University of Ottawa, December 6 - 9, 2013
The theory of divisors modulo the divisors of rational functions on a smooth complex projective variety $X$ is well understood. For example, for a smooth projective curve defined over a number field, the divisor classes of degree zero correspond to rational points on the Jacobian of $X$ and this group is finitely generated by the Mordell-Weil Theorem. For algebraic cycles of higher codimension, our understanding is much more limited. For example, for a variety defined over a number field $K$, the Chow group of cycles of codimension $r$, $r>1$, defined over $K$ modulo rational equivalence is not known to be finitely generated.
The locally symmetric varieties X associated to rational quadratic forms of signature $(n,2)$
are particularly nice test cases since they have explicitly constructed families of algebraic cycles
of all codimensions. For a fixed codimension $r$, one can form a generating series for the classes of
such special cycles. I will discuss
(1) What information is contained in the statement that such a generating series is a
(2) Borcherds' proof of modularity in the case of divisors.
(3) Recent results of Wei Zhang, Martin Raum and Jan Bruinier for higher codimension.
The story I will tell in my talk on "Political Space Curves" suggests a different question, which may get us closer to what is really going on : How exactly do mathematic(ian)s manage to generate stable knowledge ? A steady creative reinvention of truth seems to do the trick. In passing, we will see that controversies do in fact exist in mathematics, but they tend to do surprisingly little to unveil the truth.