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課程名稱 |
算術幾何專題
TOPICS IN ARITHMETIC GEOMETRY |
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開課學期 |
98-2 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
于 靖 |
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課號 |
MATH5112 |
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課程識別碼 |
221 U5530 |
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班次 |
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學分 |
3 |
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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 |
星期二2,3(9:10~11:10)星期四@,5(~13:10) |
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上課地點 |
新501新505 |
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備註 |
上課時間：
二10:30~11:45；四14:30~15:45。
總人數上限：50人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/982arithmetic |
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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 |
Modular curves as algebraic curves. Fields of definition of modular curves. Reductions of algebraic curves over global fields. Jacobians. Abelian varieties. Hecke correspondences. Eichler-Shimura relations. Drinfeld upper-half plane. Drinfeld modular curves. Rigid analytic spaces. Non-archimedean theta functions. Bruhat-Tits trees. |
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課程目標 |
Modular curves. Jacobians. Eichler-Shimura theory. Drinfeld upper-half plane. Drinfeld modular curves. |
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課程要求 |
Basic algebraic curves theory. Basic algebraic number theory, in particular basics about local fields. |
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Basic theory of elliptic curves. Algebraic curves and Riemann-Roch. Also very basic modular forms, e.g. in Chap 7 of Serre's course in arithmetic. |
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參考書目 |
F. Diamond and J. Shurman, A First Course in Modular Forms, Chap-6-9, Springer GTM, 2005.
M. Hindry and J. Silverman, Diophantine Geometry: an introduction, Springer GTM, 2000.
S. Lang, Elliptic Functions, GTM Springer, 1987.
S. Lang, Introduction to Algebraic and Abelian Functions, Chap.3, 4, 8. GTM Springer, 1982.
J.-P. Serre, A course in Arithmetic, Chap 7, GTM Springer, 1978.
J. Silverman, The Arithmetic of Elliptic Curves, GTM, Springer, 1986.
J. Silverman, Advanced topics in the Arithmetic of Elliptic Curves, GTM Springer, 1994.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1973.
M. Van der Put, and J. Martinet, Rigid analytic geometry and its applications, Progress in
Math,Birkhauser
2003. |
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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. |
Student's presentation |
100% |
We will have student's seminar at Tuesday 5:00 pm. Giving assignment and home works to be presented at these seminar sessions. |
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週次 |
日期 |
單元主題 |
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第1週 |
2/23,2/25 |
Moduli problems for elliptic curves. Modular curves. Correspondences, Hecke operators. |
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第2週 |
3/02,3/04 |
Integrality of Hecke Eigenvalues. Conjugation of normalized eigenforms. |
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第3週 |
3/09,3/11 |
Jacobians, abelian varieties. Abelian varieties associated to Hecke eigenforms. |
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第4週 |
3/16,3/18 |
Projective embeddings of complex tori. |
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第5週 |
3/23,3/25 |
Riemann's relation on period lattice, the Siegel upper half space, Riemann forms. |
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第6週 |
3/30,4/01 |
Tate elliptic curves, non-archimedean elliptic curves with multiplicative reductions. |
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第7週 |
4/06,4/08 |
Modular interpretation of modular curves. Analytic tori. |
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第8週 |
4/13,4/15 |
Drinfeld upper-half spaces, and Drinfeld modular curves. |
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第9週 |
4/20,4/22 |
Modular function fields over C. Modular function fields over Q. |
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第10週 |
4/27,4/29 |
Fields of definitions. Reduction of algebraic curves over global fields. |
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第11週 |
5/04,5/06 |
Non-archimedean theta functions. Hecke correspondences |
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第12週 |
5/11,5/13 |
The infinity-adic Abel-Jacobi theorem. Reduction of Hecke operators. |
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第13週 |
5/18,5/20 |
Rigid analytic spaces. Eichler-Shimura relations. |
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第14週 |
5/25,5/27 |
Analytic reductions. Igusa's theorem |
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第15週 |
6/01,6/03 |
Automorphisms of modular function fields. Non-archimedean norms. |
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第16週 |
6/08,6/10 |
Modular function fields. Bruhat-Tits trees. |
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第17週 |
6/15,6/17 |
Shimura reciprocity. Harmonic cochains. |
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