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SUMMARY:Analytic continuation of Hilbert modular forms and applications to
  modularity  - Payman Kassaei (King's College London)
DTSTART:20130507T151500Z
DTEND:20130507T161500Z
UID:TALK44977@talks.cam.ac.uk
CONTACT:Teruyoshi Yoshida
DESCRIPTION:In his foundational work on the theory of p-adic modular forms
 \, N. Katz observed that there is a positive lower bound for the "growth c
 ondition" of an overconvegent p-adic modular eigenform with nonzero Up-eig
 envalue. In more modern language\, this states that any such form can be a
 nalytically continued from its initial domain of definition to a not ”to
 o small” region of the rigid analytic modular curve. Years later\, K. Bu
 zzard\, by considering    these forms in their true level (i.e.\, level di
 visible by p) proved that such forms can be further extended to a certain 
 "large" region of the modular curve. These results were used by Buzzard an
 d Taylor to prove modularity lifting results which led to a proof of certa
 in cases of the Strong Artin conjecture.\n\nIt has been known for a while 
 how to extend these results to the Hilbert case when p is split in the tot
 ally real field of degree g > 1\, as the problem looks formally like a pro
 duct of g copies of the modular curve case. In the inert case\, however\, 
 a mixing happens that fundamentally changes the nature of the problem. In 
 this talk\, I will explain new results on domains of automatic analytic co
 ntinuation for overconvergent Hilbert modular forms in the case p is unram
 ified in the totally real field. These results can be used to prove many c
 ases of the strong Artin conjecture for Hilbert modular forms. Some of the
  work that will be presented is joint with Sasaki and Tian. 
LOCATION:MR13
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