PreprintarXiv licencearXiv
Arithmetic purity of strong approximation for complete intersections of large dimension
Abstract
In this paper, we prove that for a smooth complete intersection X⊂ ℙn of type (d1,…,dc) over a number field k, there exists an effective bound NAP(d1,…,dc) such that if n≥ NAP(d1,…,dc), then X satisfies arithmetic purity of strong approximation. We establish a connecting lemma that promotes weak approximation to strong approximation under a suitable connectivity hypothesis via rational curves over k. We construct the required connecting conics by combining the deformation theory of rational curves with results from the circle method.
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BibTeX
@misc{zhang2026arithmetic,
title = {{Arithmetic purity of strong approximation for complete intersections of large dimension}},
author = {Chen Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.27027},
eprint = {2608.27027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27027},
}
RIS
TY - UNPB
TI - Arithmetic purity of strong approximation for complete intersections of large dimension
AU - Chen Zhang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2608.27027
UR - https://arxiv.org/abs/2608.27027
AB - In this paper, we prove that for a smooth complete intersection X⊂ ℙ^n of type (d_1,…,d_c) over a number field k, there exists an effective bound N_AP(d_1,…,d_c) such that if n≥ N_AP(d_1,…,d_c), then X satisfies arithmetic purity of strong approximation. We establish a connecting lemma that promotes weak approximation to strong approximation under a suitable connectivity hypothesis via rational curves over k. We construct the required connecting conics by combining the deformation theory of rational curves with results from the circle method.
ER -