Program
Geometry of modular curves and arithmetic applications
Student No.:8
Time:Mon 19:00-21:00
Instructor:Yichao Tian,Weizhe Zheng  
Place:Conference Room 1, Floor 1, Jin Chun Yuan West Building (近春园西楼1层第一会议室)
Starting Date:2012-7-2
Ending Date:2012-8-27

 This can be viewed as an introduction to the geometric approach to modular forms
 

Schedule

 

Talk 0 by Yichao Tian (田一超, Morningside), July 2. Introduction.

 

Talk 1 by Yuanqing Cai (蔡园青, John Hopkins), July 9. The classical definition of modular curves as quotients of Poincare's half plane. Genus formulas.

 

Talk 2 by Jinbo Ren (任金波, Tsinghua), July 16. Adelic descriptions of modular curves. Modular forms as automorphic forms.

 

Talk 3 by Heng Du (杜衡, Fudan), July 23. Modular curves as moduli spaces of elliptic curves.

 

Talks 4 and 5 by Xiaowen Hu (胡晓文, Tsinghua), July 30 and August 6. Modular forms as sections of certain line bundles over modular curves. Geometric definition of Hecke operators and q-expansions.

 

Talk 6 by Jinbo Ren (任金波, Tsinghua), August 20. The Gauss-Manin connection and theta operators.

 

Talk 7 (to be confirmed). Computations of various cohomologies (Betti, de Rham, etale, ...) of modular curves. The Eichler-Shimura isomorphism.

 

Talk 8 (to be confirmed). Introduction to mod p geometry of modular curves. Congruence relations.

 

(more interesting topics according to the level of the students)


Remarks:
 

The participants are supposed to have some basic knowledge on algebraic geometry: algebraic curves, Riemann-Roch, fundamental group, basic definition of various cohomology theories... Each enrolled student will be required to give a seminar talk on the topics that he is interested in. Of course, we will help the students to prepare their lectures. Each of them is supposed to meet us at least once before his/her lecture for discussion.
 

References:

 

1. P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, in Modular Functions of One Variable II, Lecture Notes in Mathematics 349, Springer-Verlag, 1973, pp. 143-316. 

2. N. Katz, p-adic properties of modular schemes and modular forms, in Modular Functions of One Variable III, Lecture Notes in Mathematics 350, Springer-Verlag, 1973, pp. 69-190. 

3. H. Hida, Geometric Modular Forms and Elliptic Curves, World Scientific, 2000. 

4. F. Diamond and J. Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics 228, Springer, 2005.