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Points and their multiples on curves in powers of simple abelian varieties
Abstract
Let G be a simple abelian variety of dimension g ∈ ℕ defined over ℚalg and let C1, C2 ⊆ GN(ℂ) be irreducible closed algebraic curves with N ≥ 3. Further assume that at least one of C1 and C2 is not defined over ℚalg. Suppose that there does not exist an algebraic subgroup H1 ⊆ GN(ℂ) of dimension g such that C1 ⊆ H1 and that there does not exist an algebraic subgroup H2 ⊆ GN(ℂ) of dimension 2g such that C1 ∪ C2 ⊆ H2. Denoting 𝒩 = {n ∈ ℕ | [n]C1 ⊆ C2}, we prove that ⋃n ∈ ℕ ∖ 𝒩{x ∈ C1 | nx ∈ C2} is finite.
Related
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- Seshadri constants and hyperelliptic curves on abelian varieties
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- Intersecting a curve in an abelian variety with multiples of another curve
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BibTeX
@misc{smith2026points,
title = {{Points and their multiples on curves in powers of simple abelian varieties}},
author = {David J. Smith},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10209},
eprint = {2610.10209},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10209},
}
RIS
TY - UNPB
TI - Points and their multiples on curves in powers of simple abelian varieties
AU - David J. Smith
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.10209
UR - https://arxiv.org/abs/2610.10209
AB - Let G be a simple abelian variety of dimension g ∈ ℕ defined over ℚ^alg and let C_1, C_2 ⊆ G^N(ℂ) be irreducible closed algebraic curves with N ≥ 3. Further assume that at least one of C_1 and C_2 is not defined over ℚ^alg. Suppose that there does not exist an algebraic subgroup H_1 ⊆ G^N(ℂ) of dimension g such that C_1 ⊆ H_1 and that there does not exist an algebraic subgroup H_2 ⊆ G^N(ℂ) of dimension 2g such that C_1 ∪ C_2 ⊆ H_2. Denoting 𝒩 = {n ∈ ℕ | [n]C_1 ⊆ C_2}, we prove that ⋃_n ∈ ℕ ∖ 𝒩{x ∈ C_1 | nx ∈ C_2} is finite.
ER -