**Elliptic curves**

(Nagoya University, Spring 2024)

(Nagoya University, Spring 2024)

Details on this course will appear on this page at the end of March 2024. Please make sure to visit this site again before the Lecture starts. Feel free to contact me via email in case you have questions on this course.

この講義では楕円曲線の入門的内容を取り扱います。この講義は英語で行われます。予備知識としては、代数学の講義で取り扱われる内容を仮定します。受講学生の多くは日本語で行われた代数学の講義を受講していると思いますが、本講義の最初に代数学の知識の復習を行うことで、学生が代数学における英語の語彙を学ぶことができるようにします。また、英語が苦手な学生のために、日本語を少し取り入れた講義ノートを作成していく予定です。

Elliptic curves are central to both modern mathematics and practical applications like cryptography. In arithmetic geometry, they've been instrumental in significant breakthroughs, such as Andrew Wiles' proof of Fermat's Last Theorem. The lecture series will cover the basic theory of elliptic curves, potentially extending to advanced topics based on audience interest. We start with elliptic curves over the complex numbers, seen as quotients of the complex plane by a lattice, an approach that merges complex analysis with algebraic descriptions as cubic polynomials define these curves. Moving to algebraic geometry, we explore elliptic curves over various fields, highlighting their role as simple examples of abelian varieties - projective varieties with a group structure. Discussions will span theories over finite fields and number fields, integrating concepts from complex analysis, algebraic geometry, number theory, and their crucial use in cryptography. A detailed overview will come in March.

Based on homework assignments which you will submit at TACT.

We will create lecture notes during the semester based on the following references. A more detailed list of references will be part of the lecture notes.

The following gives a tentative overview of the topics we will cover each week.

Week 01 (04/08-04/14): Introduction and overview of Elliptic curves

Week 02 (04/15-04/21): TBA

Week 03 (04/22-04/28): TBA

Week 04 (04/29-05/05): Golden week (No Lecture on 3rd May)

Week 05 (05/06-05/12): TBA

Week 06 (05/13-05/19): TBA

Week 07 (05/20-05/26): TBA

Week 08 (05/27-06/02): TBA

Week 09 (06/03-06/09): Lecture on 5th June due to Medai-sai (No Lecture on 7th June)

Week 10 (06/10-06/16): TBA

Week 11 (06/17-06/23): TBA

Week 12 (06/24-06/30): TBA

Week 13 (07/01-07/07): TBA

Week 14 (07/08-07/14): TBA

Week 15 (07/15-07/21): TBA

Week 16 (07/22-07/28): TBA

Week 17 (07/29 - 08/04): TBA

**Materials**- TBA

**Content****(Tentative)****:**この講義では楕円曲線の入門的内容を取り扱います。この講義は英語で行われます。予備知識としては、代数学の講義で取り扱われる内容を仮定します。受講学生の多くは日本語で行われた代数学の講義を受講していると思いますが、本講義の最初に代数学の知識の復習を行うことで、学生が代数学における英語の語彙を学ぶことができるようにします。また、英語が苦手な学生のために、日本語を少し取り入れた講義ノートを作成していく予定です。

Elliptic curves are central to both modern mathematics and practical applications like cryptography. In arithmetic geometry, they've been instrumental in significant breakthroughs, such as Andrew Wiles' proof of Fermat's Last Theorem. The lecture series will cover the basic theory of elliptic curves, potentially extending to advanced topics based on audience interest. We start with elliptic curves over the complex numbers, seen as quotients of the complex plane by a lattice, an approach that merges complex analysis with algebraic descriptions as cubic polynomials define these curves. Moving to algebraic geometry, we explore elliptic curves over various fields, highlighting their role as simple examples of abelian varieties - projective varieties with a group structure. Discussions will span theories over finite fields and number fields, integrating concepts from complex analysis, algebraic geometry, number theory, and their crucial use in cryptography. A detailed overview will come in March.

**Place & Time****Lecture:****Friday 1st Period (8:45 - 10:30) @ Room****多-453****(Math building)**

**Grading & Homework submission****:**Based on homework assignments which you will submit at TACT.

**References**We will create lecture notes during the semester based on the following references. A more detailed list of references will be part of the lecture notes.

- [S] J.H. Silverman: The Arithmetic of Elliptic Curves, Springer GTM (1986).
- [K] A.W. Knapp: Elliptic Curves, Princeton, NJ: Princeton University Press, 1992.

**Course schedule (Tentative)****:**The following gives a tentative overview of the topics we will cover each week.

Week 01 (04/08-04/14): Introduction and overview of Elliptic curves

Week 02 (04/15-04/21): TBA

Week 03 (04/22-04/28): TBA

Week 04 (04/29-05/05): Golden week (No Lecture on 3rd May)

Week 05 (05/06-05/12): TBA

Week 06 (05/13-05/19): TBA

Week 07 (05/20-05/26): TBA

Week 08 (05/27-06/02): TBA

Week 09 (06/03-06/09): Lecture on 5th June due to Medai-sai (No Lecture on 7th June)

Week 10 (06/10-06/16): TBA

Week 11 (06/17-06/23): TBA

Week 12 (06/24-06/30): TBA

Week 13 (07/01-07/07): TBA

Week 14 (07/08-07/14): TBA

Week 15 (07/15-07/21): TBA

Week 16 (07/22-07/28): TBA

Week 17 (07/29 - 08/04): TBA

****Last update: 13th February 2024.