  We study the singularities (blow-ups) of the Toda lattice associated with a
real split semisimple Lie algebra $\mathfrak g$. It turns out that the total
number of blow-up points along trajectories of the Toda lattice is given by the
number of points of a Chevalley group $K({\mathbb F}_q)$ related to the maximal
compact subgroup $K$ of the group $\check G$ with $\check{\mathfrak g}={\rm
Lie}(\check G)$ over the finite field ${\mathbb F}_q$. Here $\check{\mathfrak
g}$ is the Langlands dual of ${\mathfrak g}$. The blow-ups of the Toda lattice
are given by the zero set of the $\tau$-functions. For example, the blow-ups of
the Toda lattice of A-type are determined by the zeros of the Schur polynomials
associated with rectangular Young diagrams. Those Schur polynomials are the
$\tau$-functions for the nilpotent Toda lattices. Then we conjecture that the
number of blow-ups is also given by the number of real roots of those Schur
polynomials for a specific variable. We also discuss the case of periodic Toda
lattice in connection with the real cohomology of the flag manifold associated
to an affine Kac-Moody algebra.
