数论专题选讲  070101D05008Z

学期:2020—2021学年(春)第二学期 | 课程属性:专业普及课 | 任课教师:田野
授课时间: 星期五,第9、10、11 节
授课地点: 教学楼N308
授课周次: 1、2、3、4、5、6、7、8、9、10、11、12
课程编号: 070101D05008Z 课时: 32 学分: 1.50
课程属性: 专业普及课 主讲教师:田野 助教:
英文名称: Selected Topics in Number Theory 召集人:

教学目的、要求

介绍与Heegner 点相关的数论前沿课题, 特别是在BSD 猜想中的应用。

预修课程

代数数论基础,模形式

教 材

[1] Yuan, Xinyi, Shou-Wu Zhang, and Wei Zhang.?The gross-zagier formula on shimura curves. No. 184. Princeton University Press, 2013.
[2] Kolyvagin, V. A. "Euler systems."?The Grothendieck Festschrift. Birkh?user, Boston, MA, 2007. 435-483.
[3] Zhang, Wei. "Selmer groups and the indivisibility of Heegner points."?Cambridge Journal of Mathematics?2.2 (2014): 191-253.
[4] Yun, Zhiwei, and Wei Zhang. "Shtukas and the Taylor expansion of L-functions."?Annals of Mathematics?(2017): 767-911.

主要内容

Heegner points on elliptic curves defined over $\BQ$ are constructed from CM points on modular curves (or more general, Shimura curves). It is the first systematical way of construction of rational points, and plays an important role in the study of BSD conjecture. In this course, we plan to introduce the following aspects of Heegner points:
1.Construction of Heegner Points (Tunnell-Saito, Examples and Applications)
2.Relation to L-functions (Gross-Zagier Formula)
3.Arithmetic of Heegner Points (Euler system, Iwasawa Main Conjecture, Kolyvagin Conjecture)
4.Heegner cycles over Function Fields (Quaternion Drinfeld curves, Heegner-Drinfeld cycles)
5.Open Problems

参考文献