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Unlikely intersections
in semiabelian surfaces
Daniel Bertrand and Harry Schmidt |
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Vol. 13 (2019), No. 6, 1455–1473
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DOI: 10.2140/ant.2019.13.1455
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Abstract |
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We consider a family, depending on a parameter, of multiplicative extensions of an elliptic curve with complex multiplications. They form a 3-dimensional variety which admits a dense set of special curves, known as Ribet curves, which strictly contains the torsion curves. We show that an irreducible curve in meets this set Zariski-densely only if 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 , when the parameter space is the universal one. |
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Keywords
semiabelian varieties, complex multiplication, Zilber–Pink
conjecture, mixed Shimura varieties, heights,
$o$-minimality, Ribet sections
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Mathematical Subject Classification 2010
Primary: 14K15
Secondary: 11G15, 11G50, 11U09
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Milestones
Received: 6 November 2018
Revised: 8 April 2019
Accepted: 14 May 2019
Published: 18 August 2019
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Authors |
| Daniel Bertrand | |
| Institut de Mathematiques de
Jussieu Sorbonne Université Paris France |
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| Harry Schmidt | |
| Mathematik und Informatik Universität Basel Switzerland |
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