Abstract

This is an introductory course on elliptic integrals and elliptic functions for advanced undergraduate students and graduate students who are not familiar with the subject.

An elliptic function is defined as a doubly periodic meromorphic function on the complex plane. The study of elliptic integrals started by Fagnano, Legendre, Gauss and others in the eighteenth century was turned into the theory of elliptic functions by Abel and Jacobi. Then Riemann, Weierstrass and Liouville developed the theory further by using complex analysis.

The theory of elliptic functions thus founded is 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. Examples: arc length of an ellipse, arithmetic-geometric mean, solutions of physical systems (pendulum, top, skipping rope, the KdV equation), solution of quintic equations, etc.

In this course we shall put emphasis on analytic aspects and applications.

Prerequisite

Undergraduate calculus, complex analysis.

Syllabus

Details might change depending on the wishes of the audience.

1. Introduction

2. Arc length of an ellipse

3. Arc length of a lemniscate

4. Classification of elliptic integrals

5. Arithmetic-geometric mean

6. Simple pendulum

7. Jacobi's elliptic functions (definitions)

8. Jacobi's elliptic functions (properties)

9. Simple pendulum revisited

10. Shape of a skipping rope

11. Riemann surfaces

12. Analysis on Riemann surfaces

13. Elliptic curves

14. Complex elliptic integrals

15. Conformal mapping from the upper half plane to a rectangle

16. Abel-Jacobi theorem (statement and preparation of the proof)

17. Surjectivity of the Jacobi map

18. Injectivity of the Jacobi map

19. Elliptic functions on the complex plane (definition and examples)

20. Elliptic functions on the complex plane (properties)

Reference

[1] T. Takebe, Elliptic integrals and elliptic functions (2023)

[2] V. Prasolov, Y. Solovyev, Elliptic functions and elliptic integrals (1997)

[3] E. T. Whittaker, G. N. Watson, A course of modern analysis (1902)

[4] D. Mumford, Tata lectures on Theta I (1983)

Record

Yes

Lecturer Intro.

Takashi Takebe is a researcher of mathematical physics, in particular integrable systems. He worked as a professor at the faculty of mathematics of National Research University Higher School of Economics in Moscow, Russia, till August 2023 and joined BIMSA as a research fellow in September 2023.