KTGU Mathematics Basic Lecture by Prof. Kenji Matsuki (Purdue University) will take place as follows:

Course Title

KTGU Mathematics Basic Lecture

Date & Time

January 14 - 28, 2016
Thursday, January 14, 16:00-18:00
Monday, January 18, 16:00-18:00
Thursday, January 21, 16:00-18:00
Monday, January 25, 16:00-18:00
Thursday, January 28, 16:00-18:00

Venue

108 Lecture room, Faculty of Science Bldg. #3, Kyoto University

Title

Introduction to the Theory of Elliptic Curves

Discription and Objectives

Introduce the students to the basic theory of elliptic curves with emphasis on the arithmetic properties, assuming the minimum amount of prerequisites. The subject of elliptic curves sits at the intersection of analysis, topology and number theory, i.e., almost all the areas of mathematics. As such, it has been the center of intensive studies classically and recently, ranging from the old problem of the congruent numbers, of computing the elliptic integral, to the proof of the Fermat's Last Theorem, to name a few. We give a series of five lectures, aimed at the undergraduate students, which introduces them to this fascinating subject at an elementary level with little background material required.

Introduce the students to the basic theory of elliptic curves with emphasis on the arithmetic properties, assuming the minimum amount of prerequisites. The subject of elliptic curves sits at the intersection of analysis, topology and number theory, i.e., almost all the areas of mathematics. As such, it has been the center of intensive studies classically and recently, ranging from the old problem of the congruent numbers, of computing the elliptic integral, to the proof of the Fermat's Last Theorem, to name a few. We give a series of five lectures, aimed at the undergraduate students, which introduces them to this fascinating subject at an elementary level with little background material required.

Lecture 1 (2 hours): Introduction
Lecture 2 (2 hours): Mordell-Weil theorem
Lecture 3 (2 hours): Elliptic curves over ${\Bbb C}$ (the analytic theory)
Lecture 4 (2 hours): The $j$-invariant
Lecture 5 (2 hours): Weil conjecture for elliptic curves

Requirements

It would be desirable if the student has the basic knowledge of complex analysis of one variable and the basic knowledge of algebra (groups and fields). What is required for this series of lectures as prerequisites is tried to be kept at the low level, even though the student is expected to fill in the gaps of his/her knowledge needed to understand the lectures by reading the textbooks and/or by coming to the office hours. The goal of this series of lectures is to expose the students to the basic theory of elliptic curves with the minimum amount of background knowledge at an early stage of his/her learning of mathematics.

Language

Mainly English. Some help will be given in Japanese.

Evaluation and Grading Criteria

Overall participation and performance:40%, report:60%

Textbook

The arithmetic of elliptic curves by Joseph H. Silverman, Graduate Texts in Mathematics 106, Springer

References

1. Elliptic curves by Dale Husemoller, Graduate Texts in Mathematics 111, Springer
2. Elliptic curves by Anthony W. Knapp, Mathematical Notes 40, Princeton University Press
3. Introduction to elliptic curves and modular forms by Neal Koblitz, Graduate Texts in Mathematics 97, Springer

Assignments

The reading assignment of the sections/chapters of the textbook(s), corresponding to and/or associated with the lectures may be given to the students.

Other Information (Office hours, etc)

1. The participation of the students to the lectures in the form of asking the questions is not only strongly encouraged but also reflected in the evaluation toward the final grade.
2. The students are expected to come to the office hours in order to solidify their understanding of the lectures. The time and place of the office hours will be announced later.