## Algebra & Number Theory

### Unlikely intersections in semiabelian 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.

#### Article information

Source
Algebra Number Theory, Volume 13, Number 6 (2019), 1455-1473.

Dates
Revised: 8 April 2019
Accepted: 14 May 2019
First available in Project Euclid: 21 August 2019

https://projecteuclid.org/euclid.ant/1566353015

Digital Object Identifier
doi:10.2140/ant.2019.13.1455

Mathematical Reviews number (MathSciNet)
MR3994572

Zentralblatt MATH identifier
07103981

#### Citation

Bertrand, Daniel; Schmidt, Harry. Unlikely intersections in semiabelian surfaces. Algebra Number Theory 13 (2019), no. 6, 1455--1473. doi:10.2140/ant.2019.13.1455. https://projecteuclid.org/euclid.ant/1566353015

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