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
6 - Congruence subgroups and modular curves
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
Congruence subgroups form a countable infinite class of discrete non-Abelian subgroups of SL(2,Z) and play a particularly prominent role in deriving the arithmetic properties of modular forms. In this chapter, we study various aspects of congruence subgroups, including their elliptic points, cusps, and topological properties of the associated modular curve. Jacobi theta-functions, theta-constants, and the Dedekind eta-function are used as examples of modular forms under congruence subgroups that are not modular forms under the full modular group SL(2,Z).
- Type
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- Modular Forms and String Theory , pp. 121 - 136Publisher: Cambridge University PressPrint publication year: 2024