Kenji Matsuki教授（Purdue University）によるKTGU数学基礎講義を下記の要領で行います。

科目名

KTGU数学基礎講義
＊学部生および大学院の学生には1単位認定されます。

日　時

2016年1月14日〜1月28日
1月14日(木)　16:00〜18:00
1月18日(月)　16:00〜18:00
1月21日(木)　16:00〜18:00
1月25日(月)　16:00〜18:00
1月28日(木)　16:00〜18:00

場　所

京都大学理学部３号館 108講義室

題　目

Introduction to the Theory of Elliptic Curves

授業の概要・目的

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.

授業計画と内容

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

履修要件

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.

言　語

主に英語、補足として日本語

成績評価の方法・観点及び達成度

平常点（4割）とレポート（6割）

教科書

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

参考書等

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

授業外学習（予習・復習）等

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.

その他（オフィスアワー等）

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.