Unlikely intersections in semi-abelian surfaces

TL;DR

This paper proves a special case of the Zilber-Pink conjecture for a certain mixed Shimura variety related to semi-abelian surfaces, showing that unlikely intersections occur only under specific geometric conditions.

Contribution

It establishes a new unlikely intersection result for a family of semi-abelian surfaces, confirming the Zilber-Pink conjecture in this context.

Findings

Irreducible curves intersecting dense special sets are contained in specific fibers or translates.

The result applies to the universal family over the parameter space.

Provides a proof of the Zilber-Pink conjecture for the associated mixed Shimura variety.

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.