Details
| Original language | English |
|---|---|
| Pages (from-to) | 2628-2649 |
| Number of pages | 22 |
| Journal | Compositio Mathematica |
| Volume | 156 |
| Issue number | 12 |
| Publication status | Published - 1 Feb 2021 |
| Externally published | Yes |
Abstract
In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field k. For instance, for any such group G and for any open subset U of G with codim(G \ U, G) ≥ 2, we prove that (i) if G is k-simple and k-isotropic, then U satisfies strong approximation off any finite number of places; and (ii) if G is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then U satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of G \ U, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.
Keywords
- math.NT, math.AG, 14G12, 11G35, 20G35, 11N36
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In: Compositio Mathematica, Vol. 156, No. 12, 01.02.2021, p. 2628-2649.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Arithmetic purity of strong approximation for semi-simple simply connected groups
AU - Cao, Yang
AU - Huang, Zhizhong
N1 - Funding Information: We would like to express our heartfelt thanks to the anonymous referees for their careful scrutiny and valuable suggestions. The first author is supported by a Humboldt-Forschungsstipendium. The second author is supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft.
PY - 2021/2/1
Y1 - 2021/2/1
N2 - In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field k. For instance, for any such group G and for any open subset U of G with codim(G \ U, G) ≥ 2, we prove that (i) if G is k-simple and k-isotropic, then U satisfies strong approximation off any finite number of places; and (ii) if G is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then U satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of G \ U, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.
AB - In this article we establish the arithmetic purity of strong approximation for certain semisimple simply connected linear algebraic groups and their homogeneous spaces over a number field k. For instance, for any such group G and for any open subset U of G with codim(G \ U, G) ≥ 2, we prove that (i) if G is k-simple and k-isotropic, then U satisfies strong approximation off any finite number of places; and (ii) if G is the spin group of a non-degenerate quadratic form which is not compact over archimedean places, then U satisfies strong approximation off all archimedean places. As a consequence, we prove that the same property holds for affine quadratic hypersurfaces. Our approach combines a fibration method with subgroup actions developed for induction on the codimension of G \ U, and an affine linear sieve which allows us to produce integral points with almost-prime polynomial values.
KW - math.NT
KW - math.AG
KW - 14G12, 11G35, 20G35, 11N36
UR - http://www.scopus.com/inward/record.url?scp=85101420154&partnerID=8YFLogxK
U2 - 10.48550/arXiv.1906.06967
DO - 10.48550/arXiv.1906.06967
M3 - Article
VL - 156
SP - 2628
EP - 2649
JO - Compositio Mathematica
JF - Compositio Mathematica
SN - 0010-437X
IS - 12
ER -