References:
Given elliptic surface \(\mathcal{E}\to \mathcal{C}\) over a field \(k\) of \(\mathrm{char}k =0\), there is a zero section \(O\), if we have another section \(P\) of infinite order, in [Ulmer-Urzua] they gave an explicit upper bound on the number of points where \(O\) is tangent to a multiple of \(P\).
Here \(k=\mathbb{C}\) is complex number, \(\mathcal{C}\) an irreducible smooth projective curve of genus \(g\) and \(\pi : \mathcal{E}\to \mathcal{C}\) a Jacobian elliptic surface over \(\mathcal{C}\).
Consider for now that \(\mathcal{E}\cong \mathcal{C}\times E\to \mathcal{C}\) is a constant family, then a section \(P:\mathcal{C}\to \mathcal{E}= \mathcal{C}\times E\) is equivalent to a map \(f_P : \mathcal{C}\to E\) since all the fibres are the same. \(P\) and \(f_P\) are assumed to be non-constant.
The torsion \(\mathcal{E}[n]\) consists of \(n^2\) constant sections \(\mathcal{C}\times \{p\}\subset \mathcal{C}\times E[n]\), and we can consider the set for any constant section
\[ T_c := \bigcup_{p\in E} \{ t\in \mathcal{C}: P \text{ tangent } \mathcal{C}\times \{p\} \text{ at } t \} \]
which is a subset of
\[T_{tors} = \bigcup_{p\in E_{tors}} \{ t\in \mathcal{C}: P \text{ tangent } \mathcal{C}\times \{p\} \text{ at } t \}\subset \mathcal{C}.\]
We can see that \(T_c\) is intuitively viewed as the set of points where \(f_P\) is ‘entirely horizontal’. Therefore, if the multiplicity is denoted as \(I(P,t)\), we naturally have \(I(P,t)\ge 1\) and \(t\in T_c\) if and only if \(I(P,t)\ge 2\).
This means, if we pick a non-zero invariant differential on \(E\), the pull back of this differential along \(f_P\), \(\eta_P = f_P^*(\omega)\), has the order of vanishing
\[ \mathrm{ord}_t(\eta_P) = e_f(t) - 1 = I(P,t) - 1. \]
This immediately gives
\[ |T_{tors}| \le |T_c| = \sum_{t\in \mathcal{C}} 1_{I(P,t)>1} \le \sum_{t\in \mathcal{C}} (I(P,t)-1) = \sum_{t\in \mathcal{C}} \mathrm{ord}_t(\eta_P) = \deg \eta_P = 2g-2. \]
In the general case, we can define a Betti foliation on a open subset \(U\) of \(\mathcal{E}\), generalizing the foliation of \(\mathcal{C}\times E\) by its leaves \(\mathcal{C}\times \{p\}\) that has the subsets \(\mathcal{E}[n]\) among its closed leaves. Therefore we can consider the tangencies set \(T_{Betti}\subset \mathcal{C}\) and obtain a similar estimate via a real-analytic \(1\)-form \(\eta_P\) that generically satisfy \(J(\eta_P,t)=I(P,t)-1\) at all good reduction.
\[ \sum (I(P,t)-1) = \sum_{t\in \mathcal{C}} J(\eta_P,t) = 2g-2-d \]