Vol. 13, No. 6, 2019

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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

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.

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
Received: 6 November 2018
Revised: 8 April 2019
Accepted: 14 May 2019
Published: 18 August 2019
Daniel Bertrand
Institut de Mathematiques de Jussieu
Sorbonne Université
Harry Schmidt
Mathematik und Informatik
Universität Basel