---
_id: '10765'
abstract:
- lang: eng
text: We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski
open subsets in affine quadrics of the form q(x1,...,xn)=m, where q is a non-degenerate
integral quadratic form in n>3 variables and m is a non-zero integer. This gives
asymptotic formulas for the density of integral points taking coprime polynomial
values, which is a quantitative version of the arithmetic purity of strong approximation
property off infinity for affine quadrics.
acknowledgement: "We are grateful to Mikhail Borovoi, Zeev Rudnick and Olivier Wienberg
for their interest in our\r\nwork. We would like to address our gratitude to Ulrich
Derenthal for his generous support at Leibniz Universitat Hannover. We are in debt
to Tim Browning for an enlightening discussion and to the anonymous referees for
critical comments, which lead to overall improvements of various preliminary versions
of this paper. Part of this work was carried out and reported during a visit to
the University of Science and Technology of China. We thank Yongqi Liang for offering
warm hospitality. The first author was supported by a Humboldt-Forschungsstipendium.
The second author was supported by grant DE 1646/4-2 of the Deutsche Forschungsgemeinschaft."
article_number: '108236'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yang
full_name: Cao, Yang
last_name: Cao
- first_name: Zhizhong
full_name: Huang, Zhizhong
id: 21f1b52f-2fd1-11eb-a347-a4cdb9b18a51
last_name: Huang
citation:
ama: Cao Y, Huang Z. Arithmetic purity of the Hardy-Littlewood property and geometric
sieve for affine quadrics. Advances in Mathematics. 2022;398(3). doi:10.1016/j.aim.2022.108236
apa: Cao, Y., & Huang, Z. (2022). Arithmetic purity of the Hardy-Littlewood
property and geometric sieve for affine quadrics. Advances in Mathematics.
Elsevier. https://doi.org/10.1016/j.aim.2022.108236
chicago: Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood
Property and Geometric Sieve for Affine Quadrics.” Advances in Mathematics.
Elsevier, 2022. https://doi.org/10.1016/j.aim.2022.108236.
ieee: Y. Cao and Z. Huang, “Arithmetic purity of the Hardy-Littlewood property and
geometric sieve for affine quadrics,” Advances in Mathematics, vol. 398,
no. 3. Elsevier, 2022.
ista: Cao Y, Huang Z. 2022. Arithmetic purity of the Hardy-Littlewood property and
geometric sieve for affine quadrics. Advances in Mathematics. 398(3), 108236.
mla: Cao, Yang, and Zhizhong Huang. “Arithmetic Purity of the Hardy-Littlewood Property
and Geometric Sieve for Affine Quadrics.” Advances in Mathematics, vol.
398, no. 3, 108236, Elsevier, 2022, doi:10.1016/j.aim.2022.108236.
short: Y. Cao, Z. Huang, Advances in Mathematics 398 (2022).
date_created: 2022-02-20T23:01:30Z
date_published: 2022-03-26T00:00:00Z
date_updated: 2023-08-02T14:24:18Z
day: '26'
department:
- _id: TiBr
doi: 10.1016/j.aim.2022.108236
external_id:
arxiv:
- '2003.07287'
isi:
- '000792517300014'
intvolume: ' 398'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2003.07287
month: '03'
oa: 1
oa_version: Preprint
publication: Advances in Mathematics
publication_identifier:
eissn:
- 1090-2082
issn:
- 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Arithmetic purity of the Hardy-Littlewood property and geometric sieve for
affine quadrics
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 398
year: '2022'
...
---
_id: '10033'
abstract:
- lang: eng
text: The ⊗*-monoidal structure on the category of sheaves on the Ran space is not
pro-nilpotent in the sense of [3]. However, under some connectivity assumptions,
we prove that Koszul duality induces an equivalence of categories and that this
equivalence behaves nicely with respect to Verdier duality on the Ran space and
integrating along the Ran space, i.e. taking factorization homology. Based on
ideas sketched in [4], we show that these results also offer a simpler alternative
to one of the two main steps in the proof of the Atiyah-Bott formula given in
[7] and [5].
acknowledgement: 'The author would like to express his gratitude to D. Gaitsgory,
without whose tireless guidance and encouragement in pursuing this problem, this
work would not have been possible. The author is grateful to his advisor B.C. Ngô
for many years of patient guidance and support. This paper is revised while the
author is a postdoc in Hausel group at IST Austria. We thank him and the group for
providing a wonderful research environment. The author also gratefully acknowledges
the support of the Lise Meitner fellowship “Algebro-Geometric Applications of Factorization
Homology,” Austrian Science Fund (FWF): M 2751.'
article_number: '107992'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Quoc P
full_name: Ho, Quoc P
id: 3DD82E3C-F248-11E8-B48F-1D18A9856A87
last_name: Ho
orcid: 0000-0001-6889-1418
citation:
ama: Ho QP. The Atiyah-Bott formula and connectivity in chiral Koszul duality. Advances
in Mathematics. 2021;392. doi:10.1016/j.aim.2021.107992
apa: Ho, Q. P. (2021). The Atiyah-Bott formula and connectivity in chiral Koszul
duality. Advances in Mathematics. Elsevier. https://doi.org/10.1016/j.aim.2021.107992
chicago: Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul
Duality.” Advances in Mathematics. Elsevier, 2021. https://doi.org/10.1016/j.aim.2021.107992.
ieee: Q. P. Ho, “The Atiyah-Bott formula and connectivity in chiral Koszul duality,”
Advances in Mathematics, vol. 392. Elsevier, 2021.
ista: Ho QP. 2021. The Atiyah-Bott formula and connectivity in chiral Koszul duality.
Advances in Mathematics. 392, 107992.
mla: Ho, Quoc P. “The Atiyah-Bott Formula and Connectivity in Chiral Koszul Duality.”
Advances in Mathematics, vol. 392, 107992, Elsevier, 2021, doi:10.1016/j.aim.2021.107992.
short: Q.P. Ho, Advances in Mathematics 392 (2021).
date_created: 2021-09-21T15:58:59Z
date_published: 2021-09-21T00:00:00Z
date_updated: 2023-08-14T06:54:35Z
day: '21'
ddc:
- '514'
department:
- _id: TaHa
doi: 10.1016/j.aim.2021.107992
external_id:
arxiv:
- '1610.00212'
isi:
- '000707040300031'
file:
- access_level: open_access
checksum: f3c0086d41af11db31c00014efb38072
content_type: application/pdf
creator: qho
date_created: 2021-09-21T15:58:52Z
date_updated: 2021-09-21T15:58:52Z
file_id: '10034'
file_name: 1-s2.0-S000187082100431X-main.pdf
file_size: 840635
relation: main_file
file_date_updated: 2021-09-21T15:58:52Z
has_accepted_license: '1'
intvolume: ' 392'
isi: 1
keyword:
- Chiral algebras
- Chiral homology
- Factorization algebras
- Koszul duality
- Ran space
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 26B96266-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M02751
name: Algebro-Geometric Applications of Factorization Homology
publication: Advances in Mathematics
publication_identifier:
eissn:
- 1090-2082
issn:
- 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: The Atiyah-Bott formula and connectivity in chiral Koszul duality
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 392
year: '2021'
...