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課程名稱 |
橢圓曲線、模形式及Galois表現一
Elliptic Curves, Modular Forms, and Galois Representation (I)(Ⅰ) |
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開課學期 |
106-1 |
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授課對象 |
理學院 數學系 |
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授課教師 |
于 靖 |
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課號 |
MATH5059 |
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課程識別碼 |
221 U8070 |
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班次 |
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學分 |
2.0 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期二3,4(10:20~12:10) |
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上課地點 |
天數305 |
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備註 |
總人數上限：15人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1061MATH5059_Fermat |
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課程簡介影片 |
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核心能力關聯 |
本課程尚未建立核心能力關連 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
Basic algebraic number theory, Review Galois theory, Profinite Galois groups, Galois representations, Elliptic curves, Elliptic curves mod p, Conductor, Elliptic functions, Weierstrass models, The upper-half plane, Mobius transformations, Modular groups, Modular curves, Compactification, Invariant differential forms. |
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課程目標 |
Introduce the development of mathematics in the last two century which lead to the proof of the last theorem of Fermet by Wiles-Taylor. |
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課程要求 |
Required course: Algebra, Linear Algebra, Complex Analysis, Analysis. |
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預期每週課後學習時數 |
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Office Hours |
另約時間 |
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指定閱讀 |
1. Fermat last theorem, Wikipedia.
2. L. V. Ahlfors, Complex Analysis, Chap. 7. 1979.
3. R. M. Dummit & D. S. Foote, Abstract Algebra, Chap. 14, Chap. 16, 2003.
4. K. Kato, N. Kurakawa, & T. Saito, Number Theory, I Fermat's Dream, AMS Translations. |
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參考書目 |
1. L. Ｖ. Ahlfors, Complex Analysis.
2. M. Artin, Algebra.
3. F. Diamond, J. Shurman, A first Course in Modular Forms, GTM, Springer.
4. R. M. Dummit & D. S. Foote, Abstract Algebra, 2003.
5. Y. Hellegouarch, Invitation to the mathematics of Fermat-Wiles. 2001 AP.
6. K. Kato, N. Kurakawa, & T. Saito, Number Theory, I Fermat's Dream, II Intro to Class Field Theory, III
Iwasawa Theory and Modular Forms, AMS Translations.
7. T. Saito, Fermat's last theorem, Basic tools, AMS 2009.
8. T. Saito, Fermat's last theorem, The Proof, AMS 2009.
9. J.-P. Serre, A course in Arithmetic, Springer GTM.
10. J. Silverman & J. Tate, Rational points on elliptic curves, Springer, UTM. 2015.
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Homeworks on an Overleaf site |
100% |
Exercises from Invitation to Math of Fermat-Wiles |
2. |
Presentations |
50% |
After completing homework, students may be selected to do Oral Presentations on various topics |
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週次 |
日期 |
單元主題 |
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第1週 |
9/12 |
Fermat's last theorem. The case of exponent 4.
Introducing algebraic numbers and ideals. Unique factorization of integers. |
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第2週 |
9/19 |
Class numbers, Dedekind domains. Frey curves. Elliptic curves. |
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第3週 |
9/26 |
Reduction of elliptic curves, Conductor. Differential forms. The upper half plane. |
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