Back to Departmental Colloquium: Fall 2013
Departmental Colloquium
Date: Thursday, Dec 12, 2013
Time: 4:00PM
Location: LCB 219
Tonghai Yang
University of Wisconsin, Madison
Title |
Big numbers, small factors, and Shimura varieties. |
Abstract |
It was observed 90 years ago that the famous modular j -function has the following cool property j ( 1 + - 1 6 3 2 ) = - e 1 6 3 π + 7 4 4 + t i n y = - 2 1 8 3 3 5 3 2 3 3 2 9 3 –> is a huge integer with very small prime factor, and so is j ((1+√-163)/2)-1728. In the early 80s, Gross and Zagier proved that this is a general phenomenon-the difference of values of j at two quadratic integers, of discriminants a and b, has an explicit factorization formula with prime factors less than or equal to 4ab. It turns out that this phenomenon is much more general and works for a whole family of modular functions on a Shimura variety of orthogonal type (n, 2) and unitary type (n, 1). In this talk, we will give an informal account of this story. |
Number Theory
Algebraic Geometry