Unlikely intersections in semi-abelian surfaces

International audienceWe consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety G which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. We show that an irreducible curve W in G meets this set Zariski-densely only if W lies in a fiber of the family or is a translate of a Ribet curve by a multiplica-tive 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 G, when the parameter space is the universal one