授業の目的 【日本語】
Goals of the Course(JPN) | | | 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。詳細はホームページをご覧ください。(https://www.henrikbachmann.com/ec_2024.html) |
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授業の目的 【英語】
Goals of the Course | | | The goal of this course is to give an introduction to elliptic curves. 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. |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | | 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。詳細はホームページをご覧ください。(https://www.henrikbachmann.com/ec_2024.html) |
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到達目標 【英語】
Objectives of the Course | | | The objective of the course is to provide the students with the basic understanding and application of elliptic curves and the necessary tools from algebra and algebraic geometry used in this theory. |
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授業の内容や構成
Course Content / Plan | | Elliptic curves, Weierstrass form, abelian varieties, modular forms, Mordell-Weil theorem, cryptography.
A more detailed list will appear on the course homepage. |
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履修条件
Course Prerequisites | | | Basic algebra and complex analysis will be helpful. All other techniques can be developed on the way as needed, so the course can serve both as a self-contained introduction or as a complement to a course in algebraic/arithmetic geometry. |
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関連する科目
Related Courses | | |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | The grade will be based on homework submissions. |
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教科書・テキスト
Textbook | | | We will create lecture notes during the semester. There are several reference books. |
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参考書
Reference Book | | J.H. Silverman: The Arithmetic of Elliptic Curves
A.W. Knapp: Elliptic Curves
A more complete list will be given out at the beginning of the lecture. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | The students need to recall the previous lecture before attending a lecture. |
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注意事項
Notice for Students | | | Please make sure to check the course homepage for more information: https://www.henrikbachmann.com/ec_2024.html |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions of Other department student's attendance | | |
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レベル
Level | | |
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キーワード
Keyword | | |
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履修の際のアドバイス
Advice | | |
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授業開講形態等
Lecture format, etc. | | |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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