{"id":185440,"date":"2010-10-20T00:00:00","date_gmt":"2010-10-25T15:43:01","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/selmer-ranks-of-elliptic-curves-in-families-of-quadratic-twists\/"},"modified":"2016-08-22T11:27:33","modified_gmt":"2016-08-22T18:27:33","slug":"selmer-ranks-of-elliptic-curves-in-families-of-quadratic-twists","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/selmer-ranks-of-elliptic-curves-in-families-of-quadratic-twists\/","title":{"rendered":"Selmer Ranks of Elliptic Curves in Families of Quadratic Twists"},"content":{"rendered":"<div class=\"asset-content\">\n<p>This talk will report on ongoing work with Barry Mazur that studies 2-Selmer ranks in the family of all quadratic twists of a fixed elliptic curve over a number field. Our goal is to compute the density of twists with a given 2-Selmer rank r, for every r. This has been done by Heath-Brown, Swinnerton-Dyer, and Kane for elliptic curves over Q with all 2-torsion rational. Our methods are different and work best for curves with no rational points of order 2. So far we can prove under certain hypotheses that E has &#8220;many&#8221; twists of every 2-Selmer rank, but not that the set of such twists has positive density. In this talk I will describe these results and the methods involved, and discuss a basic question about algebraic number fields that arises in trying to improve our results.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This talk will report on ongoing work with Barry Mazur that studies 2-Selmer ranks in the family of all quadratic twists of a fixed elliptic curve over a number field. Our goal is to compute the density of twists with a given 2-Selmer rank r, for every r. This has been done by Heath-Brown, Swinnerton-Dyer, [&hellip;]<\/p>\n","protected":false},"featured_media":195796,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-185440","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/6RVAdIOZr7k","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185440","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185440\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195796"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185440"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185440"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185440"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185440"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=185440"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=185440"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185440"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185440"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=185440"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=185440"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}