Specializations of elliptic surfaces, and divisibility in the Mordell-Weil group (Q655997)

Let \(\mathcal{E} \rightarrow C\) be an elliptic surface defined over a number field \(k\). Consider \(t\in C(k)\) such that its fiber \(\mathcal{E}_t\) is nonsingular and let \(\sigma_t\) be the morphism from the group of sections \(\mathcal{E}(C)\) to \(\mathcal{E}_t(k)\). The map \(\sigma_t\) is injective for almost all \(t\) and the paper deals with its cokernel. In particular, taking an elliptic surface with non constant \(j\)-invariant and a section (defined over \(k\)) \(P: C \rightarrow \mathcal{E}\), the author proves that for any couple of constants \(B_1\) and \(B_2\,\), there exists a constant \(M=M(B_1,B_2)\) which is an upper bound for the cardinality of the set \[ \left\{ Q\in \mathcal{E}_t(\overline{k})\;s.t.\;[k(Q):k]\leq B_1\quad\text{ and}\quad l^nQ=\sigma_t(P)\text{ for some } n\geq 1\,\right\} \] as \(t\) varies over the places of good reduction for \(\mathcal{E}\) with \([k(t):k]\leq B_2\,\). This theorem provides a bound (independent of \(t\)) for the \(l\)-power part of the quotient between the Mordell-Weil group and the image of the sections. The bound can be made more explicit if the section \(P\) is not \(l\)-divisible and one avoids the \textit{special primes}, i.e., 2 and the ones dividing the order of at least one pole of the \(j\)-invariant. The main ingredients for the proof are irreducibility results for the Galois orbits on the set of \(l^n\)-division points of \(P\) and the construction of a \textit{ramification tree} defined by letting \(\Gamma_1\) be the pull-back of the image \(\Gamma_0\) of the section \(P\) via the multiplication-by-\(l\) map \([l]:\mathcal{E} \rightarrow \mathcal{E}\), then pulling back \(\Gamma_1\) and so on. The details of the construction and the properties of the ramification tree are described in section 4 and, using some results in diophantine geometry and Tate uniformization, the author proves that such tree has only finitely many infinite paths. This \textit{non branching} property, together with the irreducibility of Galois orbits, provides the bounds appearing in the main theorems.