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Intersecting a curve in an abelian variety with multiples of another curve
Abstract
Levin asked what can be said about the locus of points lying on a given curve in 𝔾mn that have a non-zero integer power on another given curve in 𝔾mn for n ≥ 3. We give a definite answer to the abelian analog of Levin's question, proving what is predicted by the Zilber-Pink conjecture in this case. An important ingredient in our proof is a strengthening of a height inequality by Vojta and Rémond.
Related
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BibTeX
@misc{barroero2026intersecting,
title = {{Intersecting a curve in an abelian variety with multiples of another curve}},
author = {Fabrizio Barroero and Gabriel A. Dill and Lars Kuehne},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10212},
eprint = {2610.10212},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10212},
}
RIS
TY - UNPB
TI - Intersecting a curve in an abelian variety with multiples of another curve
AU - Fabrizio Barroero
AU - Gabriel A. Dill
AU - Lars Kuehne
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.10212
UR - https://arxiv.org/abs/2610.10212
AB - Levin asked what can be said about the locus of points lying on a given curve in 𝔾_m^n that have a non-zero integer power on another given curve in 𝔾_m^n for n ≥ 3. We give a definite answer to the abelian analog of Levin's question, proving what is predicted by the Zilber-Pink conjecture in this case. An important ingredient in our proof is a strengthening of a height inequality by Vojta and Rémond.
ER -