Title: Analytic number theory and spaces of rational curves
Abstract:
(joint work with A. Venkatesh)
Suppose $X$ is a smooth Fano variety, and $K$ a global field.
Conjectures of Batyrev-Manin and Peyre give very precise predictions
for the asymptotic behavior of the function $N_{X/K}(B)$ which
enumerates points of $X(K)$ of height at most $B$. When $K$ is $
\F_q(t)$, these conjectures are naturally related to questions about
the geometry of the space of rational curves on $X$. I will try to
promote a general philosophy that nice asymptotic behavior of the type
predicted by Batyrev-Manin should correspond with stabilization of
cohomology for spaces of rational curves. In a particularly favorable
situation, where $X$ is a very low-degree Fermat hypersurface, the
Batyrev-Manin conjecture can be verified by traditional methods of
analytic number theory, and we explain how to use this fact to prove
irreducibility of the space of rational curves on $X$.