Unlikely intersections in semiabelian surfaces

Bertrand, Daniel; Schmidt, Harry

doi:10.2140/ant.2019.13.1455

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Unlikely intersections in semiabelian surfaces

Daniel Bertrand and Harry Schmidt

Vol. 13 (2019), No. 6, 1455–1473

DOI: 10.2140/ant.2019.13.1455

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.

Keywords

semiabelian varieties, complex multiplication, Zilber–Pink conjecture, mixed Shimura varieties, heights, $o$-minimality, Ribet sections

Mathematical Subject Classification 2010

Primary: 14K15

Secondary: 11G15, 11G50, 11U09

Milestones

Received: 6 November 2018

Revised: 8 April 2019

Accepted: 14 May 2019

Published: 18 August 2019

Authors

Daniel Bertrand
Institut de Mathematiques de Jussieu Sorbonne Université Paris France

Harry Schmidt
Mathematik und Informatik Universität Basel Switzerland

Vol. 13, No. 6, 2019