Cours d'Arithmétique et de Géométrie Algébrique en vidéo
Les jeudis 4, 11, 18 et 25 Octobre 2012 de 14h30 à 16h30
Payman Kassaei (King's College London)
The seminal work of Buzzard and Taylor (Companion forms and weight one forms, Annals of Mathematics 149 (1999), 905-919) presented a powerful application of analytic continuation of overconvergent modular forms to number theory. In essence, such an application would begin with producing overconvergent automorphic forms using p-adic analytic methods, interpreting them geometrically as sections of sheaves over p-adic analytic regions in Shimura varieties, and analytically extending them over the entire Shimura variety, proving they are classical automorphic forms, and, thereby, making possible applications to number theroy. Since Buzzard and Taylor, much progress has been made concerning analytic continuation of overconvergent automorphic forms. In this course, I will begin with a survey of the classical results, and, then, focus on the more recent results on analytic continuation and classicality of over-convergent Hilbert modular forms, as well as applications, obtained by various people.