---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '12311'
abstract:
- lang: eng
  text: In this note, we prove a formula for the cancellation exponent  kv,n between
    division polynomials  ψn  and  ϕn  associated with a sequence  {nP}n∈N of points
    on an elliptic curve  E  defined over a discrete valuation field  K. The formula
    greatly generalizes the previously known special cases and treats also the case
    of non-standard Kodaira types for non-perfect residue fields.
acknowledgement: Silverman, and Paul Voutier for the comments on the earlier version
  of this paper. The first author acknowledges the support by Dioscuri programme initiated
  by the Max Planck Society, jointly managed with the National Science Centre (Poland),
  and mutually funded by the Polish Ministry of Science and Higher Education and the
  German Federal Ministry of Education and Research. The second author has been supported
  by MIUR (Italy) through PRIN 2017 ‘Geometric, algebraic and analytic methods in
  arithmetic’ and has received funding from the European Union's Horizon 2020 research
  and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Bartosz
  full_name: Naskręcki, Bartosz
  last_name: Naskręcki
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: 'Naskręcki B, Verzobio M. Common valuations of division polynomials. <i>Proceedings
    of the Royal Society of Edinburgh Section A: Mathematics</i>. 2025;155(5):1646-1660.
    doi:<a href="https://doi.org/10.1017/prm.2024.7">10.1017/prm.2024.7</a>'
  apa: 'Naskręcki, B., &#38; Verzobio, M. (2025). Common valuations of division polynomials.
    <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>. Cambridge
    University Press. <a href="https://doi.org/10.1017/prm.2024.7">https://doi.org/10.1017/prm.2024.7</a>'
  chicago: 'Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division
    Polynomials.” <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>.
    Cambridge University Press, 2025. <a href="https://doi.org/10.1017/prm.2024.7">https://doi.org/10.1017/prm.2024.7</a>.'
  ieee: 'B. Naskręcki and M. Verzobio, “Common valuations of division polynomials,”
    <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, vol.
    155, no. 5. Cambridge University Press, pp. 1646–1660, 2025.'
  ista: 'Naskręcki B, Verzobio M. 2025. Common valuations of division polynomials.
    Proceedings of the Royal Society of Edinburgh Section A: Mathematics. 155(5),
    1646–1660.'
  mla: 'Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division Polynomials.”
    <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, vol.
    155, no. 5, Cambridge University Press, 2025, pp. 1646–60, doi:<a href="https://doi.org/10.1017/prm.2024.7">10.1017/prm.2024.7</a>.'
  short: 'B. Naskręcki, M. Verzobio, Proceedings of the Royal Society of Edinburgh
    Section A: Mathematics 155 (2025) 1646–1660.'
corr_author: '1'
date_created: 2023-01-16T11:45:22Z
date_published: 2025-10-01T00:00:00Z
date_updated: 2025-12-30T06:46:17Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1017/prm.2024.7
ec_funded: 1
external_id:
  arxiv:
  - '2203.02015'
  isi:
  - '001174907100001'
file:
- access_level: open_access
  checksum: c5ec6e29aca2fb4533cb95fac409a0b2
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  creator: dernst
  date_created: 2025-12-30T06:45:47Z
  date_updated: 2025-12-30T06:45:47Z
  file_id: '20878'
  file_name: 2025_ProceedingsRoyalSocEdinburghA_Naskrecki.pdf
  file_size: 477624
  relation: main_file
  success: 1
file_date_updated: 2025-12-30T06:45:47Z
has_accepted_license: '1'
intvolume: '       155'
isi: 1
issue: '5'
keyword:
- Elliptic curves
- Néron models
- division polynomials
- height functions
- discrete valuation rings
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '10'
oa: 1
oa_version: Published Version
page: 1646-1660
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: 'Proceedings of the Royal Society of Edinburgh Section A: Mathematics'
publication_identifier:
  eissn:
  - 1473-7124
  issn:
  - 0308-2105
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Common valuations of division polynomials
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 155
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20078'
abstract:
- lang: eng
  text: 'Let A be an abelian variety defined over a number field K, E/K be an elliptic
    curve, and ϕ : A → Em be an isogeny defined over K. Let P ∈ A(K) be such that
    ϕ(P)=(Q1,..., Qm) with RankZ(⟨Q1,...,Qm⟩)=1. We will study a divisibility sequence
    related to the point P and show its relation with elliptic divisibility sequences.'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Stefan
  full_name: Barańczuk, Stefan
  last_name: Barańczuk
- first_name: Bartosz
  full_name: Naskręcki, Bartosz
  last_name: Naskręcki
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Barańczuk S, Naskręcki B, Verzobio M. Divisibility sequences related to abelian
    varieties isogenous to a power of an elliptic curve. <i>Journal of Number Theory</i>.
    2025;279:170-183. doi:<a href="https://doi.org/10.1016/j.jnt.2025.06.001">10.1016/j.jnt.2025.06.001</a>
  apa: Barańczuk, S., Naskręcki, B., &#38; Verzobio, M. (2025). Divisibility sequences
    related to abelian varieties isogenous to a power of an elliptic curve. <i>Journal
    of Number Theory</i>. Elsevier. <a href="https://doi.org/10.1016/j.jnt.2025.06.001">https://doi.org/10.1016/j.jnt.2025.06.001</a>
  chicago: Barańczuk, Stefan, Bartosz Naskręcki, and Matteo Verzobio. “Divisibility
    Sequences Related to Abelian Varieties Isogenous to a Power of an Elliptic Curve.”
    <i>Journal of Number Theory</i>. Elsevier, 2025. <a href="https://doi.org/10.1016/j.jnt.2025.06.001">https://doi.org/10.1016/j.jnt.2025.06.001</a>.
  ieee: S. Barańczuk, B. Naskręcki, and M. Verzobio, “Divisibility sequences related
    to abelian varieties isogenous to a power of an elliptic curve,” <i>Journal of
    Number Theory</i>, vol. 279. Elsevier, pp. 170–183, 2025.
  ista: Barańczuk S, Naskręcki B, Verzobio M. 2025. Divisibility sequences related
    to abelian varieties isogenous to a power of an elliptic curve. Journal of Number
    Theory. 279, 170–183.
  mla: Barańczuk, Stefan, et al. “Divisibility Sequences Related to Abelian Varieties
    Isogenous to a Power of an Elliptic Curve.” <i>Journal of Number Theory</i>, vol.
    279, Elsevier, 2025, pp. 170–83, doi:<a href="https://doi.org/10.1016/j.jnt.2025.06.001">10.1016/j.jnt.2025.06.001</a>.
  short: S. Barańczuk, B. Naskręcki, M. Verzobio, Journal of Number Theory 279 (2025)
    170–183.
corr_author: '1'
date_created: 2025-07-27T22:01:25Z
date_published: 2025-07-23T00:00:00Z
date_updated: 2025-09-30T14:09:38Z
day: '23'
department:
- _id: TiBr
doi: 10.1016/j.jnt.2025.06.001
external_id:
  arxiv:
  - '2309.09699'
  isi:
  - '001541172400002'
intvolume: '       279'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1016/j.jnt.2025.06.001
month: '07'
oa: 1
oa_version: Published Version
page: 170-183
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: epub_ahead
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Divisibility sequences related to abelian varieties isogenous to a power of
  an elliptic curve
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 279
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '20222'
abstract:
- lang: eng
  text: Let X be a smooth projective hypersurface defined over Q. We provide new bounds
    for rational points of bounded height on X. In particular, we show that if X is
    a smooth projective hypersurface in Pn with n  4 and degree d  50, then the set
    of rational points on X of height bounded by B have cardinality On,d,ε (Bn−2+ε
    ). If X is smooth and has degree d  6, we improve the dimension growth conjecture
    bound. We achieve an analogue result for affine hypersurfaces whose projective
    closure is smooth.
acknowledgement: "While working on this paper, the author was supported by the European
  Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
  Grant Agreement No. 101034413. The author is very grateful to Tim Browning for suggesting
  the problem and for many useful discussions. We thank the anonymous referees for
  their many helpful comments, which improved the exposition of the paper. We are
  also grateful to Gal Binyamini for their interest in this work and for drawing our
  attention to the aforementioned paper [1].\r\nWe shared an early version of this
  paper with Per Salberger, who mentioned that he announced a new bound for smooth
  threefolds in P4 during a talk in 2019 (see [7] for the abstract). This result has
  not been published."
article_number: rnaf249
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Verzobio M. Counting rational points on smooth hypersurfaces with high degree.
    <i>International Mathematics Research Notices</i>. 2025;2025(16). doi:<a href="https://doi.org/10.1093/imrn/rnaf249">10.1093/imrn/rnaf249</a>
  apa: Verzobio, M. (2025). Counting rational points on smooth hypersurfaces with
    high degree. <i>International Mathematics Research Notices</i>. Oxford University
    Press. <a href="https://doi.org/10.1093/imrn/rnaf249">https://doi.org/10.1093/imrn/rnaf249</a>
  chicago: Verzobio, Matteo. “Counting Rational Points on Smooth Hypersurfaces with
    High Degree.” <i>International Mathematics Research Notices</i>. Oxford University
    Press, 2025. <a href="https://doi.org/10.1093/imrn/rnaf249">https://doi.org/10.1093/imrn/rnaf249</a>.
  ieee: M. Verzobio, “Counting rational points on smooth hypersurfaces with high degree,”
    <i>International Mathematics Research Notices</i>, vol. 2025, no. 16. Oxford University
    Press, 2025.
  ista: Verzobio M. 2025. Counting rational points on smooth hypersurfaces with high
    degree. International Mathematics Research Notices. 2025(16), rnaf249.
  mla: Verzobio, Matteo. “Counting Rational Points on Smooth Hypersurfaces with High
    Degree.” <i>International Mathematics Research Notices</i>, vol. 2025, no. 16,
    rnaf249, Oxford University Press, 2025, doi:<a href="https://doi.org/10.1093/imrn/rnaf249">10.1093/imrn/rnaf249</a>.
  short: M. Verzobio, International Mathematics Research Notices 2025 (2025).
corr_author: '1'
date_created: 2025-08-24T22:01:31Z
date_published: 2025-08-01T00:00:00Z
date_updated: 2025-09-30T14:26:34Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1093/imrn/rnaf249
ec_funded: 1
external_id:
  arxiv:
  - '2503.19451'
  isi:
  - '001549126000001'
file:
- access_level: open_access
  checksum: 482ae2be98841ee446cf2bdfcd79f86f
  content_type: application/pdf
  creator: dernst
  date_created: 2025-09-02T07:55:05Z
  date_updated: 2025-09-02T07:55:05Z
  file_id: '20275'
  file_name: 2025_IMRN_Verzobio.pdf
  file_size: 540263
  relation: main_file
  success: 1
file_date_updated: 2025-09-02T07:55:05Z
has_accepted_license: '1'
intvolume: '      2025'
isi: 1
issue: '16'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-nd/4.0/
month: '08'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: International Mathematics Research Notices
publication_identifier:
  eissn:
  - 1687-0247
  issn:
  - 1073-7928
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Counting rational points on smooth hypersurfaces with high degree
tmp:
  image: /images/cc_by_nc_nd.png
  legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
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  short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 2025
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
_id: '19407'
abstract:
- lang: eng
  text: We discuss, in a non-Archimedean setting, the distribution of the coefficients
    of L-polynomials of curves of genus g over  Fq . Among other results, this allows
    us to prove that the  Q-vector space spanned by such characteristic polynomials
    has dimension g + 1. We also state a conjecture about the Archimedean distribution
    of the number of rational points of curves over finite fields.
acknowledgement: We thank Umberto Zannier for bringing the problem to our attention,
  for many useful suggestions, and especially for pointing out the relevance of the
  equidistribution results of Katz–Sarnak, noting that they imply the case  q≫g0 of
  theorem 1.4. In addition, the first author would like to thank Umberto Zannier for
  his guidance during his undergraduate studies, on a topic that ultimately inspired
  much of the work in this article. We are grateful to J. Kaczorowski and A. Perelli
  for sharing their work [Reference Kaczorowski and Perelli28] before publication.
  We thank Christophe Ritzenthaler and Elisa Lorenzo García for their interesting
  comments on the first version of this article, Zhao Yu Ma for a comment about remark
  3.12, and the anonymous referees for their helpful suggestions.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Francesco
  full_name: Ballini, Francesco
  last_name: Ballini
- first_name: Davide
  full_name: Lombardo, Davide
  last_name: Lombardo
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: 'Ballini F, Lombardo D, Verzobio M. On the L-polynomials of curves over finite
    fields. <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>.
    2025. doi:<a href="https://doi.org/10.1017/prm.2025.7">10.1017/prm.2025.7</a>'
  apa: 'Ballini, F., Lombardo, D., &#38; Verzobio, M. (2025). On the L-polynomials
    of curves over finite fields. <i>Proceedings of the Royal Society of Edinburgh
    Section A: Mathematics</i>. Cambridge University Press. <a href="https://doi.org/10.1017/prm.2025.7">https://doi.org/10.1017/prm.2025.7</a>'
  chicago: 'Ballini, Francesco, Davide Lombardo, and Matteo Verzobio. “On the L-Polynomials
    of Curves over Finite Fields.” <i>Proceedings of the Royal Society of Edinburgh
    Section A: Mathematics</i>. Cambridge University Press, 2025. <a href="https://doi.org/10.1017/prm.2025.7">https://doi.org/10.1017/prm.2025.7</a>.'
  ieee: 'F. Ballini, D. Lombardo, and M. Verzobio, “On the L-polynomials of curves
    over finite fields,” <i>Proceedings of the Royal Society of Edinburgh Section
    A: Mathematics</i>. Cambridge University Press, 2025.'
  ista: 'Ballini F, Lombardo D, Verzobio M. 2025. On the L-polynomials of curves over
    finite fields. Proceedings of the Royal Society of Edinburgh Section A: Mathematics.'
  mla: 'Ballini, Francesco, et al. “On the L-Polynomials of Curves over Finite Fields.”
    <i>Proceedings of the Royal Society of Edinburgh Section A: Mathematics</i>, Cambridge
    University Press, 2025, doi:<a href="https://doi.org/10.1017/prm.2025.7">10.1017/prm.2025.7</a>.'
  short: 'F. Ballini, D. Lombardo, M. Verzobio, Proceedings of the Royal Society of
    Edinburgh Section A: Mathematics (2025).'
corr_author: '1'
date_created: 2025-03-16T23:01:25Z
date_published: 2025-02-06T00:00:00Z
date_updated: 2025-09-30T11:00:35Z
day: '06'
department:
- _id: TiBr
doi: 10.1017/prm.2025.7
external_id:
  isi:
  - '001414690400001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1017/prm.2025.7
month: '02'
oa: 1
oa_version: Published Version
publication: 'Proceedings of the Royal Society of Edinburgh Section A: Mathematics'
publication_identifier:
  eissn:
  - 1473-7124
  issn:
  - 0308-2105
publication_status: epub_ahead
publisher: Cambridge University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the L-polynomials of curves over finite fields
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
year: '2025'
...
---
OA_place: publisher
OA_type: diamond
_id: '21003'
abstract:
- lang: eng
  text: We extend work of Heath-Brown and Salberger, based on the determinant method,
    to provide a uniform upper bound for the number of integral points of bounded
    height on an affine surface, which are subject to a polynomial congruence condition.
    This is applied to get a new uniform bound for points on diagonal quadric surfaces,
    and to a problem about the representation of integers as a sum of four unlike
    powers.
acknowledgement: "Supported by FWF grant (DOI 10.55776/P36278), Supported by European
  Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
  Grant\r\nAgreement No. 101034413."
article_number: '12'
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: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Browning TD, Verzobio M. Counting integer points on affine surfaces with a
    side condition. <i>Discrete Analysis</i>. 2025;2025. doi:<a href="https://doi.org/10.19086/da.143787">10.19086/da.143787</a>
  apa: 'Browning, T. D., &#38; Verzobio, M. (2025). Counting integer points on affine
    surfaces with a side condition. <i>Discrete Analysis</i>. Cambridge: Alliance
    of Diamond Open Access Journals. <a href="https://doi.org/10.19086/da.143787">https://doi.org/10.19086/da.143787</a>'
  chicago: 'Browning, Timothy D, and Matteo Verzobio. “Counting Integer Points on
    Affine Surfaces with a Side Condition.” <i>Discrete Analysis</i>. Cambridge: Alliance
    of Diamond Open Access Journals, 2025. <a href="https://doi.org/10.19086/da.143787">https://doi.org/10.19086/da.143787</a>.'
  ieee: 'T. D. Browning and M. Verzobio, “Counting integer points on affine surfaces
    with a side condition,” <i>Discrete Analysis</i>, vol. 2025. Cambridge: Alliance
    of Diamond Open Access Journals, 2025.'
  ista: Browning TD, Verzobio M. 2025. Counting integer points on affine surfaces
    with a side condition. Discrete Analysis. 2025, 12.
  mla: 'Browning, Timothy D., and Matteo Verzobio. “Counting Integer Points on Affine
    Surfaces with a Side Condition.” <i>Discrete Analysis</i>, vol. 2025, 12, Cambridge:
    Alliance of Diamond Open Access Journals, 2025, doi:<a href="https://doi.org/10.19086/da.143787">10.19086/da.143787</a>.'
  short: T.D. Browning, M. Verzobio, Discrete Analysis 2025 (2025).
corr_author: '1'
date_created: 2026-01-18T23:02:44Z
date_published: 2025-09-01T00:00:00Z
date_updated: 2026-02-12T08:03:12Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.19086/da.143787
ec_funded: 1
external_id:
  arxiv:
  - '2408.11453'
file:
- access_level: open_access
  checksum: 3d38e850b40f3e1abbfd30073bd4388a
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  date_created: 2026-02-12T07:50:47Z
  date_updated: 2026-02-12T07:50:47Z
  file_id: '21214'
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  file_size: 393625
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  success: 1
file_date_updated: 2026-02-12T07:50:47Z
has_accepted_license: '1'
intvolume: '      2025'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Discrete Analysis
publication_identifier:
  eissn:
  - 2397-3129
publication_status: published
publisher: 'Cambridge: Alliance of Diamond Open Access Journals'
scopus_import: '1'
status: public
title: Counting integer points on affine surfaces with a side condition
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2025
year: '2025'
...
---
_id: '12312'
abstract:
- lang: eng
  text: "Let $\\ell$ be a prime number. We classify the subgroups $G$ of $\\operatorname{Sp}_4(\\mathbb{F}_\\ell)$
    and $\\operatorname{GSp}_4(\\mathbb{F}_\\ell)$ that act irreducibly on $\\mathbb{F}_\\ell^4$,
    but such that every element of $G$ fixes an $\\mathbb{F}_\\ell$-vector subspace
    of dimension 1. We use this classification to prove that the local-global principle
    for isogenies of degree $\\ell$ between abelian surfaces over number fields holds
    in many cases -- in particular, whenever the abelian surface has non-trivial endomorphisms
    and $\\ell$ is large enough with respect to the field of definition. Finally,
    we prove that there exist arbitrarily large primes $\\ell$ for which some abelian
    surface\r\n$A/\\mathbb{Q}$ fails the local-global principle for isogenies of degree
    $\\ell$."
acknowledgement: "It is a pleasure to thank Samuele Anni for his interest in this
  project and for several discussions on the topic of this paper, which led in particular
  to Remark 6.30 and to a better understanding of the difficulties with [6]. We also
  thank John Cullinan for correspondence about [6] and Barinder Banwait for his many
  insightful comments on the first version of this paper. Finally, we thank the referee
  for their thorough reading of the manuscript.\r\nOpen access funding provided by
  Università di Pisa within the CRUI-CARE Agreement. The authors have been partially
  supported by MIUR (Italy) through PRIN 2017 “Geometric, algebraic and analytic methods
  in arithmetic\" and PRIN 2022 “Semiabelian varieties, Galois representations and
  related Diophantine problems\", and by the University of Pisa through PRA 2018-19
  and 2022 “Spazi di moduli, rappresentazioni e strutture combinatorie\". The first
  author is a member of the INdAM group GNSAGA."
article_number: '18'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Davide
  full_name: Lombardo, Davide
  last_name: Lombardo
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Lombardo D, Verzobio M. On the local-global principle for isogenies of abelian
    surfaces. <i>Selecta Mathematica</i>. 2024;30(2). doi:<a href="https://doi.org/10.1007/s00029-023-00908-0">10.1007/s00029-023-00908-0</a>
  apa: Lombardo, D., &#38; Verzobio, M. (2024). On the local-global principle for
    isogenies of abelian surfaces. <i>Selecta Mathematica</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s00029-023-00908-0">https://doi.org/10.1007/s00029-023-00908-0</a>
  chicago: Lombardo, Davide, and Matteo Verzobio. “On the Local-Global Principle for
    Isogenies of Abelian Surfaces.” <i>Selecta Mathematica</i>. Springer Nature, 2024.
    <a href="https://doi.org/10.1007/s00029-023-00908-0">https://doi.org/10.1007/s00029-023-00908-0</a>.
  ieee: D. Lombardo and M. Verzobio, “On the local-global principle for isogenies
    of abelian surfaces,” <i>Selecta Mathematica</i>, vol. 30, no. 2. Springer Nature,
    2024.
  ista: Lombardo D, Verzobio M. 2024. On the local-global principle for isogenies
    of abelian surfaces. Selecta Mathematica. 30(2), 18.
  mla: Lombardo, Davide, and Matteo Verzobio. “On the Local-Global Principle for Isogenies
    of Abelian Surfaces.” <i>Selecta Mathematica</i>, vol. 30, no. 2, 18, Springer
    Nature, 2024, doi:<a href="https://doi.org/10.1007/s00029-023-00908-0">10.1007/s00029-023-00908-0</a>.
  short: D. Lombardo, M. Verzobio, Selecta Mathematica 30 (2024).
corr_author: '1'
date_created: 2023-01-16T11:45:53Z
date_published: 2024-01-26T00:00:00Z
date_updated: 2025-08-05T13:26:34Z
day: '26'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1007/s00029-023-00908-0
external_id:
  arxiv:
  - '2206.15240'
  isi:
  - '001148959100001'
file:
- access_level: open_access
  checksum: ae75441420aabd80c5828bce38272ba1
  content_type: application/pdf
  creator: dernst
  date_created: 2024-07-22T09:33:58Z
  date_updated: 2024-07-22T09:33:58Z
  file_id: '17298'
  file_name: 2024_SelectaMath_Lombardo.pdf
  file_size: 1301415
  relation: main_file
  success: 1
file_date_updated: 2024-07-22T09:33:58Z
has_accepted_license: '1'
intvolume: '        30'
isi: 1
issue: '2'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
publication: Selecta Mathematica
publication_identifier:
  eissn:
  - 1420-9020
  issn:
  - 1022-1824
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the local-global principle for isogenies of abelian surfaces
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 30
year: '2024'
...
---
OA_place: publisher
OA_type: hybrid
_id: '17323'
abstract:
- lang: eng
  text: We investigate strong divisibility sequences and produce lower and upper bounds
    for the density of integers in the sequence that only have (somewhat) large prime
    factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility
    sequences, discussing the limitations of our methods. At the end of the paper,
    there is an appendix by Sandro Bettin on divisor closed sets that we use to study
    the density of prime terms that appear in strong divisibility sequences.
acknowledgement: The authors are very grateful to Andrew Granville, Dimitris Koukoulopoulos,
  Davide Lombardo,Florian Luca, Igor Shparlinski and Joni Teräväinen for useful comments.
  While working on thispaper, the first author was supported by a FWF Grant (DOI 10.55776/P36278)
  and the secondauthor was supported by the European Union’s Horizon 2020 research
  and innovation programunder the Marie Skłodowska-Curie Grant Agreement Number 101034413.
article_number: e12269
article_processing_charge: Yes (via OA deal)
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: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Browning TD, Verzobio M. Strong divisibility sequences and sieve methods. <i>Mathematika</i>.
    2024;70(4). doi:<a href="https://doi.org/10.1112/mtk.12269">10.1112/mtk.12269</a>
  apa: Browning, T. D., &#38; Verzobio, M. (2024). Strong divisibility sequences and
    sieve methods. <i>Mathematika</i>. London Mathematical Society. <a href="https://doi.org/10.1112/mtk.12269">https://doi.org/10.1112/mtk.12269</a>
  chicago: Browning, Timothy D, and Matteo Verzobio. “Strong Divisibility Sequences
    and Sieve Methods.” <i>Mathematika</i>. London Mathematical Society, 2024. <a
    href="https://doi.org/10.1112/mtk.12269">https://doi.org/10.1112/mtk.12269</a>.
  ieee: T. D. Browning and M. Verzobio, “Strong divisibility sequences and sieve methods,”
    <i>Mathematika</i>, vol. 70, no. 4. London Mathematical Society, 2024.
  ista: Browning TD, Verzobio M. 2024. Strong divisibility sequences and sieve methods.
    Mathematika. 70(4), e12269.
  mla: Browning, Timothy D., and Matteo Verzobio. “Strong Divisibility Sequences and
    Sieve Methods.” <i>Mathematika</i>, vol. 70, no. 4, e12269, London Mathematical
    Society, 2024, doi:<a href="https://doi.org/10.1112/mtk.12269">10.1112/mtk.12269</a>.
  short: T.D. Browning, M. Verzobio, Mathematika 70 (2024).
corr_author: '1'
date_created: 2024-07-28T22:01:08Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2025-09-08T08:44:11Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1112/mtk.12269
ec_funded: 1
external_id:
  isi:
  - '001273912800001'
file:
- access_level: open_access
  checksum: 0b1518bdc1a901413005c19202cfa497
  content_type: application/pdf
  creator: dernst
  date_created: 2025-01-13T11:06:25Z
  date_updated: 2025-01-13T11:06:25Z
  file_id: '18842'
  file_name: 2024_Mathematika_Browning.pdf
  file_size: 273006
  relation: main_file
  success: 1
file_date_updated: 2025-01-13T11:06:25Z
has_accepted_license: '1'
intvolume: '        70'
isi: 1
issue: '4'
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Mathematika
publication_identifier:
  eissn:
  - 2041-7942
  issn:
  - 0025-5793
publication_status: published
publisher: London Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Strong divisibility sequences and sieve methods
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: 70
year: '2024'
...
---
_id: '12313'
abstract:
- lang: eng
  text: Let P be a nontorsion point on an elliptic curve defined over a number field
    K and consider the sequence {Bn}n∈N of the denominators of x(nP). We prove that
    every term of the sequence of the Bn has a primitive divisor for n greater than
    an effectively computable constant that we will explicitly compute. This constant
    will depend only on the model defining the curve.
acknowledgement: "This paper is part of the author’s PhD thesis at Università of Pisa.
  Moreover, this\r\nproject has received funding from the European Union’s Horizon
  2020 research\r\nand innovation programme under the Marie Skłodowska-Curie Grant
  Agreement\r\nNo. 101034413. I thank the referee for many helpful comments."
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Verzobio M. Some effectivity results for primitive divisors of elliptic divisibility 
    sequences. <i>Pacific Journal of Mathematics</i>. 2023;325(2):331-351. doi:<a
    href="https://doi.org/10.2140/pjm.2023.325.331">10.2140/pjm.2023.325.331</a>
  apa: Verzobio, M. (2023). Some effectivity results for primitive divisors of elliptic
    divisibility  sequences. <i>Pacific Journal of Mathematics</i>. Mathematical Sciences
    Publishers. <a href="https://doi.org/10.2140/pjm.2023.325.331">https://doi.org/10.2140/pjm.2023.325.331</a>
  chicago: Verzobio, Matteo. “Some Effectivity Results for Primitive Divisors of Elliptic
    Divisibility  Sequences.” <i>Pacific Journal of Mathematics</i>. Mathematical
    Sciences Publishers, 2023. <a href="https://doi.org/10.2140/pjm.2023.325.331">https://doi.org/10.2140/pjm.2023.325.331</a>.
  ieee: M. Verzobio, “Some effectivity results for primitive divisors of elliptic
    divisibility  sequences,” <i>Pacific Journal of Mathematics</i>, vol. 325, no.
    2. Mathematical Sciences Publishers, pp. 331–351, 2023.
  ista: Verzobio M. 2023. Some effectivity results for primitive divisors of elliptic
    divisibility  sequences. Pacific Journal of Mathematics. 325(2), 331–351.
  mla: Verzobio, Matteo. “Some Effectivity Results for Primitive Divisors of Elliptic
    Divisibility  Sequences.” <i>Pacific Journal of Mathematics</i>, vol. 325, no.
    2, Mathematical Sciences Publishers, 2023, pp. 331–51, doi:<a href="https://doi.org/10.2140/pjm.2023.325.331">10.2140/pjm.2023.325.331</a>.
  short: M. Verzobio, Pacific Journal of Mathematics 325 (2023) 331–351.
corr_author: '1'
date_created: 2023-01-16T11:46:19Z
date_published: 2023-11-03T00:00:00Z
date_updated: 2025-04-14T07:54:54Z
day: '03'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.2140/pjm.2023.325.331
ec_funded: 1
external_id:
  arxiv:
  - '2001.02987'
  isi:
  - '001104766900001'
file:
- access_level: open_access
  checksum: b6218d16a72742d8bb38d6fc3c9bb8c6
  content_type: application/pdf
  creator: dernst
  date_created: 2023-11-13T09:50:41Z
  date_updated: 2023-11-13T09:50:41Z
  file_id: '14525'
  file_name: 2023_PacificJourMaths_Verzobio.pdf
  file_size: 389897
  relation: main_file
  success: 1
file_date_updated: 2023-11-13T09:50:41Z
has_accepted_license: '1'
intvolume: '       325'
isi: 1
issue: '2'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 331-351
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Pacific Journal of Mathematics
publication_identifier:
  eissn:
  - 0030-8730
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Some effectivity results for primitive divisors of elliptic divisibility  sequences
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 325
year: '2023'
...
---
_id: '12308'
abstract:
- lang: eng
  text: Let P and Q be two points on an elliptic curve defined over a number field
    K. For α∈End(E), define Bα to be the OK-integral ideal generated by the denominator
    of x(α(P)+Q). Let O be a subring of End(E), that is a Dedekind domain. We will
    study the sequence {Bα}α∈O. We will show that, for all but finitely many α∈O,
    the ideal Bα has a primitive divisor when P is a non-torsion point and there exist
    two endomorphisms g≠0 and f so that f(P)=g(Q). This is a generalization of previous
    results on elliptic divisibility sequences.
article_number: '37'
article_processing_charge: No
article_type: original
author:
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Verzobio M. Primitive divisors of sequences associated to elliptic curves with
    complex multiplication. <i>Research in Number Theory</i>. 2021;7(2). doi:<a href="https://doi.org/10.1007/s40993-021-00267-9">10.1007/s40993-021-00267-9</a>
  apa: Verzobio, M. (2021). Primitive divisors of sequences associated to elliptic
    curves with complex multiplication. <i>Research in Number Theory</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s40993-021-00267-9">https://doi.org/10.1007/s40993-021-00267-9</a>
  chicago: Verzobio, Matteo. “Primitive Divisors of Sequences Associated to Elliptic
    Curves with Complex Multiplication.” <i>Research in Number Theory</i>. Springer
    Nature, 2021. <a href="https://doi.org/10.1007/s40993-021-00267-9">https://doi.org/10.1007/s40993-021-00267-9</a>.
  ieee: M. Verzobio, “Primitive divisors of sequences associated to elliptic curves
    with complex multiplication,” <i>Research in Number Theory</i>, vol. 7, no. 2.
    Springer Nature, 2021.
  ista: Verzobio M. 2021. Primitive divisors of sequences associated to elliptic curves
    with complex multiplication. Research in Number Theory. 7(2), 37.
  mla: Verzobio, Matteo. “Primitive Divisors of Sequences Associated to Elliptic Curves
    with Complex Multiplication.” <i>Research in Number Theory</i>, vol. 7, no. 2,
    37, Springer Nature, 2021, doi:<a href="https://doi.org/10.1007/s40993-021-00267-9">10.1007/s40993-021-00267-9</a>.
  short: M. Verzobio, Research in Number Theory 7 (2021).
corr_author: '1'
date_created: 2023-01-16T11:44:39Z
date_published: 2021-05-20T00:00:00Z
date_updated: 2024-10-09T21:05:08Z
day: '20'
doi: 10.1007/s40993-021-00267-9
extern: '1'
intvolume: '         7'
issue: '2'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s40993-021-00267-9
month: '05'
oa: 1
oa_version: Published Version
publication: Research in Number Theory
publication_identifier:
  issn:
  - 2522-0160
  - 2363-9555
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Primitive divisors of sequences associated to elliptic curves with complex
  multiplication
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2021'
...
---
_id: '12309'
abstract:
- lang: eng
  text: Take a rational elliptic curve defined by the equation y2=x3+ax in minimal
    form and consider the sequence Bn of the denominators of the abscissas of the
    iterate of a non-torsion point. We show that B5m has a primitive divisor for every
    m. Then, we show how to generalize this method to the terms of the form Bmp with
    p a prime congruent to 1 modulo 4.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Verzobio M. Primitive divisors of elliptic divisibility sequences for elliptic
    curves with j=1728. <i>Acta Arithmetica</i>. 2021;198(2):129-168. doi:<a href="https://doi.org/10.4064/aa191016-30-7">10.4064/aa191016-30-7</a>
  apa: Verzobio, M. (2021). Primitive divisors of elliptic divisibility sequences
    for elliptic curves with j=1728. <i>Acta Arithmetica</i>. Institute of Mathematics,
    Polish Academy of Sciences. <a href="https://doi.org/10.4064/aa191016-30-7">https://doi.org/10.4064/aa191016-30-7</a>
  chicago: Verzobio, Matteo. “Primitive Divisors of Elliptic Divisibility Sequences
    for Elliptic Curves with J=1728.” <i>Acta Arithmetica</i>. Institute of Mathematics,
    Polish Academy of Sciences, 2021. <a href="https://doi.org/10.4064/aa191016-30-7">https://doi.org/10.4064/aa191016-30-7</a>.
  ieee: M. Verzobio, “Primitive divisors of elliptic divisibility sequences for elliptic
    curves with j=1728,” <i>Acta Arithmetica</i>, vol. 198, no. 2. Institute of Mathematics,
    Polish Academy of Sciences, pp. 129–168, 2021.
  ista: Verzobio M. 2021. Primitive divisors of elliptic divisibility sequences for
    elliptic curves with j=1728. Acta Arithmetica. 198(2), 129–168.
  mla: Verzobio, Matteo. “Primitive Divisors of Elliptic Divisibility Sequences for
    Elliptic Curves with J=1728.” <i>Acta Arithmetica</i>, vol. 198, no. 2, Institute
    of Mathematics, Polish Academy of Sciences, 2021, pp. 129–68, doi:<a href="https://doi.org/10.4064/aa191016-30-7">10.4064/aa191016-30-7</a>.
  short: M. Verzobio, Acta Arithmetica 198 (2021) 129–168.
corr_author: '1'
date_created: 2023-01-16T11:44:54Z
date_published: 2021-01-04T00:00:00Z
date_updated: 2024-10-09T21:05:08Z
day: '04'
doi: 10.4064/aa191016-30-7
extern: '1'
external_id:
  arxiv:
  - '2001.09634'
intvolume: '       198'
issue: '2'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2001.09634
month: '01'
oa: 1
oa_version: Preprint
page: 129-168
publication: Acta Arithmetica
publication_identifier:
  issn:
  - 0065-1036
  - 1730-6264
publication_status: published
publisher: Institute of Mathematics, Polish Academy of Sciences
quality_controlled: '1'
scopus_import: '1'
status: public
title: Primitive divisors of elliptic divisibility sequences for elliptic curves with
  j=1728
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 198
year: '2021'
...
---
_id: '12314'
abstract:
- lang: eng
  text: "In literature, there are two different definitions of elliptic divisibility\r\nsequences.
    The first one says that a sequence of integers $\\{h_n\\}_{n\\geq 0}$\r\nis an
    elliptic divisibility sequence if it verifies the recurrence relation\r\n$h_{m+n}h_{m-n}h_{r}^2=h_{m+r}h_{m-r}h_{n}^2-h_{n+r}h_{n-r}h_{m}^2$
    for every\r\nnatural number $m\\geq n\\geq r$. The second definition says that
    a sequence of\r\nintegers $\\{\\beta_n\\}_{n\\geq 0}$ is an elliptic divisibility
    sequence if it is\r\nthe sequence of the square roots (chosen with an appropriate
    sign) of the\r\ndenominators of the abscissas of the iterates of a point on a
    rational elliptic\r\ncurve. It is well-known that the two sequences are not equivalent.
    Hence, given\r\na sequence of the denominators $\\{\\beta_n\\}_{n\\geq 0}$, in
    general does not\r\nhold\r\n$\\beta_{m+n}\\beta_{m-n}\\beta_{r}^2=\\beta_{m+r}\\beta_{m-r}\\beta_{n}^2-\\beta_{n+r}\\beta_{n-r}\\beta_{m}^2$\r\nfor
    $m\\geq n\\geq r$. We will prove that the recurrence relation above holds for\r\n$\\{\\beta_n\\}_{n\\geq
    0}$ under some conditions on the indexes $m$, $n$, and $r$."
article_number: '2102.07573'
article_processing_charge: No
arxiv: 1
author:
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Verzobio M. A recurrence relation for elliptic divisibility sequences. <i>arXiv</i>.
    doi:<a href="https://doi.org/10.48550/arXiv.2102.07573">10.48550/arXiv.2102.07573</a>
  apa: Verzobio, M. (n.d.). A recurrence relation for elliptic divisibility sequences.
    <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2102.07573">https://doi.org/10.48550/arXiv.2102.07573</a>
  chicago: Verzobio, Matteo. “A Recurrence Relation for Elliptic Divisibility Sequences.”
    <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2102.07573">https://doi.org/10.48550/arXiv.2102.07573</a>.
  ieee: M. Verzobio, “A recurrence relation for elliptic divisibility sequences,”
    <i>arXiv</i>. .
  ista: Verzobio M. A recurrence relation for elliptic divisibility sequences. arXiv,
    2102.07573.
  mla: Verzobio, Matteo. “A Recurrence Relation for Elliptic Divisibility Sequences.”
    <i>ArXiv</i>, 2102.07573, doi:<a href="https://doi.org/10.48550/arXiv.2102.07573">10.48550/arXiv.2102.07573</a>.
  short: M. Verzobio, ArXiv (n.d.).
corr_author: '1'
date_created: 2023-01-16T11:46:36Z
date_published: 2021-02-15T00:00:00Z
date_updated: 2024-10-09T21:05:07Z
day: '15'
doi: 10.48550/arXiv.2102.07573
extern: '1'
external_id:
  arxiv:
  - '2102.07573'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: ' https://doi.org/10.48550/arXiv.2102.07573'
month: '02'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
status: public
title: A recurrence relation for elliptic divisibility sequences
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '12310'
abstract:
- lang: eng
  text: Let  be a sequence of points on an elliptic curve defined over a number field
    K. In this paper, we study the denominators of the x-coordinates of this sequence.
    We prove that, if Q is a torsion point of prime order, then for n large enough
    there always exists a primitive divisor. Later on, we show the link between the
    study of the primitive divisors and a Lang-Trotter conjecture. Indeed, given two
    points P and Q on the elliptic curve, we prove a lower bound for the number of
    primes p such that P is in the orbit of Q modulo p.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Matteo
  full_name: Verzobio, Matteo
  id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
  last_name: Verzobio
  orcid: 0000-0002-0854-0306
citation:
  ama: Verzobio M. Primitive divisors of sequences associated to elliptic curves.
    <i>Journal of Number Theory</i>. 2020;209(4):378-390. doi:<a href="https://doi.org/10.1016/j.jnt.2019.09.003">10.1016/j.jnt.2019.09.003</a>
  apa: Verzobio, M. (2020). Primitive divisors of sequences associated to elliptic
    curves. <i>Journal of Number Theory</i>. Elsevier. <a href="https://doi.org/10.1016/j.jnt.2019.09.003">https://doi.org/10.1016/j.jnt.2019.09.003</a>
  chicago: Verzobio, Matteo. “Primitive Divisors of Sequences Associated to Elliptic
    Curves.” <i>Journal of Number Theory</i>. Elsevier, 2020. <a href="https://doi.org/10.1016/j.jnt.2019.09.003">https://doi.org/10.1016/j.jnt.2019.09.003</a>.
  ieee: M. Verzobio, “Primitive divisors of sequences associated to elliptic curves,”
    <i>Journal of Number Theory</i>, vol. 209, no. 4. Elsevier, pp. 378–390, 2020.
  ista: Verzobio M. 2020. Primitive divisors of sequences associated to elliptic curves.
    Journal of Number Theory. 209(4), 378–390.
  mla: Verzobio, Matteo. “Primitive Divisors of Sequences Associated to Elliptic Curves.”
    <i>Journal of Number Theory</i>, vol. 209, no. 4, Elsevier, 2020, pp. 378–90,
    doi:<a href="https://doi.org/10.1016/j.jnt.2019.09.003">10.1016/j.jnt.2019.09.003</a>.
  short: M. Verzobio, Journal of Number Theory 209 (2020) 378–390.
corr_author: '1'
date_created: 2023-01-16T11:45:07Z
date_published: 2020-04-01T00:00:00Z
date_updated: 2024-10-09T21:05:15Z
day: '01'
doi: 10.1016/j.jnt.2019.09.003
extern: '1'
external_id:
  arxiv:
  - '1906.00632'
intvolume: '       209'
issue: '4'
keyword:
- Algebra and Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.1906.00632
month: '04'
oa: 1
oa_version: Preprint
page: 378-390
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Primitive divisors of sequences associated to elliptic curves
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 209
year: '2020'
...