Finite Orbits in Surfaces with a Double Elliptic Fibration and torsion values of sections

We consider surfaces with two elliptic fibrations, each of them provided with a section. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a similar direction of (and is motivated by) recent work by Cantat–Dujardin, although we use very different methods and obtain related but different results. As a sample of application of similar arguments, we prove a new case of the Zilber–Pink conjecture, namely Theorem 1.5, for certain schemes over a 2-dimensional base, which was known to lead to substantial difficulties. Most results rely, among other things, on recent theorems by Bakker and the second author of “Ax–Schanuel type”; we also relate a functional condition with a theorem of Shioda on unramified sections of the Legendre scheme. For one of our proofs, we also use recent height inequalities by Dimitrov–Gao–Habegger (or those by Yuan–Zhang). Finally, in an appendix, we show that the Relative Manin–Mumford Conjecture over the complex number field is equivalent to its version over the field of algebraic numbers.