## Duke Mathematical Journal

Published by Duke University Press since its inception in 1935, the Duke Mathematical Journal is one of the world's leading mathematical journals. DMJ emphasizes the most active and influential areas of current mathematics. Advance publication of articles online is available.

## Top downloads over the last seven days

The fundamental lemma of Jacquet and RallisVolume 156, Number 2 (2011)
Counterexamples to a conjecture of WoodsVolume 166, Number 13 (2017)
On the zeros of $\zeta'(s)$ near the critical lineVolume 110, Number 3 (2001)
CM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$Volume 166, Number 13 (2017)
Representation stability and finite linear groupsVolume 166, Number 13 (2017)
• ISSN: 0012-7094 (print), 1547-7398 (electronic)
• Publisher: Duke University Press
• Discipline(s): Mathematics
• Full text available in Euclid: 1935--
• Access: By subscription only
• Euclid URL: https://projecteuclid.org/dmj

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MR Citation Database MCQ (2016): 2.23
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Eigenfactor: Duke Mathematical Journal
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### Featured article

#### Bounded height in pencils of finitely generated subgroups

Volume 166, Number 13 (2017)
##### Abstract

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 ${{\mathbb{G}_{m}^{r}}_{/\mathcal{C}}}$ over a curve $\mathcal{C}$ defined over $\overline{\mathbb{Q}}$, and let $\Gamma$ 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 $P\in\mathcal{C}(\overline{\mathbb{Q}})$ such that, for some $\gamma\in\Gamma$ which does not generically lie in $V$, $\gamma(P)$ lies in the fiber $V_{P}$. 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.