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arXiv · 2607.15741

Arithmetic unlikely intersections in powers of the multiplicative group

Francesco Campagna·Gabriel Andreas Dill·Robert Wilms
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Abstract

Inspired by work of Bugeaud-Corvaja-Zannier, we formulate a conjecture about unlikely intersections in powers of the multiplicative group over the ring of integers in a number field. Broadly speaking, if an intersection with a subgroup scheme is unlikely for dimension reasons, its ``size" should not be too big compared to the ``complexity" of the subgroup scheme. We first obtain some results on likely intersections that serve as a benchmark for the unlikely case and generalize work of Barroero-Capuano-M\'erai-Ostafe-Sha. We then show that our conjecture in dimension $1$ follows from work of Corvaja-Zannier, we obtain some partial result in dimension $2$, and we present some open problems that are special cases of the conjecture.

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Francesco Campagna, Gabriel Andreas Dill, Robert Wilms. 2026-07-17. Arithmetic unlikely intersections in powers of the multiplicative group. https://arxiv.org/abs/2607.15741

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