---
_id: '13091'
abstract:
- lang: eng
text: We use a function field version of the Hardy–Littlewood circle method to study
the locus of free rational curves on an arbitrary smooth projective hypersurface
of sufficiently low degree. On the one hand this allows us to bound the dimension
of the singular locus of the moduli space of rational curves on such hypersurfaces
and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin
conjecture in terms of slopes with respect to the tangent bundle.
acknowledgement: The authors are grateful to Paul Nelson, Per Salberger and Jason
Starr for useful comments. While working on this paper the first author was supported
by EPRSC grant EP/P026710/1. The research was partially conducted during the period
the second author served as a Clay Research Fellow, and partially conducted during
the period he was supported by Dr. Max Rössler, the Walter Haefner Foundation and
the ETH Zurich Foundation.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
full_name: Browning, Timothy D
id: 35827D50-F248-11E8-B48F-1D18A9856A87
last_name: Browning
orcid: 0000-0002-8314-0177
- first_name: Will
full_name: Sawin, Will
last_name: Sawin
citation:
ama: Browning TD, Sawin W. Free rational curves on low degree hypersurfaces and
the circle method. Algebra & Number Theory. 2023;17(3):719-748. doi:10.2140/ant.2023.17.719
apa: Browning, T. D., & Sawin, W. (2023). Free rational curves on low degree
hypersurfaces and the circle method. Algebra & Number Theory. Mathematical
Sciences Publishers. https://doi.org/10.2140/ant.2023.17.719
chicago: Browning, Timothy D, and Will Sawin. “Free Rational Curves on Low Degree
Hypersurfaces and the Circle Method.” Algebra & Number Theory. Mathematical
Sciences Publishers, 2023. https://doi.org/10.2140/ant.2023.17.719.
ieee: T. D. Browning and W. Sawin, “Free rational curves on low degree hypersurfaces
and the circle method,” Algebra & Number Theory, vol. 17, no. 3. Mathematical
Sciences Publishers, pp. 719–748, 2023.
ista: Browning TD, Sawin W. 2023. Free rational curves on low degree hypersurfaces
and the circle method. Algebra & Number Theory. 17(3), 719–748.
mla: Browning, Timothy D., and Will Sawin. “Free Rational Curves on Low Degree Hypersurfaces
and the Circle Method.” Algebra & Number Theory, vol. 17, no. 3, Mathematical
Sciences Publishers, 2023, pp. 719–48, doi:10.2140/ant.2023.17.719.
short: T.D. Browning, W. Sawin, Algebra & Number Theory 17 (2023) 719–748.
corr_author: '1'
das_tickbox: '0'
date_created: 2023-05-28T22:01:02Z
date_published: 2023-04-12T00:00:00Z
date_updated: 2026-07-29T10:11:06Z
day: '12'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/ant.2023.17.719
external_id:
arxiv:
- '1810.06882'
isi:
- '000996014700004'
file:
- access_level: open_access
checksum: 5d5d67b235905650e33cf7065d7583b4
content_type: application/pdf
creator: dernst
date_created: 2023-05-30T08:05:22Z
date_updated: 2023-05-30T08:05:22Z
file_id: '13101'
file_name: 2023_AlgebraNumberTheory_Browning.pdf
file_size: 1430719
relation: main_file
success: 1
file_date_updated: 2023-05-30T08:05:22Z
fulldoi: https://doi.org/10.2140/ant.2023.17.719
has_accepted_license: '1'
intvolume: ' 17'
isi: 1
issue: '3'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 719-748
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
grant_number: EP-P026710-2
name: Between rational and integral points
publication: Algebra & Number Theory
publication_identifier:
eissn:
- 1944-7833
issn:
- 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Free rational curves on low degree hypersurfaces and the circle method
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 17
year: '2023'
...
---
_id: '14717'
abstract:
- lang: eng
text: We count primitive lattices of rank d inside Zn as their covolume tends to
infinity, with respect to certain parameters of such lattices. These parameters
include, for example, the subspace that a lattice spans, namely its projection
to the Grassmannian; its homothety class and its equivalence class modulo rescaling
and rotation, often referred to as a shape. We add to a prior work of Schmidt
by allowing sets in the spaces of parameters that are general enough to conclude
the joint equidistribution of these parameters. In addition to the primitive d-lattices
Λ themselves, we also consider their orthogonal complements in Zn, A1, and show
that the equidistribution occurs jointly for Λ and A1. Finally, our asymptotic
formulas for the number of primitive lattices include an explicit bound on the
error term.
acknowledgement: This work was done when both authors were visiting Institute of Science
and Technology (IST) Austria. T.H. was being supported by Engineering and Physical
Sciences Research Council grant EP/P026710/1. Y.K. had a great time there and is
grateful for the hospitality. The appendix to this paper is largely based on a mini
course T.H. had given at IST in February 2020.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Tal
full_name: Horesh, Tal
id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
last_name: Horesh
- first_name: Yakov
full_name: Karasik, Yakov
last_name: Karasik
citation:
ama: Horesh T, Karasik Y. Equidistribution of primitive lattices in ℝn. Quarterly
Journal of Mathematics. 2023;74(4):1253-1294. doi:10.1093/qmath/haad008
apa: Horesh, T., & Karasik, Y. (2023). Equidistribution of primitive lattices
in ℝn. Quarterly Journal of Mathematics. Oxford University Press. https://doi.org/10.1093/qmath/haad008
chicago: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices
in ℝn.” Quarterly Journal of Mathematics. Oxford University Press, 2023.
https://doi.org/10.1093/qmath/haad008.
ieee: T. Horesh and Y. Karasik, “Equidistribution of primitive lattices in ℝn,”
Quarterly Journal of Mathematics, vol. 74, no. 4. Oxford University Press,
pp. 1253–1294, 2023.
ista: Horesh T, Karasik Y. 2023. Equidistribution of primitive lattices in ℝn. Quarterly
Journal of Mathematics. 74(4), 1253–1294.
mla: Horesh, Tal, and Yakov Karasik. “Equidistribution of Primitive Lattices in
ℝn.” Quarterly Journal of Mathematics, vol. 74, no. 4, Oxford University
Press, 2023, pp. 1253–94, doi:10.1093/qmath/haad008.
short: T. Horesh, Y. Karasik, Quarterly Journal of Mathematics 74 (2023) 1253–1294.
corr_author: '1'
das_tickbox: '0'
date_created: 2023-12-31T23:01:03Z
date_published: 2023-12-01T00:00:00Z
date_updated: 2026-07-29T10:22:09Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1093/qmath/haad008
external_id:
arxiv:
- '2012.04508'
isi:
- '001005945400001'
file:
- access_level: open_access
checksum: bf29baa9eae8500f3374dbcb80712687
content_type: application/pdf
creator: dernst
date_created: 2024-01-02T07:37:09Z
date_updated: 2024-01-02T07:37:09Z
file_id: '14720'
file_name: 2023_QuarterlyJourMath_Horesh.pdf
file_size: 724748
relation: main_file
success: 1
file_date_updated: 2024-01-02T07:37:09Z
fulldoi: https://doi.org/10.1093/qmath/haad008
has_accepted_license: '1'
intvolume: ' 74'
isi: 1
issue: '4'
language:
- iso: eng
month: '12'
oa: 1
oa_version: Published Version
page: 1253-1294
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
grant_number: EP-P026710-2
name: Between rational and integral points
publication: Quarterly Journal of Mathematics
publication_identifier:
eissn:
- 1464-3847
issn:
- 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Equidistribution of primitive lattices in ℝn
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 74
year: '2023'
...
---
_id: '18179'
abstract:
- lang: eng
text: 'Linnik type problems concern the distribution of projections of integral
points on the unit sphere as their norm increases, and different generalizations
of this phenomenon. Our work addresses a question of this type: we prove the uniform
distribution of the projections of primitive Z2 points in the p-adic unit sphere,
as their (real) norm tends to infinity. The proof is via counting lattice points
in semi-simple S-arithmetic groups.'
- lang: fre
text: 'Les problèmes de type Linnik concernent la distribution des projections des
points entiers sur la sphère unitaire lorsque leur norme augmente et différentes
généralisations de ce phénomène. Notre travail s’intéresse à une question de ce
type : nous prouvons la distribution uniforme des projections des points primitifs
de Z2 sur la sphère unitaire p-adique lorsque leur norme (réelle) tend vers l’infini.
La preuve se fait en comptant les points d’un réseau dans des S-groupes arithmétiques
semi-simples.'
acknowledgement: The second author is supported by EPRSC grant EP/P026710/1.
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Antonin
full_name: Guilloux, Antonin
last_name: Guilloux
- first_name: Tal
full_name: Horesh, Tal
id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
last_name: Horesh
citation:
ama: Guilloux A, Horesh T. p-adic directions of primitive vectors. Publications
mathématiques de Besançon - Algèbre et Théorie des nombres. 2023;2023:85-107.
doi:10.5802/pmb.50
apa: Guilloux, A., & Horesh, T. (2023). p-adic directions of primitive vectors.
Publications Mathématiques de Besançon - Algèbre et Théorie Des Nombres.
Presses Universitaires de Franche-Comté. https://doi.org/10.5802/pmb.50
chicago: Guilloux, Antonin, and Tal Horesh. “P-Adic Directions of Primitive Vectors.”
Publications Mathématiques de Besançon - Algèbre et Théorie Des Nombres.
Presses Universitaires de Franche-Comté, 2023. https://doi.org/10.5802/pmb.50.
ieee: A. Guilloux and T. Horesh, “p-adic directions of primitive vectors,” Publications
mathématiques de Besançon - Algèbre et Théorie des nombres, vol. 2023. Presses
Universitaires de Franche-Comté, pp. 85–107, 2023.
ista: Guilloux A, Horesh T. 2023. p-adic directions of primitive vectors. Publications
mathématiques de Besançon - Algèbre et Théorie des nombres. 2023, 85–107.
mla: Guilloux, Antonin, and Tal Horesh. “P-Adic Directions of Primitive Vectors.”
Publications Mathématiques de Besançon - Algèbre et Théorie Des Nombres,
vol. 2023, Presses Universitaires de Franche-Comté, 2023, pp. 85–107, doi:10.5802/pmb.50.
short: A. Guilloux, T. Horesh, Publications Mathématiques de Besançon - Algèbre
et Théorie Des Nombres 2023 (2023) 85–107.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-10-06T22:01:13Z
date_published: 2023-06-15T00:00:00Z
date_updated: 2026-07-29T10:40:38Z
day: '15'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.5802/pmb.50
external_id:
arxiv:
- '2103.10889'
file:
- access_level: open_access
checksum: cefc47a2cf55a87f8e4960197f73353b
content_type: application/pdf
creator: dernst
date_created: 2024-10-07T11:32:32Z
date_updated: 2024-10-07T11:32:32Z
file_id: '18186'
file_name: 2023_MathBesancon_Guilloux.pdf
file_size: 1399390
relation: main_file
success: 1
file_date_updated: 2024-10-07T11:32:32Z
fulldoi: https://doi.org/10.5802/pmb.50
has_accepted_license: '1'
intvolume: ' 2023'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 85-107
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
grant_number: EP-P026710-2
name: Between rational and integral points
publication: Publications mathématiques de Besançon - Algèbre et Théorie des nombres
publication_identifier:
eissn:
- 2592-6616
issn:
- 2804-8504
publication_status: published
publisher: Presses Universitaires de Franche-Comté
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: p-adic directions of primitive vectors
tmp:
image: /image/cc_by_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nd/4.0/legalcode
name: Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)
short: CC BY-ND (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 2023
year: '2023'
...
---
_id: '9199'
abstract:
- lang: eng
text: "We associate a certain tensor product lattice to any primitive integer lattice
and ask about its typical shape. These lattices are related to the tangent bundle
of Grassmannians and their study is motivated by Peyre's programme on \"freeness\"
for rational points of bounded height on Fano\r\nvarieties."
acknowledgement: The authors are very grateful to Will Sawin for useful remarks about
this topic. While working on this paper the first two authors were supported by
EPSRC grant EP/P026710/1, and the first and last authors by FWF grant P 32428-N35.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
full_name: Browning, Timothy D
id: 35827D50-F248-11E8-B48F-1D18A9856A87
last_name: Browning
orcid: 0000-0002-8314-0177
- first_name: Tal
full_name: Horesh, Tal
id: C8B7BF48-8D81-11E9-BCA9-F536E6697425
last_name: Horesh
- first_name: Florian Alexander
full_name: Wilsch, Florian Alexander
id: 560601DA-8D36-11E9-A136-7AC1E5697425
last_name: Wilsch
orcid: 0000-0001-7302-8256
citation:
ama: Browning TD, Horesh T, Wilsch FA. Equidistribution and freeness on Grassmannians.
Algebra & Number Theory. 2022;16(10):2385-2407. doi:10.2140/ant.2022.16.2385
apa: Browning, T. D., Horesh, T., & Wilsch, F. A. (2022). Equidistribution and
freeness on Grassmannians. Algebra & Number Theory. Mathematical Sciences
Publishers. https://doi.org/10.2140/ant.2022.16.2385
chicago: Browning, Timothy D, Tal Horesh, and Florian Alexander Wilsch. “Equidistribution
and Freeness on Grassmannians.” Algebra & Number Theory. Mathematical
Sciences Publishers, 2022. https://doi.org/10.2140/ant.2022.16.2385.
ieee: T. D. Browning, T. Horesh, and F. A. Wilsch, “Equidistribution and freeness
on Grassmannians,” Algebra & Number Theory, vol. 16, no. 10. Mathematical
Sciences Publishers, pp. 2385–2407, 2022.
ista: Browning TD, Horesh T, Wilsch FA. 2022. Equidistribution and freeness on Grassmannians.
Algebra & Number Theory. 16(10), 2385–2407.
mla: Browning, Timothy D., et al. “Equidistribution and Freeness on Grassmannians.”
Algebra & Number Theory, vol. 16, no. 10, Mathematical Sciences Publishers,
2022, pp. 2385–407, doi:10.2140/ant.2022.16.2385.
short: T.D. Browning, T. Horesh, F.A. Wilsch, Algebra & Number Theory 16 (2022)
2385–2407.
corr_author: '1'
das_tickbox: '0'
date_created: 2021-02-25T09:56:57Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2026-08-06T11:07:27Z
day: '01'
department:
- _id: TiBr
doi: 10.2140/ant.2022.16.2385
external_id:
arxiv:
- '2102.11552'
isi:
- '000961514100004'
fulldoi: https://doi.org/10.2140/ant.2022.16.2385
intvolume: ' 16'
isi: 1
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2102.11552
month: '12'
oa: 1
oa_version: Preprint
page: 2385-2407
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
grant_number: EP-P026710-2
name: Between rational and integral points
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P32428
name: New frontiers of the Manin conjecture
publication: Algebra & Number Theory
publication_identifier:
eissn:
- 1944-7833
issn:
- 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Equidistribution and freeness on Grassmannians
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 16
year: '2022'
...
---
_id: '9260'
abstract:
- lang: eng
text: We study the density of rational points on a higher-dimensional orbifold (Pn−1,Δ)
when Δ is a Q-divisor involving hyperplanes. This allows us to address a question
of Tanimoto about whether the set of rational points on such an orbifold constitutes
a thin set. Our approach relies on the Hardy–Littlewood circle method to first
study an asymptotic version of Waring’s problem for mixed powers. In doing so
we make crucial use of the recent resolution of the main conjecture in Vinogradov’s
mean value theorem, due to Bourgain–Demeter–Guth and Wooley.
acknowledgement: While working on this paper the authors were both supported by EPSRC
grant EP/P026710/1, and the second author received additional support from the NWO
Veni Grant 016.Veni.192.047. Thanks are due to Marta Pieropan, Arne Smeets and Sho
Tanimoto for useful conversations related to this topic, and to the anonymous referee
for numerous helpful suggestions.
article_processing_charge: No
article_type: original
author:
- first_name: Timothy D
full_name: Browning, Timothy D
id: 35827D50-F248-11E8-B48F-1D18A9856A87
last_name: Browning
orcid: 0000-0002-8314-0177
- first_name: Shuntaro
full_name: Yamagishi, Shuntaro
last_name: Yamagishi
citation:
ama: Browning TD, Yamagishi S. Arithmetic of higher-dimensional orbifolds and a
mixed Waring problem. Mathematische Zeitschrift. 2021;299:1071–1101. doi:10.1007/s00209-021-02695-w
apa: Browning, T. D., & Yamagishi, S. (2021). Arithmetic of higher-dimensional
orbifolds and a mixed Waring problem. Mathematische Zeitschrift. Springer
Nature. https://doi.org/10.1007/s00209-021-02695-w
chicago: Browning, Timothy D, and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
Orbifolds and a Mixed Waring Problem.” Mathematische Zeitschrift. Springer
Nature, 2021. https://doi.org/10.1007/s00209-021-02695-w.
ieee: T. D. Browning and S. Yamagishi, “Arithmetic of higher-dimensional orbifolds
and a mixed Waring problem,” Mathematische Zeitschrift, vol. 299. Springer
Nature, pp. 1071–1101, 2021.
ista: Browning TD, Yamagishi S. 2021. Arithmetic of higher-dimensional orbifolds
and a mixed Waring problem. Mathematische Zeitschrift. 299, 1071–1101.
mla: Browning, Timothy D., and Shuntaro Yamagishi. “Arithmetic of Higher-Dimensional
Orbifolds and a Mixed Waring Problem.” Mathematische Zeitschrift, vol.
299, Springer Nature, 2021, pp. 1071–1101, doi:10.1007/s00209-021-02695-w.
short: T.D. Browning, S. Yamagishi, Mathematische Zeitschrift 299 (2021) 1071–1101.
das_tickbox: '0'
date_created: 2021-03-21T23:01:21Z
date_published: 2021-03-05T00:00:00Z
date_updated: 2026-08-06T11:14:26Z
day: '05'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s00209-021-02695-w
external_id:
isi:
- '000625573800002'
file:
- access_level: open_access
checksum: 8ed9f49568806894744096dbbca0ad7b
content_type: application/pdf
creator: dernst
date_created: 2021-03-22T12:41:26Z
date_updated: 2021-03-22T12:41:26Z
file_id: '9279'
file_name: 2021_MathZeitschrift_Browning.pdf
file_size: 492685
relation: main_file
success: 1
file_date_updated: 2021-03-22T12:41:26Z
fulldoi: https://doi.org/10.1007/s00209-021-02695-w
has_accepted_license: '1'
intvolume: ' 299'
isi: 1
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 1071–1101
project:
- _id: 26A8D266-B435-11E9-9278-68D0E5697425
grant_number: EP-P026710-2
name: Between rational and integral points
publication: Mathematische Zeitschrift
publication_identifier:
eissn:
- 1432-1823
issn:
- 0025-5874
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Arithmetic of higher-dimensional orbifolds and a mixed Waring problem
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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 299
year: '2021'
...