Translator Disclaimer
2019 Unlikely intersections in semiabelian surfaces
Daniel Bertrand, Harry Schmidt
Algebra Number Theory 13(6): 1455-1473 (2019). 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.

Citation

Download Citation

Daniel Bertrand. Harry Schmidt. "Unlikely intersections in semiabelian surfaces." Algebra Number Theory 13 (6) 1455 - 1473, 2019. https://doi.org/10.2140/ant.2019.13.1455

Information

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

zbMATH: 07103981
MathSciNet: MR3994572
Digital Object Identifier: 10.2140/ant.2019.13.1455

Subjects:
Primary: 14K15
Secondary: 11G15 , 11G50 , 11U09

Keywords: $o$-minimality , Complex Multiplication , heights , mixed Shimura varieties , Ribet sections , semiabelian varieties , Zilber–Pink conjecture

Rights: Copyright © 2019 Mathematical Sciences Publishers

JOURNAL ARTICLE
19 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.13 • No. 6 • 2019
MSP
Back to Top