PreprintCC BY-NC-ND 4.0HAL
ARITHMETIC UNLIKELY INTERSECTIONS IN POWERS OF THE MULTIPLICATIVE GROUP
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-Mérai-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.
Related
- Degeneracy loci and unlikely intersections in abelian schemes
- Regularity of Symbolic Powers of Polymatroidal Ideals and Supermodular Prime Intersections
- Arithmetic purity of strong approximation for complete intersections of large dimension
- On the arithmetic rank and cohomological dimension of some residual intersections
- Truncated pretentious distances of multiplicative arithmetic functions
BibTeX
@misc{campagna2026arithmetic,
title = {{ARITHMETIC UNLIKELY INTERSECTIONS IN POWERS OF THE MULTIPLICATIVE GROUP}},
author = {Francesco Campagna and Gabriel A Dill and Robert Wilms},
howpublished = {HAL},
year = {2026},
month = jul,
url = {https://hal.science/hal-05779467v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - ARITHMETIC UNLIKELY INTERSECTIONS IN POWERS OF THE MULTIPLICATIVE GROUP
AU - Francesco Campagna
AU - Gabriel A Dill
AU - Robert Wilms
PY - 2026
DA - 2026/07/17
PB - HAL
UR - https://hal.science/hal-05779467v1
AB - 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-Mérai-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.
ER -