  We shall consider sections of an elliptic scheme $\mathcal{E}$ over a(n
affine) base curve $B$, and study the points of $B$ where the section takes a
torsion value. In particular, we shall relate the distribution in $B$ of these
points with the canonical height of the section, proving an integral formula
involving a measure on $B$ coming from the so-called Betti map of the section.
We shall show that this measure is the same appearing in dynamical issues
related to the section.
  This analysis will also involve the multiplicity with which a torsion value
is attained, which is an independent problem. We shall prove finiteness
theorems for the points where the multiplicity is higher than expected. Such
multiplicity has also a relation with Diophantine Approximation and
quasi-integral points on $\mathcal{E}$ (over the affine ring of $B$), and in
the last part of the paper we shall exploit this viewpoint, proving an
effective result in the spirit of Siegel's theorem on integral points.
