[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-116916-en":3,"doc-seo-116916-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},116916,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Machine learning invariants of arithmetic curves - Prediction of BSD-type invariants","The paper investigates how standard machine learning methods can be trained to predict arithmetic invariants of low-genus curves. Using datasets on the order of 10^5, the study demonstrates classification performance for elliptic curves, targeting Birch–Swinnerton–Dyer related quantities such as rank and torsion subgroup, and for genus 2 curves using analogous invariants. The trained models classify curves according to these invariants with high accuracy (exceeding 0.97) and reach up to 0.998 for tasks like distinguishing torsion orders and recognizing integral points.","Journal of Symbolic Computation 115 (2023) 478–491  \nContents lists available at ScienceDirect  \nJournal of Symbolic Computation  \n[www.elsevier.com/locate/jsc](www.elsevier.com/locate/jsc)  \nMachine learning invariants of arithmetic curves Yang-Hui He a,b,c , Kyu-Hwan Lee d , Thomas Oliver e  \na London Institute, Royal Institution, 21 Albemarle St, London W1S 4BS, UK b Merton College, University of Oxford, OX14JD, UK  \nc School of Physics, NanKai University, Tianjin, 300071, PR China  \nd Department of Mathematics, University of Connecticut, Storrs, CT, 06269-1009, USA e SCEDT, Teesside University, Middlesbrough, TS1 3BX, UK  \n\n| a r t i c l e i n f o | a b s t r a c t\u003Cbr>We show that standard machine learning algorithms may be trained to predict certain invariants of low genus arithmetic curves. Using datasets of size around 105 , we demonstrate the utility of machine learning in classification problems pertaining to the BSD invariants of an elliptic curve (including its rank and torsion subgroup), and the analogous invariants of a genus 2 curve. Our results show that a trained machine can efficiently classify curves according to these invariants with high accuracies (> 0.97). For problems such as distinguishing between torsion orders, and the recognition of integral points, the accuracies can reach 0.998.\u003Cbr>© 2022 The Author(s). Published by Elsevier Ltd. This is an open\u003Cbr>access article under the CC BY license ([http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)). |\n| --- | --- |\n| Article history:\u003Cbr>Received 7 December 2021\u003Cbr>Received in revised form 29 June 2022 Accepted 15 August 2022\u003Cbr>Available online 22 August 2022 |  |\n| Keywords:\u003Cbr>Machine-learning\u003Cbr>Arithmetic geometry\u003Cbr>Elliptic curves\u003Cbr>Hyper-elliptic curves\u003Cbr>Birch-Swinnerton-Dyer conjecture |  |\n\nContents  \n1. Introduction ................................................................ 479  \nAcknowledgements ........................................................... 481  \n2. Notation .................................................................. 481  \n3. Methodology ............................................................... 481  \n3.1. Euler factors .......................................................... 481  \n3.2. Generic experimental strategy .............................................. 483  \n3.3. Further specifications .................................................... 484  \n4. Elliptic curves .............................................................. 484  \n4.1. Rank ................................................................ 485  \n[E-mail addresses:](E-mail addresses: hey@maths.ox.ac.uk)[ hey@maths.ox.ac.uk](E-mail addresses: hey@maths.ox.ac.uk) (Y.-H. He), [khlee@math.uconn.edu](khlee@math.uconn.edu) (K.-H. Lee), [T.Oliver@tees.ac.uk](T.Oliver@tees.ac.uk) (T. Oliver).  \n[https://doi.org/10.1016/j.jsc.2022.08.017](https://doi.org/10.1016/j.jsc.2022.08.017)  \n0747-7171/© 2022 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ([http://creativecommons.org/licenses/by/4.0/](http://creativecommons.org/licenses/by/4.0/)).  \nY.-H. He, K.-H. Lee and T. Oliver Journal of Symbolic Computation 115 (2023) 478 –491  \n4.2. Torsion order .......................................................... 485  \n4.3. Torsion structure ....................................................... 486  \n4.4. Integral points ......................................................... 486  \n4.5. Tate–Shafarevich group ................................................... 486  \n4.6. Interpretation of naive Bayesian models ....................................... 487  \n5. Genus 2 curves ............................................................. 489  \n5.1. Rank ................................................................ 489  \n5.2. Torsion order .......................................................... 489  \n5.3. Rational points ................................................","cbCaidzLRUkrLlw1","https://ap.wps.com/l/cbCaidzLRUkrLlw1","pdf",432630,1,14,"English","en",105,"# Introduction\n## Notation\n## Methodology\n### Euler factors\n### Generic experimental strategy\n### Further specifications\n## Elliptic curves\n### Rank\n### Torsion order\n### Torsion structure\n### Integral points\n### Tate–Shafarevich group\n## Genus 2 curves\n### Rank\n### Torsion order\n### Rational points\n### Trivial Tate–Shafarevich group\n# Conclusions and outlook","[{\"question\":\"What invariants does the paper focus on predicting with machine learning?\",\"answer\":\"It targets BSD-related invariants for elliptic curves, including rank and torsion subgroup, and analogous invariants for genus 2 curves.\"},{\"question\":\"How accurate are the trained machine learning models for these classifications?\",\"answer\":\"The results show high accuracies above 0.97, with some tasks reaching 0.998.\"},{\"question\":\"Which specific tasks achieve the highest performance in the study?\",\"answer\":\"Distinguishing between torsion orders and recognizing integral points attain the top accuracies reported, up to 0.998.\"}]","Machine learning invariants of arithmetic curves - Prediction of BSD-type invariants | PDF",1785672510,35,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"machine-learning-invariants-of-arithmetic-curves-prediction-of-bsd-type-invariants","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-invariants-of-arithmetic-curves-prediction-of-bsd-type-invariants/116916/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What invariants does the paper focus on predicting with machine learning?","Question",{"text":75,"@type":76},"It targets BSD-related invariants for elliptic curves, including rank and torsion subgroup, and analogous invariants for genus 2 curves.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How accurate are the trained machine learning models for these classifications?",{"text":80,"@type":76},"The results show high accuracies above 0.97, with some tasks reaching 0.998.",{"name":82,"@type":73,"acceptedAnswer":83},"Which specific tasks achieve the highest performance in the study?",{"text":84,"@type":76},"Distinguishing between torsion orders and recognizing integral points attain the top accuracies reported, up to 0.998.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]