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 $X$.