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DTSTAMP:20250707T112031Z
DESCRIPTION:Title: Elliptic curves and automorphic forms\n\nAbstract: Ellip
 tic curves are some of the simplest examples of algebraic varieties that c
 arry a group structure. The interplay between this group structure and Dio
 phantine problems has led to some of the most interesting questions in num
 ber theory. It is a remarkable fact that in order to study many arithmetic
  properties of elliptic curves\, it seems necessary to relate them to them
  objects much more analytic in nature called automorphic forms. For ellipt
 ic curves defined over the rationals\, a proof of many cases of this corre
 spondence was the subject of the famous work of Wiles that implied Fermat'
 s Last Theorem. The Langlands philosophy predicts that such a corresponden
 ce should exist for any finite field extension of the rationals\, but thin
 gs become more mysterious because a certain connection to algebraic geomet
 ry disappears. I will discuss some solutions to this problem over so-calle
 d CM fields\, such as the field of Gaussian numbers. Different parts of th
 is are joint work with Caraiani\, Calegari\, Gee\, Helm\, Khare\, Le Hung\
 , Newton\, Scholze\, Taylor\, and Thorne.\n
DTSTART:20181212T210000Z
DTEND:20181212T220000Z
LOCATION:Room 1104\, Burnside Hall\, CA\, QC\, Montreal\, H3A 0B9\, 805 rue
  Sherbrooke Ouest
SUMMARY:Patrick Allen - Urbana-Champaign
URL:/mathstat/channels/event/patrick-allen-urbana-cham
 paign-292381
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