Departmental Colloquium
Stephen Kudla
University of Toronto
Cycles on ball quotients and modular forms
Abstract: The locally symmetric varieties $Y=\Gamma\backslash D$ where $D$ is the unit ball in $\mathbb C^n$ and $\Gamma$ is an arithmetic
subgroup of $U(n,1)$ have a rich geometry and arithmetic. Such quotients have many `special' algebraic cycles
arising as quotients $Z_x = \Gamma_x\backslash D_x$ of sub-balls $D_x \subset D$. By results of old joint work with
John Millson, the generating series for the cohomology classes determined by suitable collections of such cycles
are modular forms for unitary groups $U(r,r)$. Recently, in joint work with Michael Rapoport, we utilize the fact that
ball quotients can be viewed as moduli spaces of abelian varieties to define arithmetic analogues of the cycles $Z_x$.
We conjecture that the classes in arithmetic Chow groups defined by such cycles are, again, the Fourier
coefficients of certain modular forms. I will discuss some evidence for such a conjecture, especially in the
case of arithmetic $0$-cycles.
The talk will be followed by wine and cheese reception in SEO 300. The Atkin Memorial lecture will be followed by a workshop "Arithmetic cycles on Shimura varieties and automorphic forms" on Saturday May 1, 2010. The speakers are Ben Howard (Boston College), Stephen Kudla (Toronto), Kartik Prasanna (Michigan), Ulrich Terstiege (Harvard), and Tonghai Yang (Wisconsin). For more information see
http://www.math.uic.edu/~rtakloo/atkin2010.html.
Friday April 30, 2010 at 3:00 PM in SEO 636