish that, in general, each component of a vector-valued modular form is a modular form for a congruence subgroup.
- Type
- Chapter
- Information
- Modular Forms and String Theory , pp. 155 - 167Publisher: Cambridge University PressPrint publication year: 2024