  For the study of the Mordell-Weil group of an elliptic curve ${\bf E}$ over a
complex function field of a projective curve $B$, the first author introduced
the use of differential-geometric methods arising from K\"ahler metrics on
$\mathcal H \times \mathbb C$ invariant under the action of the semi-direct
product ${\rm SL}(2,\mathbb R) \ltimes \mathbb R^2$. To a properly chosen
geometric model $\pi: \mathcal E \to B$ of ${\bf E}$ as an elliptic surface and
a non-torsion holomorphic section $\sigma: B \to \mathcal E$ there is an
associated ``verticality'' $\eta_\sigma$ of $\sigma$ related to the locally
defined Betti map. The first-order linear differential equation satisfied by
$\eta_\sigma$, expressed in terms of invariant metrics, is made use of to count
the zeros of $\eta_\sigma$, in the case when the regular locus $B^0\subset B$
of $\pi: \mathcal E \to B$ admits a classifying map $f_0$ into a modular curve
for elliptic curves with level-$k$ structure, $k \ge 3$, explicitly and
linearly in terms of the degree of the ramification divisor $R_{f_0}$ of the
classifying map, and the degree of the log-canonical line bundle of $B^0$ in
$B$. Our method highlights ${\rm deg}(R_{f_0})$ in the estimates, and recovers
the effective estimate obtained by a different method of Ulmer-Urz\'ua on the
multiplicities of the Betti map associated to a non-torsion section, noting
that the finiteness of zeros of $\eta_\sigma$ was due to
Corvaja-Demeio-Masser-Zannier. The role of $R_{f_0}$ is natural in the subject
given that in the case of an elliptic modular surface there is no non-torsion
section by a theorem of Shioda, for which a differential-geometric proof had
been given by the first author. Our approach sheds light on the study of
non-torsion sections of certain abelian schemes.
