  For a sub-Riemannian manifold provided with a smooth volume, we relate the
small time asymptotics of the heat kernel at a point $y$ of the cut locus from
$x$ with roughly "how much" $y$ is conjugate to $x$. This is done under the
hypothesis that all minimizers connecting $x$ to $y$ are strongly normal, i.e.\
all pieces of the trajectory are not abnormal. Our result is a refinement of
the one of Leandre $4t\log p_t(x,y)\to -d^2(x,y)$ for $t\to 0$, in which only
the leading exponential term is detected. Our results are obtained by extending
an idea of Molchanov from the Riemannian to the sub-Riemannian case, and some
details we get appear to be new even in the Riemannian context. These results
permit us to obtain properties of the sub-Riemannian distance starting from
those of the heat kernel and vice versa. For the Grushin plane endowed with the
Euclidean volume we get the expansion $p_t(x,y)\sim t^{-5/4}\exp(-d^2(x,y)/4t)$
where $y$ is reached from a Riemannian point $x$ by a minimizing geodesic which
is conjugate at $y$.
