---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '21385'
abstract:
- lang: eng
text: 'We prove that the average size of a mixed character sum (math. formular)
(for a suitable smooth function w) is on the order of √x for all irrational real
θ satisfying a weak Diophantine condition, where χ is drawn from the family of
Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was
proved by Harper that the average size is o(√x) for rational θ. Certain quadratic
Diophantine equations play a key role in the present paper. '
acknowledgement: "We thank Ofir Gorodetsky, Andrew Granville, Adam Harper, Youness
Lamzouri,\r\nKannan Soundararajan, Ping Xi, and Matt Young for their interest, helpful
discussions, and comments. Special thanks are due to Jonathan Bober, Oleksiy Klurman,\r\nand
Besfort Shala for sending us a letter about Question 1.3, and to Hung Bui\r\nfor
informing us of [7]. V.W. thanks Stanford University for its hospitality and is
supported by the European Union’s Horizon 2020 research and innovation program\r\nunder
the Marie Skłodowska–Curie Grant Agreement No. 101034413. M.X. is supported by a
Simons Junior Fellowship from the Simons Society of Fellows at the\r\nSimons Foundation."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Victor
full_name: Wang, Victor
id: 76096395-aea4-11ed-a680-ab8ebbd3f1b9
last_name: Wang
orcid: 0000-0002-0704-7026
- first_name: Max
full_name: Xu, Max
last_name: Xu
citation:
ama: 'Wang V, Xu M. Average sizes of mixed character sums. Proceedings of the
Royal Society of Edinburgh: Section A Mathematics. 2026:1-15. doi:10.1017/prm.2026.10123'
apa: 'Wang, V., & Xu, M. (2026). Average sizes of mixed character sums. Proceedings
of the Royal Society of Edinburgh: Section A Mathematics. Cambridge University
Press. https://doi.org/10.1017/prm.2026.10123'
chicago: 'Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” Proceedings
of the Royal Society of Edinburgh: Section A Mathematics. Cambridge University
Press, 2026. https://doi.org/10.1017/prm.2026.10123.'
ieee: 'V. Wang and M. Xu, “Average sizes of mixed character sums,” Proceedings
of the Royal Society of Edinburgh: Section A Mathematics. Cambridge University
Press, pp. 1–15, 2026.'
ista: 'Wang V, Xu M. 2026. Average sizes of mixed character sums. Proceedings of
the Royal Society of Edinburgh: Section A Mathematics., 1–15.'
mla: 'Wang, Victor, and Max Xu. “Average Sizes of Mixed Character Sums.” Proceedings
of the Royal Society of Edinburgh: Section A Mathematics, Cambridge University
Press, 2026, pp. 1–15, doi:10.1017/prm.2026.10123.'
short: 'V. Wang, M. Xu, Proceedings of the Royal Society of Edinburgh: Section A
Mathematics (2026) 1–15.'
corr_author: '1'
date_created: 2026-03-02T10:09:23Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-03-02T14:05:47Z
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1017/prm.2026.10123
ec_funded: 1
external_id:
arxiv:
- '2411.14181'
has_accepted_license: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1017/prm.2026.10123
month: '01'
oa: 1
oa_version: Published Version
page: 1-15
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: epub_ahead
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: Average sizes of mixed character sums
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
year: '2026'
...
---
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. Proceedings
of the Royal Society of Edinburgh Section A: Mathematics. 2025;155(5):1646-1660.
doi:10.1017/prm.2024.7'
apa: 'Naskręcki, B., & Verzobio, M. (2025). Common valuations of division polynomials.
Proceedings of the Royal Society of Edinburgh Section A: Mathematics. Cambridge
University Press. https://doi.org/10.1017/prm.2024.7'
chicago: 'Naskręcki, Bartosz, and Matteo Verzobio. “Common Valuations of Division
Polynomials.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics.
Cambridge University Press, 2025. https://doi.org/10.1017/prm.2024.7.'
ieee: 'B. Naskręcki and M. Verzobio, “Common valuations of division polynomials,”
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 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.”
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, vol.
155, no. 5, Cambridge University Press, 2025, pp. 1646–60, doi:10.1017/prm.2024.7.'
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'
...
---
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. Proceedings of the Royal Society of Edinburgh Section A: Mathematics.
2025. doi:10.1017/prm.2025.7'
apa: '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. Cambridge University Press. https://doi.org/10.1017/prm.2025.7'
chicago: 'Ballini, Francesco, Davide Lombardo, and Matteo Verzobio. “On the L-Polynomials
of Curves over Finite Fields.” Proceedings of the Royal Society of Edinburgh
Section A: Mathematics. Cambridge University Press, 2025. https://doi.org/10.1017/prm.2025.7.'
ieee: 'F. Ballini, D. Lombardo, and M. Verzobio, “On the L-polynomials of curves
over finite fields,” Proceedings of the Royal Society of Edinburgh Section
A: Mathematics. 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.”
Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Cambridge
University Press, 2025, doi:10.1017/prm.2025.7.'
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: 2026-06-18T18:15:49Z
day: '06'
ddc:
- '500'
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...