• betti map
• definability
• elliptic schemes
• heights
• o-minimality
• torsion

An elliptic scheme is a surface E with a morphism pi: E -> B to a curve B such that all but finitely many fibers are elliptic curves. The points on the fibers can be parametrized using the Betti map (introduced by Manin in 1963). It was proven in a paper of Corvaja, Demeio, Masser, Zannier (2018) that, given a finitely generated group G of sections for pi defined over Qbar, there are finitely many points p of B such that, for some s \in G, s(p) is torsion and s is tangent to the fiber of the Betti map in s(p); this is motivated by questions of diophantine approximation related to quasi-integral points on function fields. A key ingredient is a result of o-minimality (Jones, Schimdt, 2018): the Betti map is definable in the structure R_{an,exp}. We extend the theorem to sections defined over C, using a method that can be fruitful for other generalizations from Qbar to C, for instance for recent work of Barroero and Capuano. The exposition is self-contained and we review the basics of elliptic schemes, heights (to deal with issues over Qbar) and o-minimality (to extend the results to C).