# 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.

## Key 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.