Unlikely intersections in semi-abelian surfaces

Abstract

We 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 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 $G$, when the parameter space is the universal one.

20 pages. Appendix added, with a proof of the Zilber-Pink for the Poincaré-biextension over a CM elliptic curve

References in corpus (1)

• CM relations in fibered powers of elliptic families

Cited by in corpus (1)

• Pink's conjecture on unlikely intersections and families of semi-abelian varieties