2025/2026
The Modular Forms and the Elliptic Functions
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Open to:
students of all HSE University campuses
Instructors:
Andrey Levin
Language:
English
ECTS credits:
3
Contact hours:
36
Course Syllabus
Abstract
"The concept of elliptic functions appeared at 19th century as inversion of some integrals of geometric and physical origin. They (and auxiliary in this context modular objects) were subject of comprehensive study during two consecutive centuries. These higher transcendental functions are important tools in the Number Theory in addition to the Geometry and Physics.
At the other hand, they are the starting point for many branches of Algebraic and Arithmetic Geometries."
Learning Objectives
- To learn the theories of double periodic functions and covariant with respect to fraction-linear action of the integer unimodular matrices group functions. These theories are developed almost as complite as theories of meromorphic and trigonometric functions.
Expected Learning Outcomes
- Application of the group-theoretic constructions in geometry.
- Creating the vision of mathematics as unified science and interrelations between its different bruches.
- Demonstration of possible variability of the basic defenition for adoption to explicit problems.
- Demonstration of the averaging method in different contexts.
- Introduction to the basic theory of geometric groups.
- Realization of the concept of compactification for explicit example.
- Realization of the technique of the universal object in the description of configuration of points in a linear space.
Course Contents
- Elliptic Integrals and Elliptic Functions.
- 2-Lattices in the Complex Plane.
- Degeneration of the Lattices and Comactification.
- Modular Forms in One Variable
- The Eisenstein Series
- Double coset construction
- Automorphic Triples of Groups
