**Elliptic curves**

(Nagoya University, Spring 2024)

(Nagoya University, Spring 2024)

We will meet for the first time on April 12th in Room 453 of the math building. Please make sure that you will be part of this TACT group. If you want to audit this lecture please contact me so I can add you to it.

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。

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.

Below you can find a list of standard books. For a start, I recommend [K], [LR], [ST], [M], and for advanced topics [S1] and [S2]. The lecture will probably follow [ST].

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 I

Week 02 (04/15-04/21): Overview of Elliptic curves II

Week 03 (04/22-04/28): Projective curves

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

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

Week 06 (05/13-05/19): The group E(Q) and points of finite order

Week 07 (05/20-05/26): The Nagell Lutz Theorem

Week 08 (05/27-06/02): Mordell's Theorem I & Descent Theorem

Week 09 (06/03-06/09): Meidai-sai (No Lecture on 7th June)

Week 10 (06/10-06/16): Mordell's Theorem II

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

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

Week 13 (07/01-07/07): Curves over Finite Fields & Factorization of integers

Week 14 (07/08-07/14): Elliptic Curve Cryptography

Week 15 (07/15-07/21): Reductions modulo p & L-Functions

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

Week 17 (07/29 - 08/04): Modular forms

**Materials**- Course information
- Handwritten lecture notes: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lecture 11
- Homework: Homework 1, Homework 2, Homework 3
- Sage Notebook Lecture 8

**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。

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**Below you can find a list of standard books. For a start, I recommend [K], [LR], [ST], [M], and for advanced topics [S1] and [S2]. The lecture will probably follow [ST].

- [K] A. W. Knapp: Elliptic Curves, Mathematical Notes, No. 40. Princeton University Press, Princeton, NJ, 1992.
- [LR] A. Lozano-Robledo: Elliptic Curves, Modular Forms, and Their L-functions, Student Mathematical Library, No. 58. American Mathematical Society, Providence, RI, 2011.
- [M] J.S. Milne: Elliptic Curves, World Scientific Publishing Co., Singapore, 2021.
- [S1] J. H. Silverman: The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, No. 106. Springer-Verlag, New York, 1986.
- [S2] J. H. Silverman: Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, No. 151. Springer-Verlag, New York, 1994.
- [ST] J. H. Silverman, J. Tate: Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics. Springer-Verlag, New York, 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 I

Week 02 (04/15-04/21): Overview of Elliptic curves II

Week 03 (04/22-04/28): Projective curves

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

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

Week 06 (05/13-05/19): The group E(Q) and points of finite order

Week 07 (05/20-05/26): The Nagell Lutz Theorem

Week 08 (05/27-06/02): Mordell's Theorem I & Descent Theorem

Week 09 (06/03-06/09): Meidai-sai (No Lecture on 7th June)

Week 10 (06/10-06/16): Mordell's Theorem II

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

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

Week 13 (07/01-07/07): Curves over Finite Fields & Factorization of integers

Week 14 (07/08-07/14): Elliptic Curve Cryptography

Week 15 (07/15-07/21): Reductions modulo p & L-Functions

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

Week 17 (07/29 - 08/04): Modular forms

****Last update: 2nd August 2024.