Program
 Modular functions and their CM values Student No.： 50 Time： Mon/Wed 15:10-17:00, 2014-06-30 ~ 2014-08-06 Instructor： Tonghai Yang  [University of Wisconsin] Place： Conference Room 3, Floor 2, Jin Chun Yuan West Building Starting Date： 2014-6-30 Ending Date： 2014-8-6

Description:

This course is aimed at high level undergraduates and lower level graduate students. It is a little more research oriented. It will be a combination of lectures and discussions. Active involvement is essential. Students should also be willing to accept some unknown (to them) facts.

The course is focusing around understanding and generalizing the following interesting fact. Let $j(z)$ be the famous $j$-invariant (modular function), compute $j(\frac{1+\sqrt{-163}}2)$ and \$j(\frac{1+\sqrt{-163}}2) –j(\sqrt{-1}), (you may assume that they are both integers). You will find them huge but their prime factorization has only very small prime factors. Why? Are they accidental or is there some underlying beautiful theory behind it? That is the focus of this short course.

Continuing discussion of these topics after the short course is welcome via various means.

Prerequisite:

Algebraic number theory and a little complex analysis.

Knowing a little about classical modular forms, modular curves, and/or complex multiplication will be a plus.

Reference:

For background on modular form and modular curves, pick any graduate text book on this subject. For complex multiplication, see J. Silverman Advanced topics in the arithmetic of elliptic curves, or Shimura’s book on complex multiplication.

For the topics itself, see for example

1.       B. Gross and D. Zagier, On singular moduli. J. Reine Angew. Math.   355 (1985), 191—220

2.       Borcherds forms and generalizations of singular moduli. J. Reine Angew. Math. 629(2009), 1–36

3.     J. Bruinier and T.H. Yang, Faltings heights of CM cycles and derivatives of L-functions.Invent. Math. 177 (2009), no. 3, 631–681