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.
- The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows:
1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session;
2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam;
3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental session.
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, in particular the Weierstrass ℘ function
- Weierstrass ζ and σ functions
- Elliptic curves and their representation as Weierstrass cubics
- Möbius transformations and modular curve
- Modular forms and functions, in particular Eisenstein series and the j-function
- 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.