[{"das_tickbox":"1","isi":1,"date_updated":"2026-07-23T11:33:57Z","file":[{"date_created":"2026-07-23T11:32:51Z","content_type":"application/pdf","access_level":"open_access","relation":"main_file","file_size":754810,"checksum":"34e6e965a2b30a258e0d4103350a68fd","file_name":"2026_JourNumberTheory_Baranczuk.pdf","creator":"dernst","success":1,"date_updated":"2026-07-23T11:32:51Z","file_id":"22396"}],"oa":1,"oa_version":"Published Version","publication":"Journal of Number Theory","file_date_updated":"2026-07-23T11:32:51Z","intvolume":"       279","_id":"20078","dataavailabilitystatement":"No data was used for the research described in the article.","month":"02","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2025-07-27T22:01:25Z","department":[{"_id":"TiBr"}],"article_type":"original","abstract":[{"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.","lang":"eng"}],"title":"Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve","quality_controlled":"1","license":"https://creativecommons.org/licenses/by/4.0/","status":"public","publisher":"Elsevier","doi":"10.1016/j.jnt.2025.06.001","OA_type":"hybrid","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>. 2026;279:170-183. doi:<a href=\"https://doi.org/10.1016/j.jnt.2025.06.001\">10.1016/j.jnt.2025.06.001</a>","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, 2026, pp. 170–83, doi:<a href=\"https://doi.org/10.1016/j.jnt.2025.06.001\">10.1016/j.jnt.2025.06.001</a>.","ista":"Barańczuk S, Naskręcki B, Verzobio M. 2026. Divisibility sequences related to abelian varieties isogenous to a power of an elliptic curve. Journal of Number Theory. 279, 170–183.","short":"S. Barańczuk, B. Naskręcki, M. Verzobio, Journal of Number Theory 279 (2026) 170–183.","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, 2026.","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, 2026. <a href=\"https://doi.org/10.1016/j.jnt.2025.06.001\">https://doi.org/10.1016/j.jnt.2025.06.001</a>.","apa":"Barańczuk, S., Naskręcki, B., &#38; Verzobio, M. (2026). 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>"},"publication_status":"published","OA_place":"publisher","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"year":"2026","volume":279,"researchdata_availability":"no","keyword":["Divisibility sequences","Abelian varieties","Elliptic divisibility sequences","Isogenies","Primitive divisors"],"language":[{"iso":"eng"}],"external_id":{"isi":["001541172400002"],"arxiv":["2309.09699"]},"date_published":"2026-02-01T00:00:00Z","arxiv":1,"has_accepted_license":"1","ddc":["500"],"fulldoi":"https://doi.org/10.1016/j.jnt.2025.06.001","type":"journal_article","PlanS_conform":"1","article_processing_charge":"Yes (via OA deal)","day":"01","scopus_import":"1","publication_identifier":{"issn":["0022-314X"]},"author":[{"first_name":"Stefan","full_name":"Barańczuk, Stefan","last_name":"Barańczuk"},{"last_name":"Naskręcki","first_name":"Bartosz","full_name":"Naskręcki, Bartosz"},{"orcid":"0000-0002-0854-0306","last_name":"Verzobio","full_name":"Verzobio, Matteo","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb"}],"page":"170-183","corr_author":"1","supplementarymaterial":"no"},{"corr_author":"1","article_number":"12","supplementarymaterial":"no","publication_identifier":{"eissn":["2397-3129"]},"scopus_import":"1","author":[{"orcid":"0000-0002-8314-0177","last_name":"Browning","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D","full_name":"Browning, Timothy D"},{"full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","first_name":"Matteo","last_name":"Verzobio","orcid":"0000-0002-0854-0306"}],"day":"01","type":"journal_article","article_processing_charge":"No","has_accepted_license":"1","ddc":["510"],"fulldoi":"https://doi.org/10.19086/da.143787","date_published":"2025-09-01T00:00:00Z","external_id":{"arxiv":["2408.11453"]},"arxiv":1,"language":[{"iso":"eng"}],"OA_place":"publisher","publication_status":"published","citation":{"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.","short":"T.D. Browning, M. Verzobio, Discrete Analysis 2025 (2025).","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>.","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>.","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>","ista":"Browning TD, Verzobio M. 2025. Counting integer points on affine surfaces with a side condition. Discrete Analysis. 2025, 12."},"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":2025,"researchdata_availability":"no","year":"2025","doi":"10.19086/da.143787","OA_type":"diamond","status":"public","publisher":"Cambridge: Alliance of Diamond Open Access Journals","project":[{"name":"Rational curves via function field analytic number theory","_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278"},{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"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.","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."}],"title":"Counting integer points on affine surfaces with a side condition","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2026-01-18T23:02:44Z","department":[{"_id":"TiBr"}],"article_type":"original","ec_funded":1,"_id":"21003","month":"09","publication":"Discrete Analysis","oa_version":"Published Version","file_date_updated":"2026-02-12T07:50:47Z","intvolume":"      2025","das_tickbox":"0","file":[{"relation":"main_file","file_size":393625,"checksum":"3d38e850b40f3e1abbfd30073bd4388a","date_created":"2026-02-12T07:50:47Z","content_type":"application/pdf","access_level":"open_access","success":1,"date_updated":"2026-02-12T07:50:47Z","file_id":"21214","creator":"dernst","file_name":"2025_DiscreteAnalysis_Browning.pdf"}],"date_updated":"2026-07-16T08:48:35Z","oa":1},{"fulldoi":"https://doi.org/10.1017/prm.2025.7","ddc":["500"],"external_id":{"isi":["001414690400001"]},"date_published":"2025-02-06T00:00:00Z","language":[{"iso":"eng"}],"corr_author":"1","supplementarymaterial":"no","publication_identifier":{"issn":["0308-2105"],"eissn":["1473-7124"]},"scopus_import":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1017/prm.2025.7"}],"author":[{"last_name":"Ballini","full_name":"Ballini, Francesco","first_name":"Francesco"},{"last_name":"Lombardo","full_name":"Lombardo, Davide","first_name":"Davide"},{"last_name":"Verzobio","orcid":"0000-0002-0854-0306","full_name":"Verzobio, Matteo","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb"}],"day":"06","type":"journal_article","article_processing_charge":"Yes (via OA deal)","date_created":"2025-03-16T23:01:25Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","article_type":"original","department":[{"_id":"TiBr"}],"_id":"19407","month":"02","oa_version":"Published Version","publication":"Proceedings of the Royal Society of Edinburgh Section A: Mathematics","das_tickbox":"0","oa":1,"isi":1,"date_updated":"2026-07-16T09:31:02Z","OA_place":"publisher","citation":{"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.","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>","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>.","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>.","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>","short":"F. Ballini, D. Lombardo, M. Verzobio, Proceedings of the Royal Society of Edinburgh Section A: Mathematics (2025).","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."},"publication_status":"epub_ahead","year":"2025","researchdata_availability":"no","doi":"10.1017/prm.2025.7","OA_type":"hybrid","quality_controlled":"1","publisher":"Cambridge University Press","status":"public","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.","title":"On the L-polynomials of curves over finite fields","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."}]},{"month":"08","_id":"20222","ec_funded":1,"department":[{"_id":"TiBr"}],"article_type":"original","date_created":"2025-08-24T22:01:31Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","file":[{"success":1,"file_id":"20275","date_updated":"2025-09-02T07:55:05Z","creator":"dernst","file_name":"2025_IMRN_Verzobio.pdf","relation":"main_file","file_size":540263,"checksum":"482ae2be98841ee446cf2bdfcd79f86f","date_created":"2025-09-02T07:55:05Z","content_type":"application/pdf","access_level":"open_access"}],"date_updated":"2026-07-17T11:55:00Z","isi":1,"oa":1,"das_tickbox":"0","file_date_updated":"2025-09-02T07:55:05Z","intvolume":"      2025","publication":"International Mathematics Research Notices","oa_version":"Published Version","OA_type":"hybrid","doi":"10.1093/imrn/rnaf249","volume":2025,"tmp":{"image":"/images/cc_by_nc_nd.png","short":"CC BY-NC-ND (4.0)","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)"},"year":"2025","researchdata_availability":"no","publication_status":"published","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>","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>.","ista":"Verzobio M. 2025. Counting rational points on smooth hypersurfaces with high degree. International Mathematics Research Notices. 2025(16), rnaf249.","short":"M. Verzobio, International Mathematics Research Notices 2025 (2025).","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.","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>.","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>"},"OA_place":"publisher","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."}],"title":"Counting rational points on smooth hypersurfaces with high degree","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.","issue":"16","status":"public","project":[{"call_identifier":"H2020","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","name":"IST-BRIDGE: International postdoctoral program"}],"publisher":"Oxford University Press","quality_controlled":"1","license":"https://creativecommons.org/licenses/by-nc-nd/4.0/","ddc":["510"],"fulldoi":"https://doi.org/10.1093/imrn/rnaf249","has_accepted_license":"1","language":[{"iso":"eng"}],"arxiv":1,"date_published":"2025-08-01T00:00:00Z","external_id":{"isi":["001549126000001"],"arxiv":["2503.19451"]},"author":[{"orcid":"0000-0002-0854-0306","last_name":"Verzobio","full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","first_name":"Matteo"}],"scopus_import":"1","publication_identifier":{"issn":["1073-7928"],"eissn":["1687-0247"]},"supplementarymaterial":"no","corr_author":"1","article_number":"rnaf249","article_processing_charge":"Yes (via OA deal)","type":"journal_article","day":"01"},{"month":"10","_id":"12311","ec_funded":1,"article_type":"original","department":[{"_id":"TiBr"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2023-01-16T11:45:22Z","oa":1,"isi":1,"date_updated":"2026-10-21T11:01:35Z","file":[{"access_level":"open_access","date_created":"2025-12-30T06:45:47Z","content_type":"application/pdf","file_size":477624,"checksum":"c5ec6e29aca2fb4533cb95fac409a0b2","relation":"main_file","creator":"dernst","file_name":"2025_ProceedingsRoyalSocEdinburghA_Naskrecki.pdf","success":1,"file_id":"20878","date_updated":"2025-12-30T06:45:47Z"}],"das_tickbox":"0","intvolume":"       155","file_date_updated":"2025-12-30T06:45:47Z","publication":"Proceedings of the Royal Society of Edinburgh Section A: Mathematics","oa_version":"Published Version","OA_type":"hybrid","doi":"10.1017/prm.2024.7","researchdata_availability":"no","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"year":"2025","volume":155,"publication_status":"published","citation":{"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>.","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>","short":"B. Naskręcki, M. Verzobio, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 155 (2025) 1646–1660.","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.","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>","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>."},"OA_place":"publisher","title":"Common valuations of division polynomials","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.","publisher":"Cambridge University Press","project":[{"name":"IST-BRIDGE: International postdoctoral program","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","grant_number":"101034413","call_identifier":"H2020"}],"status":"public","issue":"5","quality_controlled":"1","fulldoi":"https://doi.org/10.1017/prm.2024.7","ddc":["510"],"has_accepted_license":"1","language":[{"iso":"eng"}],"keyword":["Elliptic curves","Néron models","division polynomials","height functions","discrete valuation rings"],"arxiv":1,"date_published":"2025-10-01T00:00:00Z","external_id":{"arxiv":["2203.02015"],"isi":["001174907100001"]},"author":[{"last_name":"Naskręcki","full_name":"Naskręcki, Bartosz","first_name":"Bartosz"},{"last_name":"Verzobio","orcid":"0000-0002-0854-0306","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","full_name":"Verzobio, Matteo"}],"scopus_import":"1","publication_identifier":{"issn":["0308-2105"],"eissn":["1473-7124"]},"supplementarymaterial":"no","page":"1646-1660","corr_author":"1","article_processing_charge":"Yes (via OA deal)","PlanS_conform":"1","type":"journal_article","day":"01"},{"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"researchdata_availability":"no","volume":30,"year":"2024","citation":{"short":"D. Lombardo, M. Verzobio, Selecta Mathematica 30 (2024).","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.","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>.","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>","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>","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>.","ista":"Lombardo D, Verzobio M. 2024. On the local-global principle for isogenies of abelian surfaces. Selecta Mathematica. 30(2), 18."},"publication_status":"published","doi":"10.1007/s00029-023-00908-0","publisher":"Springer Nature","issue":"2","status":"public","quality_controlled":"1","abstract":[{"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 $\\ell$.","lang":"eng"}],"title":"On the local-global principle for isogenies of abelian surfaces","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_type":"original","department":[{"_id":"TiBr"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2023-01-16T11:45:53Z","month":"01","_id":"12312","intvolume":"        30","file_date_updated":"2024-07-22T09:33:58Z","oa_version":"Published Version","publication":"Selecta Mathematica","oa":1,"isi":1,"date_updated":"2026-10-21T11:01:56Z","file":[{"success":1,"file_id":"17298","date_updated":"2024-07-22T09:33:58Z","file_name":"2024_SelectaMath_Lombardo.pdf","creator":"dernst","relation":"main_file","file_size":1301415,"checksum":"ae75441420aabd80c5828bce38272ba1","date_created":"2024-07-22T09:33:58Z","content_type":"application/pdf","access_level":"open_access"}],"das_tickbox":"0","supplementarymaterial":"no","article_number":"18","corr_author":"1","author":[{"last_name":"Lombardo","first_name":"Davide","full_name":"Lombardo, Davide"},{"first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","full_name":"Verzobio, Matteo","orcid":"0000-0002-0854-0306","last_name":"Verzobio"}],"publication_identifier":{"issn":["1022-1824"],"eissn":["1420-9020"]},"scopus_import":"1","day":"26","article_processing_charge":"Yes (via OA deal)","type":"journal_article","fulldoi":"https://doi.org/10.1007/s00029-023-00908-0","ddc":["510"],"has_accepted_license":"1","arxiv":1,"date_published":"2024-01-26T00:00:00Z","external_id":{"arxiv":["2206.15240"],"isi":["001148959100001"]},"language":[{"iso":"eng"}]},{"author":[{"orcid":"0000-0002-8314-0177","last_name":"Browning","full_name":"Browning, Timothy D","id":"35827D50-F248-11E8-B48F-1D18A9856A87","first_name":"Timothy D"},{"orcid":"0000-0002-0854-0306","last_name":"Verzobio","full_name":"Verzobio, Matteo","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb"}],"publication_identifier":{"issn":["0025-5793"],"eissn":["2041-7942"]},"scopus_import":"1","supplementarymaterial":"yes","corr_author":"1","article_number":"e12269","article_processing_charge":"Yes (via OA deal)","type":"journal_article","day":"01","ddc":["510"],"fulldoi":"https://doi.org/10.1112/mtk.12269","has_accepted_license":"1","language":[{"iso":"eng"}],"external_id":{"isi":["001273912800001"]},"date_published":"2024-10-01T00:00:00Z","OA_type":"hybrid","doi":"10.1112/mtk.12269","tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"volume":70,"year":"2024","researchdata_availability":"no","publication_status":"published","citation":{"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>.","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.","short":"T.D. Browning, M. Verzobio, Mathematika 70 (2024)."},"OA_place":"publisher","title":"Strong divisibility sequences and sieve methods","abstract":[{"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.","lang":"eng"}],"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.","status":"public","issue":"4","project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","grant_number":"P36278","name":"Rational curves via function field analytic number theory"},{"grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","call_identifier":"H2020","name":"IST-BRIDGE: International postdoctoral program"}],"publisher":"London Mathematical Society","quality_controlled":"1","month":"10","_id":"17323","ec_funded":1,"department":[{"_id":"TiBr"}],"article_type":"original","date_created":"2024-07-28T22:01:08Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_updated":"2026-10-21T11:04:53Z","file":[{"file_id":"18842","date_updated":"2025-01-13T11:06:25Z","success":1,"file_name":"2024_Mathematika_Browning.pdf","creator":"dernst","checksum":"0b1518bdc1a901413005c19202cfa497","file_size":273006,"relation":"main_file","access_level":"open_access","content_type":"application/pdf","date_created":"2025-01-13T11:06:25Z"}],"isi":1,"oa":1,"das_tickbox":"0","file_date_updated":"2025-01-13T11:06:25Z","intvolume":"        70","publication":"Mathematika","oa_version":"Published Version"},{"type":"journal_article","article_processing_charge":"Yes (in subscription journal)","day":"03","scopus_import":"1","publication_identifier":{"eissn":["0030-8730"]},"author":[{"full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","first_name":"Matteo","last_name":"Verzobio","orcid":"0000-0002-0854-0306"}],"corr_author":"1","page":"331-351","supplementarymaterial":"no","language":[{"iso":"eng"}],"date_published":"2023-11-03T00:00:00Z","external_id":{"arxiv":["2001.02987"],"isi":["001104766900001"]},"arxiv":1,"has_accepted_license":"1","ddc":["510"],"fulldoi":"https://doi.org/10.2140/pjm.2023.325.331","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.","title":"Some effectivity results for primitive divisors of elliptic divisibility  sequences","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","issue":"2","status":"public","project":[{"call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program"}],"publisher":"Mathematical Sciences Publishers","doi":"10.2140/pjm.2023.325.331","citation":{"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>.","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>","ista":"Verzobio M. 2023. Some effectivity results for primitive divisors of elliptic divisibility  sequences. Pacific Journal of Mathematics. 325(2), 331–351.","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.","short":"M. Verzobio, Pacific Journal of Mathematics 325 (2023) 331–351.","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>."},"publication_status":"published","volume":325,"tmp":{"image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"researchdata_availability":"no","year":"2023","das_tickbox":"0","file":[{"file_id":"14525","date_updated":"2023-11-13T09:50:41Z","success":1,"creator":"dernst","file_name":"2023_PacificJourMaths_Verzobio.pdf","checksum":"b6218d16a72742d8bb38d6fc3c9bb8c6","file_size":389897,"relation":"main_file","access_level":"open_access","content_type":"application/pdf","date_created":"2023-11-13T09:50:41Z"}],"isi":1,"date_updated":"2026-07-29T10:07:53Z","oa":1,"oa_version":"Published Version","publication":"Pacific Journal of Mathematics","file_date_updated":"2023-11-13T09:50:41Z","intvolume":"       325","ec_funded":1,"_id":"12313","month":"11","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2023-01-16T11:46:19Z","department":[{"_id":"TiBr"}],"article_type":"original"},{"article_number":"37","corr_author":"1","author":[{"orcid":"0000-0002-0854-0306","last_name":"Verzobio","full_name":"Verzobio, Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","first_name":"Matteo"}],"publication_identifier":{"issn":["2522-0160","2363-9555"]},"scopus_import":"1","main_file_link":[{"url":"https://doi.org/10.1007/s40993-021-00267-9","open_access":"1"}],"day":"20","article_processing_charge":"No","type":"journal_article","fulldoi":"https://doi.org/10.1007/s40993-021-00267-9","date_published":"2021-05-20T00:00:00Z","language":[{"iso":"eng"}],"keyword":["Algebra and Number Theory"],"volume":7,"year":"2021","publication_status":"published","citation":{"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.","short":"M. Verzobio, Research in Number Theory 7 (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>.","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>"},"doi":"10.1007/s40993-021-00267-9","publisher":"Springer Nature","status":"public","issue":"2","quality_controlled":"1","title":"Primitive divisors of sequences associated to elliptic curves with complex multiplication","abstract":[{"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.","lang":"eng"}],"article_type":"original","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-01-16T11:44:39Z","month":"05","extern":"1","_id":"12308","intvolume":"         7","oa_version":"Published Version","publication":"Research in Number Theory","oa":1,"date_updated":"2024-10-09T21:05:08Z"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-01-16T11:44:54Z","article_type":"original","_id":"12309","extern":"1","month":"01","oa_version":"Preprint","publication":"Acta Arithmetica","intvolume":"       198","date_updated":"2024-10-09T21:05:08Z","oa":1,"publication_status":"published","citation":{"short":"M. Verzobio, Acta Arithmetica 198 (2021) 129–168.","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.","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>.","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>","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>","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>.","ista":"Verzobio M. 2021. Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728. Acta Arithmetica. 198(2), 129–168."},"year":"2021","volume":198,"doi":"10.4064/aa191016-30-7","quality_controlled":"1","issue":"2","status":"public","publisher":"Institute of Mathematics, Polish Academy of Sciences","abstract":[{"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.","lang":"eng"}],"title":"Primitive divisors of elliptic divisibility sequences for elliptic curves with j=1728","fulldoi":"https://doi.org/10.4064/aa191016-30-7","date_published":"2021-01-04T00:00:00Z","external_id":{"arxiv":["2001.09634"]},"arxiv":1,"keyword":["Algebra and Number Theory"],"language":[{"iso":"eng"}],"corr_author":"1","page":"129-168","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2001.09634","open_access":"1"}],"scopus_import":"1","publication_identifier":{"issn":["0065-1036","1730-6264"]},"author":[{"orcid":"0000-0002-0854-0306","last_name":"Verzobio","full_name":"Verzobio, Matteo","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb"}],"day":"04","type":"journal_article","article_processing_charge":"No"},{"extern":"1","_id":"12314","month":"02","fulldoi":"https://doi.org/10.48550/arXiv.2102.07573","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-01-16T11:46:36Z","oa":1,"date_updated":"2024-10-09T21:05:07Z","language":[{"iso":"eng"}],"external_id":{"arxiv":["2102.07573"]},"date_published":"2021-02-15T00:00:00Z","publication":"arXiv","oa_version":"Preprint","arxiv":1,"doi":"10.48550/arXiv.2102.07573","main_file_link":[{"url":" https://doi.org/10.48550/arXiv.2102.07573","open_access":"1"}],"author":[{"last_name":"Verzobio","orcid":"0000-0002-0854-0306","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","full_name":"Verzobio, Matteo"}],"article_number":"2102.07573","citation":{"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>. .","short":"M. Verzobio, ArXiv (n.d.).","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>.","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>"},"publication_status":"submitted","corr_author":"1","year":"2021","type":"preprint","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 $r$."}],"title":"A recurrence relation for elliptic divisibility sequences","article_processing_charge":"No","day":"15","status":"public"},{"date_updated":"2024-10-09T21:05:15Z","oa":1,"intvolume":"       209","oa_version":"Preprint","publication":"Journal of Number Theory","month":"04","extern":"1","_id":"12310","article_type":"original","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2023-01-16T11:45:07Z","title":"Primitive divisors of sequences associated to elliptic curves","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."}],"status":"public","issue":"4","publisher":"Elsevier","quality_controlled":"1","doi":"10.1016/j.jnt.2019.09.003","volume":209,"year":"2020","citation":{"ista":"Verzobio M. 2020. Primitive divisors of sequences associated to elliptic curves. Journal of Number Theory. 209(4), 378–390.","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>","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>.","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>.","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>","short":"M. Verzobio, Journal of Number Theory 209 (2020) 378–390.","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."},"publication_status":"published","language":[{"iso":"eng"}],"keyword":["Algebra and Number Theory"],"arxiv":1,"external_id":{"arxiv":["1906.00632"]},"date_published":"2020-04-01T00:00:00Z","fulldoi":"https://doi.org/10.1016/j.jnt.2019.09.003","article_processing_charge":"No","type":"journal_article","day":"01","author":[{"full_name":"Verzobio, Matteo","first_name":"Matteo","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","orcid":"0000-0002-0854-0306","last_name":"Verzobio"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1906.00632"}],"scopus_import":"1","publication_identifier":{"issn":["0022-314X"]},"corr_author":"1","page":"378-390"}]