Algebra & Number Theory

Unlikely intersections in semiabelian surfaces

Daniel Bertrand and Harry Schmidt

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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 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

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

Received: 6 November 2018
Revised: 8 April 2019
Accepted: 14 May 2019
First available in Project Euclid: 21 August 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx]
Secondary: 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 11G50: Heights [See also 14G40, 37P30] 11U09: Model theory [See also 03Cxx]

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


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.

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