Research Seminar "Elliptic Integrals and Elliptic Functions"

2021/2022

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: 1, 2 module

Open to: students of all HSE University campuses

Instructors: Takashi Takebe

Language: English

ECTS credits: 3

Contact hours: 30

Full Syllabus

Abstract

The study of elliptic integrals started in the eighteenth century was turned into the theory of elliptic functions in the nineteenth century, which eventually became a prototype of today’s algebraic geometry. On the other hand elliptic functions and elliptic integrals appear in various problems in mathematics as well as in physics. In this research seminar (lecture style) we shall study elliptic integrals and elliptic functions with emphasis on analytic aspects and applications. PREREQUISITES: Calculus course (first/second year courses without the Lebesgue integral), Complex analysis (till the argument principle).

Learning Objectives

Learn fundamental definitions and properties of elliptic integrals and elliptic functions.
Learn basics of Riemann surfaces and analysis on them.

Expected Learning Outcomes

classification of elliptic integrals
Construction and basic properties of the Weierstrass elliptic function
Definition and properties of theta functions
Definitions and local properties of integrals on elliptic curves
Jacobian elliptic functions as ratios of theta functions, their properties
Learn construction and topology of elliptic curves
Learn real Jacobian functions and skills of application.
Learn what is a Riemann surface, why it is necessary.
Perspective of the whole theory
Proof of Abel-Jacobi theorem
Skills of computation and application of real elliptic integrals
Theory of elliptic functions as doubly periodic meromorphic functions

Course Contents

Introduction
Real elliptic integrals
Classification of elliptic integrals
Real elliptic functions
Riemann surfaces of algebraic functions
Elliptic curves
Complex elliptic integrals
Abel-Jacobi theorem
General theory of elliptic functions
Weierstrass ℘-function
Theta functions
Complex Jacobian functions

Assessment Elements

Homework1
Homework2

Interim Assessment

2021/2022 2nd module
Homework max{(points obtained/maximum possible points)*15,10}

Bibliography

Recommended Core Bibliography

Elliptic functions, Lang, S., 1987
Fundamenta nova theoriae functionum ellipticarum, Jacobi, C. G. J., 2013
Jan Nekovar. (n.d.). ELLIPTIC FUNCTIONS AND ELLIPTIC CURVES (A Classical Introduction). Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.EACE09C4
N I Akhiezer. (2018). Elements of the theory of elliptic functions. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1790158

Course Syllabus