[{"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"}],"intvolume":"         7","publisher":"Springer Nature","publication":"Research in Number Theory","date_published":"2021-05-20T00:00:00Z","quality_controlled":"1","corr_author":"1","article_processing_charge":"No","extern":"1","scopus_import":"1","type":"journal_article","volume":7,"main_file_link":[{"url":"https://doi.org/10.1007/s40993-021-00267-9","open_access":"1"}],"date_updated":"2024-10-09T21:05:08Z","author":[{"orcid":"0000-0002-0854-0306","last_name":"Verzobio","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb","first_name":"Matteo","full_name":"Verzobio, Matteo"}],"year":"2021","oa":1,"article_type":"original","language":[{"iso":"eng"}],"doi":"10.1007/s40993-021-00267-9","status":"public","date_created":"2023-01-16T11:44:39Z","publication_identifier":{"issn":["2522-0160","2363-9555"]},"issue":"2","month":"05","day":"20","citation":{"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).","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.","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>","ista":"Verzobio M. 2021. Primitive divisors of sequences associated to elliptic curves with complex multiplication. Research in Number Theory. 7(2), 37.","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>","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>."},"keyword":["Algebra and Number Theory"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"12308","fulldoi":"https://doi.org/10.1007/s40993-021-00267-9","article_number":"37","title":"Primitive divisors of sequences associated to elliptic curves with complex multiplication","publication_status":"published","oa_version":"Published Version"},{"publication_status":"published","OA_type":"hybrid","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"oa_version":"Published Version","title":"A density of ramified primes","fulldoi":"https://doi.org/10.1007/s40993-021-00295-5","article_number":"1","_id":"19489","citation":{"ista":"Chan S, McMeekin C, Milovic D. 2021. A density of ramified primes. Research in Number Theory. 8, 1.","ama":"Chan S, McMeekin C, Milovic D. A density of ramified primes. <i>Research in Number Theory</i>. 2021;8. doi:<a href=\"https://doi.org/10.1007/s40993-021-00295-5\">10.1007/s40993-021-00295-5</a>","chicago":"Chan, Stephanie, Christine McMeekin, and Djordjo Milovic. “A Density of Ramified Primes.” <i>Research in Number Theory</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s40993-021-00295-5\">https://doi.org/10.1007/s40993-021-00295-5</a>.","short":"S. Chan, C. McMeekin, D. Milovic, Research in Number Theory 8 (2021).","mla":"Chan, Stephanie, et al. “A Density of Ramified Primes.” <i>Research in Number Theory</i>, vol. 8, 1, Springer Nature, 2021, doi:<a href=\"https://doi.org/10.1007/s40993-021-00295-5\">10.1007/s40993-021-00295-5</a>.","ieee":"S. Chan, C. McMeekin, and D. Milovic, “A density of ramified primes,” <i>Research in Number Theory</i>, vol. 8. Springer Nature, 2021.","apa":"Chan, S., McMeekin, C., &#38; Milovic, D. (2021). A density of ramified primes. <i>Research in Number Theory</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40993-021-00295-5\">https://doi.org/10.1007/s40993-021-00295-5</a>"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"15","month":"11","publication_identifier":{"eissn":["2363-9555"],"issn":["2522-0160"]},"date_created":"2025-04-05T10:50:51Z","status":"public","doi":"10.1007/s40993-021-00295-5","language":[{"iso":"eng"}],"article_type":"original","external_id":{"arxiv":["2005.10188"]},"oa":1,"main_file_link":[{"url":"https://doi.org/10.1007/s40993-021-00295-5","open_access":"1"}],"year":"2021","date_updated":"2025-07-10T11:51:46Z","author":[{"id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","orcid":"0000-0001-8467-4106","last_name":"Chan","first_name":"Yik Tung","full_name":"Chan, Yik Tung"},{"last_name":"McMeekin","first_name":"Christine","full_name":"McMeekin, Christine"},{"first_name":"Djordjo","full_name":"Milovic, Djordjo","last_name":"Milovic"}],"type":"journal_article","scopus_import":"1","volume":8,"extern":"1","article_processing_charge":"No","arxiv":1,"ddc":["510"],"quality_controlled":"1","has_accepted_license":"1","date_published":"2021-11-15T00:00:00Z","OA_place":"publisher","publication":"Research in Number Theory","intvolume":"         8","publisher":"Springer Nature","abstract":[{"lang":"eng","text":"Let K be a cyclic number field of odd degree over \r\n𝑄 with odd narrow class number, such that 2 is inert in 𝐾/𝑄. We define a family of number fields {𝐾(𝑝)}𝑝, depending on K and indexed by the rational primes p that split completely in 𝐾/𝑄, in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in 𝐾(𝑝)/𝑄 is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension 𝐾/𝑄. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals."}]},{"external_id":{"arxiv":["1807.08986"]},"oa":1,"researchdata_availability":"no","doi":"10.1007/s40993-018-0146-6","language":[{"iso":"eng"}],"article_type":"original","supplementarymaterial":"no","date_created":"2022-03-18T12:09:48Z","publication_identifier":{"issn":["2522-0160"],"eissn":["2363-9555"]},"das_tickbox":"0","status":"public","month":"01","day":"02","citation":{"ista":"Ionica S, Kılıçer P, Lauter K, Lorenzo García E, Manzateanu M-A, Massierer M, Vincent C. 2019. Modular invariants for genus 3 hyperelliptic curves. Research in Number Theory. 5, 9.","chicago":"Ionica, Sorina, Pınar Kılıçer, Kristin Lauter, Elisa Lorenzo García, Maria-Adelina Manzateanu, Maike Massierer, and Christelle Vincent. “Modular Invariants for Genus 3 Hyperelliptic Curves.” <i>Research in Number Theory</i>. Springer Nature, 2019. <a href=\"https://doi.org/10.1007/s40993-018-0146-6\">https://doi.org/10.1007/s40993-018-0146-6</a>.","ama":"Ionica S, Kılıçer P, Lauter K, et al. Modular invariants for genus 3 hyperelliptic curves. <i>Research in Number Theory</i>. 2019;5. doi:<a href=\"https://doi.org/10.1007/s40993-018-0146-6\">10.1007/s40993-018-0146-6</a>","ieee":"S. Ionica <i>et al.</i>, “Modular invariants for genus 3 hyperelliptic curves,” <i>Research in Number Theory</i>, vol. 5. Springer Nature, 2019.","apa":"Ionica, S., Kılıçer, P., Lauter, K., Lorenzo García, E., Manzateanu, M.-A., Massierer, M., &#38; Vincent, C. (2019). Modular invariants for genus 3 hyperelliptic curves. <i>Research in Number Theory</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s40993-018-0146-6\">https://doi.org/10.1007/s40993-018-0146-6</a>","mla":"Ionica, Sorina, et al. “Modular Invariants for Genus 3 Hyperelliptic Curves.” <i>Research in Number Theory</i>, vol. 5, 9, Springer Nature, 2019, doi:<a href=\"https://doi.org/10.1007/s40993-018-0146-6\">10.1007/s40993-018-0146-6</a>.","short":"S. Ionica, P. Kılıçer, K. Lauter, E. Lorenzo García, M.-A. Manzateanu, M. Massierer, C. Vincent, Research in Number Theory 5 (2019)."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","keyword":["Algebra and Number Theory"],"article_number":"9","fulldoi":"https://doi.org/10.1007/s40993-018-0146-6","title":"Modular invariants for genus 3 hyperelliptic curves","_id":"10874","publication_status":"published","oa_version":"Preprint","abstract":[{"text":"In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.","lang":"eng"}],"intvolume":"         5","publisher":"Springer Nature","date_published":"2019-01-02T00:00:00Z","publication":"Research in Number Theory","quality_controlled":"1","article_processing_charge":"No","acknowledgement":"The authors would like to thank the Lorentz Center in Leiden for hosting the Women in Numbers Europe 2 workshop and providing a productive and enjoyable environment for our initial work on this project. We are grateful to the organizers of WIN-E2, Irene Bouw, Rachel Newton and Ekin Ozman, for making this conference and this collaboration possible. We\r\nthank Irene Bouw and Christophe Ritzenhaler for helpful discussions. Ionica acknowledges support from the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. Most of Kılıçer’s work was carried out during her stay in Universiteit Leiden and Carl von Ossietzky Universität Oldenburg. Massierer was supported by the Australian Research Council (DP150101689). Vincent is supported by the National Science Foundation under Grant No. DMS-1802323 and by the Thomas Jefferson Fund of the Embassy of France in the United States and the FACE Foundation. ","arxiv":1,"scopus_import":"1","type":"journal_article","volume":5,"department":[{"_id":"TiBr"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1807.08986"}],"year":"2019","author":[{"last_name":"Ionica","full_name":"Ionica, Sorina","first_name":"Sorina"},{"first_name":"Pınar","full_name":"Kılıçer, Pınar","last_name":"Kılıçer"},{"full_name":"Lauter, Kristin","first_name":"Kristin","last_name":"Lauter"},{"last_name":"Lorenzo García","full_name":"Lorenzo García, Elisa","first_name":"Elisa"},{"id":"be8d652e-a908-11ec-82a4-e2867729459c","last_name":"Manzateanu","first_name":"Maria-Adelina","full_name":"Manzateanu, Maria-Adelina"},{"first_name":"Maike","full_name":"Massierer, Maike","last_name":"Massierer"},{"last_name":"Vincent","first_name":"Christelle","full_name":"Vincent, Christelle"}],"date_updated":"2026-08-06T12:10:57Z"}]