{"id":2779,"date":"2022-08-10T17:17:52","date_gmt":"2022-08-11T00:17:52","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=2779"},"modified":"2022-09-02T10:34:15","modified_gmt":"2022-09-02T17:34:15","slug":"monodromy-groups-of-belyi-lattes-maps-edray-goins-pomona-college","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/monodromy-groups-of-belyi-lattes-maps-edray-goins-pomona-college\/","title":{"rendered":"Monodromy groups of Belyi Lattes maps (Edray Goins, Pomona College)"},"content":{"rendered":"<div dir=\"ltr\">\n<div class=\"\">\n<div class=\"\">\n<div class=\"\">An elliptic curve $ E: y^2 + a_1 \\, x \\, y + a_3 \\, y = x^3 + a_2 \\, x^2 + a_1 \\, x + a_6 $ is a cubic equation which has two curious properties: (1) the curve is nonsingular, so that we can draw tangent lines to every point $ P = (x,y) $ on the curve; and (2) the collection of complex points, namely $ E(\\mathbb C) $, forms an abelian group under a certain binary operation $ \\bigoplus: E(\\mathbb C) \\times E(\\mathbb C) \\to E(\\mathbb C) $. \u00a0 In particular, for every positive integer $N$, the map $ P \\mapsto [N] P $ which adds a point $ P \\in E(\\mathbb C) $ to itself $N$ times is a group homomorphism. \u00a0 A rational map $\\gamma: \\mathbb P^1(\\mathbb C) \\to \\mathbb P^1(\\mathbb C) $ from the Riemann Sphere to itself is said to be a Latt\\`{e}s Map if there are &#8220;well-behaved&#8221; maps $ \\phi: E(\\mathbb C) \\to \\mathbb P^1(\\mathbb C) $ and $\\psi: E(\\mathbb C) \\to E(\\mathbb C) $ such that $\\gamma \\circ \\phi = \\phi \\circ \\psi$. \u00a0We are interested in those Latt\\`{e}s Maps $\\gamma$ which are also Bely\\u{\\i} Maps, that is, the only critical values are $ 0 $, $ 1 $, and $ \\infty $. \u00a0Work of Zeytin classifies all such maps: For example, if $ E: y^2 = x^3 + 1 $ then $ \\phi: (x,y) \\mapsto (y+1)\/2 $ while $\\psi = [N] $ for some positive integer $N$.<\/div>\n<div class=\"\"><\/div>\n<div class=\"\">We would like to know more about Bely\\u{\\i} Latt\\`{e}s Maps $\\gamma$. \u00a0What can we say about such maps? \u00a0What are their Dessin d&#8217;Enfants? \u00a0In some cases, this is a bipartite graph with $ 3 \\, N^2 $ vertices. \u00a0What are their monodromy groups? Sometimes this is a group of size $ 3 \\, N^2 $. \u00a0In this talk, we explain the complete answers to these questions, exploiting the relationship between fundamental groups of Riemann surfaces and Galois groups of function fields. \u00a0This work is conducted as part of the Pomona Research in Mathematics Experience (DMS-2113782).<\/div>\n<div class=\"\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>An elliptic curve $ E: y^2 + a_1 \\, x \\, y + a_3 \\, y = x^3 + a_2 \\, x^2 + a_1 \\, x + a_6 $ is [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[13],"class_list":["post-2779","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-antc-seminar","cat_antc-seminar"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - 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