---
_id: '8742'
abstract:
- lang: eng
  text: We develop a version of Ekedahl’s geometric sieve for integral quadratic forms
    of rank at least five. As one ranges over the zeros of such quadratic forms, we
    use the sieve to compute the density of coprime values of polynomials, and furthermore,
    to address a question about local solubility in families of varieties parameterised
    by the zeros.
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: Roger
  full_name: Heath-Brown, Roger
  last_name: Heath-Brown
citation:
  ama: Browning TD, Heath-Brown R. The geometric sieve for quadrics. <i>Forum Mathematicum</i>.
    2021;33(1):147-165. doi:<a href="https://doi.org/10.1515/forum-2020-0074">10.1515/forum-2020-0074</a>
  apa: Browning, T. D., &#38; Heath-Brown, R. (2021). The geometric sieve for quadrics.
    <i>Forum Mathematicum</i>. De Gruyter. <a href="https://doi.org/10.1515/forum-2020-0074">https://doi.org/10.1515/forum-2020-0074</a>
  chicago: Browning, Timothy D, and Roger Heath-Brown. “The Geometric Sieve for Quadrics.”
    <i>Forum Mathematicum</i>. De Gruyter, 2021. <a href="https://doi.org/10.1515/forum-2020-0074">https://doi.org/10.1515/forum-2020-0074</a>.
  ieee: T. D. Browning and R. Heath-Brown, “The geometric sieve for quadrics,” <i>Forum
    Mathematicum</i>, vol. 33, no. 1. De Gruyter, pp. 147–165, 2021.
  ista: Browning TD, Heath-Brown R. 2021. The geometric sieve for quadrics. Forum
    Mathematicum. 33(1), 147–165.
  mla: Browning, Timothy D., and Roger Heath-Brown. “The Geometric Sieve for Quadrics.”
    <i>Forum Mathematicum</i>, vol. 33, no. 1, De Gruyter, 2021, pp. 147–65, doi:<a
    href="https://doi.org/10.1515/forum-2020-0074">10.1515/forum-2020-0074</a>.
  short: T.D. Browning, R. Heath-Brown, Forum Mathematicum 33 (2021) 147–165.
date_created: 2020-11-08T23:01:25Z
date_published: 2021-01-01T00:00:00Z
date_updated: 2025-04-15T07:39:01Z
day: '01'
department:
- _id: TiBr
doi: 10.1515/forum-2020-0074
external_id:
  arxiv:
  - '2003.09593'
  isi:
  - '000604750900008'
intvolume: '        33'
isi: 1
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2003.09593
month: '01'
oa: 1
oa_version: Preprint
page: 147-165
project:
- _id: 26AEDAB2-B435-11E9-9278-68D0E5697425
  call_identifier: FWF
  grant_number: P32428
  name: New frontiers of the Manin conjecture
publication: Forum Mathematicum
publication_identifier:
  eissn:
  - 1435-5337
  issn:
  - 0933-7741
publication_status: published
publisher: De Gruyter
quality_controlled: '1'
scopus_import: '1'
status: public
title: The geometric sieve for quadrics
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 33
year: '2021'
...
---
OA_place: repository
OA_type: green
_id: '257'
abstract:
- lang: eng
  text: For suitable pairs of diagonal quadratic forms in eight variables we use the
    circle method to investigate the density of simultaneous integer solutions and
    relate this to the problem of estimating linear correlations among sums of two
    squares.
acknowledgement: While working on this paper the first author was supported by ERC
  grant 306457 and the second author was supported by SwarnaJayanti Fellowship 2011–12,
  DST, Government of India.
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: Ritabrata
  full_name: Munshi, Ritabrata
  last_name: Munshi
citation:
  ama: Browning TD, Munshi R. Pairs of diagonal quadratic forms and linear correlations
    among sums of two squares. <i>Forum Mathematicum</i>. 2015;27(4):2025-2050. doi:<a
    href="https://doi.org/10.1515/forum-2013-6024">10.1515/forum-2013-6024</a>
  apa: Browning, T. D., &#38; Munshi, R. (2015). Pairs of diagonal quadratic forms
    and linear correlations among sums of two squares. <i>Forum Mathematicum</i>.
    De Gruyter. <a href="https://doi.org/10.1515/forum-2013-6024">https://doi.org/10.1515/forum-2013-6024</a>
  chicago: Browning, Timothy D, and Ritabrata Munshi. “Pairs of Diagonal Quadratic
    Forms and Linear Correlations among Sums of Two Squares.” <i>Forum Mathematicum</i>.
    De Gruyter, 2015. <a href="https://doi.org/10.1515/forum-2013-6024">https://doi.org/10.1515/forum-2013-6024</a>.
  ieee: T. D. Browning and R. Munshi, “Pairs of diagonal quadratic forms and linear
    correlations among sums of two squares,” <i>Forum Mathematicum</i>, vol. 27, no.
    4. De Gruyter, pp. 2025–2050, 2015.
  ista: Browning TD, Munshi R. 2015. Pairs of diagonal quadratic forms and linear
    correlations among sums of two squares. Forum Mathematicum. 27(4), 2025–2050.
  mla: Browning, Timothy D., and Ritabrata Munshi. “Pairs of Diagonal Quadratic Forms
    and Linear Correlations among Sums of Two Squares.” <i>Forum Mathematicum</i>,
    vol. 27, no. 4, De Gruyter, 2015, pp. 2025–50, doi:<a href="https://doi.org/10.1515/forum-2013-6024">10.1515/forum-2013-6024</a>.
  short: T.D. Browning, R. Munshi, Forum Mathematicum 27 (2015) 2025–2050.
date_created: 2018-12-11T11:45:28Z
date_published: 2015-07-10T00:00:00Z
date_updated: 2026-05-19T13:08:06Z
day: '10'
doi: 10.1515/forum-2013-6024
extern: '1'
external_id:
  arxiv:
  - '1302.2434'
intvolume: '        27'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1302.2434
month: '07'
oa: 1
oa_version: Preprint
page: 2025 - 2050
publication: Forum Mathematicum
publication_identifier:
  eissn:
  - 1435-5337
  issn:
  - 0933-7741
publication_status: published
publisher: De Gruyter
publist_id: '7645'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Pairs of diagonal quadratic forms and linear correlations among sums of two
  squares
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 27
year: '2015'
...
---
OA_place: repository
OA_type: green
_id: '22049'
abstract:
- lang: eng
  text: 'We consider the minimal mass m0 required for solutions to the mass-critical
    nonlinear Schrödinger (NLS) equation iut + Δu = μ|u|^4/d u to blow up. If m0 is
    finite, we show that there exists a minimal-mass solution blowing up (in the sense
    of an infinite spacetime norm) in both time directions, whose orbit in  is compact
    after quotienting out by the symmetries of the equation. A similar result is obtained
    for spherically symmetric solutions. Similar results were previously obtained
    by Keraani, [Keraani S.: On the blow-up phenomenon of the critical nonlinear Schrödinger
    equation. J. Funct. Anal. 235 (2006), 171–192], in dimensions 1, 2 and Begout
    and Vargas, [Begout P., Vargas A.: Mass concentration phenomena for the L2-critical
    nonlinear Schrödinger equation, preprint], in dimensions d ≥ 3 for the mass-critical
    NLS and by Kenig and Merle, [Kenig C., Merle F.: Global well-posedness, scattering,
    and blowup for the energy-critical, focusing, non-linear Schrödinger equation
    in the radial case, preprint], in the energy-critical case. In a subsequent paper
    we shall use this compactness result to establish global existence and scattering
    in  for the defocusing NLS in three and higher dimensions with spherically symmetric
    data.'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Terence
  full_name: Tao, Terence
  last_name: Tao
- first_name: Monica
  full_name: Visan, Monica
  id: 056daca0-b8d1-11f0-964f-f91054abf8ca
  last_name: Visan
- first_name: Xiaoyi
  full_name: Zhang, Xiaoyi
  last_name: Zhang
citation:
  ama: Tao T, Vişan M, Zhang X. Minimal-mass blowup solutions of the mass-critical
    NLS. <i>Forum Mathematicum</i>. 2008;20(5):881-919. doi:<a href="https://doi.org/10.1515/forum.2008.042">10.1515/forum.2008.042</a>
  apa: Tao, T., Vişan, M., &#38; Zhang, X. (2008). Minimal-mass blowup solutions of
    the mass-critical NLS. <i>Forum Mathematicum</i>. De Gruyter. <a href="https://doi.org/10.1515/forum.2008.042">https://doi.org/10.1515/forum.2008.042</a>
  chicago: Tao, Terence, Monica Vişan, and Xiaoyi Zhang. “Minimal-Mass Blowup Solutions
    of the Mass-Critical NLS.” <i>Forum Mathematicum</i>. De Gruyter, 2008. <a href="https://doi.org/10.1515/forum.2008.042">https://doi.org/10.1515/forum.2008.042</a>.
  ieee: T. Tao, M. Vişan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical
    NLS,” <i>Forum Mathematicum</i>, vol. 20, no. 5. De Gruyter, pp. 881–919, 2008.
  ista: Tao T, Vişan M, Zhang X. 2008. Minimal-mass blowup solutions of the mass-critical
    NLS. Forum Mathematicum. 20(5), 881–919.
  mla: Tao, Terence, et al. “Minimal-Mass Blowup Solutions of the Mass-Critical NLS.”
    <i>Forum Mathematicum</i>, vol. 20, no. 5, De Gruyter, 2008, pp. 881–919, doi:<a
    href="https://doi.org/10.1515/forum.2008.042">10.1515/forum.2008.042</a>.
  short: T. Tao, M. Vişan, X. Zhang, Forum Mathematicum 20 (2008) 881–919.
das_tickbox: '1'
date_created: 2026-06-19T07:53:12Z
date_published: 2008-11-03T00:00:00Z
date_updated: 2026-06-25T08:15:22Z
day: '03'
doi: 10.1515/forum.2008.042
extern: '1'
external_id:
  arxiv:
  - math/0609690
intvolume: '        20'
issue: '5'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.math/0609690
mathsc:
- 35Q55
month: '11'
oa: 1
oa_version: Preprint
page: 881-919
publication: Forum Mathematicum
publication_identifier:
  eissn:
  - 1435-5337
  issn:
  - 0933-7741
publication_status: published
publisher: De Gruyter
quality_controlled: '1'
scopus_import: '1'
status: public
title: Minimal-mass blowup solutions of the mass-critical NLS
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 20
year: '2008'
...