---
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
  content_type: application/pdf
  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
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'
...