{"id":47,"date":"2018-01-08T16:38:26","date_gmt":"2018-01-08T16:38:26","guid":{"rendered":"https:\/\/math-sites.uncg.edu\/number-theory\/?p=47"},"modified":"2025-01-24T19:37:57","modified_gmt":"2025-01-24T19:37:57","slug":"quadratic-form","status":"publish","type":"post","link":"https:\/\/math-sites.uncg.edu\/number-theory\/quadratic-form\/","title":{"rendered":"Well-rounded forms and the Voronoi tessellation"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-48 size-large\" src=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/quadratic-form-tree-1024x512.png\" alt=\"Well-rounded forms and the Voronoi tessellation\" width=\"1024\" height=\"512\" srcset=\"https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/quadratic-form-tree-1024x512.png 1024w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/quadratic-form-tree-300x150.png 300w, https:\/\/math-sites.uncg.edu\/number-theory\/wp-content\/uploads\/sites\/6\/2018\/01\/quadratic-form-tree-768x384.png 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>One identifies the space of binary quadratic forms with the upper half plane $$\\mathfrak{h}$$. The set of well-rounded forms corresponds to an infinite tree (red). The vertices of this tree correspond to perfect forms, and dualizing gives the Voronoi tessellation of h by hyperbolic triangles (blue). This structure can be used in the study of modular forms. In joint work with F. Hajir and P. Gunnells, D. Yasaki studied a 7-dimensional analogue of this structure in the investigation of cusp forms over a CM quartic field in <a href=\"https:\/\/math-sites.uncg.edu\/sites\/yasaki\/publications\/gunnells-hajir-yasaki_final.pdf\">Modular forms and elliptic curves over the field of fifth roots of unity<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>One identifies the space of binary quadratic forms with the upper half plane $$\\mathfrak{h}$$. The set of well-rounded forms corresponds to an&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-47","post","type-post","status-publish","format-standard","hentry","category-number-theory"],"acf":[],"_links":{"self":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/comments?post=47"}],"version-history":[{"count":3,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/47\/revisions"}],"predecessor-version":[{"id":240,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/posts\/47\/revisions\/240"}],"wp:attachment":[{"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/media?parent=47"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/categories?post=47"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math-sites.uncg.edu\/number-theory\/wp-json\/wp\/v2\/tags?post=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}