MAU34207 Elliptic functions and modular forms
| Module Code | MAU34207 |
|---|---|
| Module Title | Elliptic functions and modular forms |
| Semester taught | Semester 1 |
| ECTS Credits | 5 |
| Module Lecturer | Prof. Katrin Wendland |
| Module Prerequisites | MAU23204 Introduction to complex analysis |
Assessment Details
- This module is examined in a 2-hour examination at the end of Semester 1.
- Continuous assessment contributes 10% towards the overall mark.
- Re-assessment, if needed, consists of 100% exam.
Contact Hours
11 weeks of teaching with 3 lectures per week.
Learning Outcomes
On successful completion of this module, students will be able to
- Apply basic theorems on elliptic functions, that is, meromorphic doubly periodic functions on the complex plane.
- Use these theorems to calculate examples of elliptic functions.
- Describe elliptic curves as Weierstrass cubics.
- Describe the moduli space of elliptic curves.
- Explain the concept of modular forms and functions.
- Demonstrate that modular forms of a given weight form a finite-dimensional vector space.
- Be familiar with basic examples of modular forms and functions, including the Eisenstein series, the discriminant function and the modular j-function.
Module Content
- Elliptic functions.
- Weierstrass ℘, ζ and σ functions
- Elliptic curves.
- Weierstrass cubics.
- Möbius transformations.
- Modular forms and functions, including standard examples.
- Algebra of modular forms.
Recommended Reading
- J.H. Bruinier, G. v.d. Geer, G. Harder, D. Zagier, The 1-2-3 of Modular Forms, Springer 2008.
- F. Diamond, J. Shurman, A First Course in Modular Forms, Springer 2005.
- J.-P. Serre, A course in arithmetic, Springer 1973.