15 September 2017 Bounded height in pencils of finitely generated subgroups
F. Amoroso, D. Masser, U. Zannier
Duke Math. J. 166(13): 2599-2642 (15 September 2017). DOI: 10.1215/00127094-2017-0009


In this article we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell–Lang theorem by replacing finiteness with emptiness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the result is as follows. Consider the torus scheme Gmr/C over a curve C defined over Q¯, and let Γ be a subgroup scheme generated by finitely many sections (satisfying some necessary conditions). Further, let V be any subscheme. Then there is a bound for the height of the points PC(Q¯) such that, for some γΓ which does not generically lie in V, γ(P) lies in the fiber VP. We further offer some direct Diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.


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F. Amoroso. D. Masser. U. Zannier. "Bounded height in pencils of finitely generated subgroups." Duke Math. J. 166 (13) 2599 - 2642, 15 September 2017. https://doi.org/10.1215/00127094-2017-0009


Received: 7 July 2016; Revised: 21 December 2016; Published: 15 September 2017
First available in Project Euclid: 18 July 2017

zbMATH: 06797414
MathSciNet: MR3703436
Digital Object Identifier: 10.1215/00127094-2017-0009

Primary: 11GXX
Secondary: 11DXX

Keywords: Diophantine geometry , specialization problems , unlikely intersections

Rights: Copyright © 2017 Duke University Press


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Vol.166 • No. 13 • 15 September 2017
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