We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. We show that an irreducible curve in meets this set Zariski-densely only if lies in a fiber of the family or is a translate of a Ribet curve by a multiplicative section. We further deduce from this result a proof of the Zilber–Pink conjecture (over number fields) for the mixed Shimura variety attached to the threefold , when the parameter space is the universal one.
"Unlikely intersections in semiabelian surfaces." Algebra Number Theory 13 (6) 1455 - 1473, 2019. https://doi.org/10.2140/ant.2019.13.1455