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SUMMARY:S_5 Galois extensions of totally real fields and automorphy - Shek
 har Khare (UCLA)
DTSTART:20161109T143000Z
DTEND:20161109T153000Z
UID:TALK67456@talks.cam.ac.uk
CONTACT:Jack Thorne
DESCRIPTION:We consider S_5 extensions of totally real fields F that are t
 otally odd. These arise as splitting fields of quintic polynomials over F 
 not all of whose roots are real.\n\nNoting the isomorphism S_5=PGL_2(F_5)\
 , one can ask if these arise as splitting fields of the 5-torsion of an el
 liptic curve defined over F\, or more generally from the 5-torsion of an a
 belian variety defined over F with real multiplication. One can also ask i
 f such S_5 extensions arise from Hilbert modular forms. The case when the 
 images of complex conjugations are even permutations (so conjugate to (12)
 (34)) is understood\, while the case of odd permutations is still open.\n\
 nThe case of S_5 extensions is also interesting from the point of view of 
 automorphy lifting results of Wiles\, Taylor-Wiles et al as when the fixed
  field of PSL_2(F_5) is given by F(zeta_5) this falls in a blind spot of t
 he Taylor-Wiles patching method. We will describe joint work with Jack Tho
 rne which combines patching with an argument using p-adic approximations t
 o overcome this blind spot.
LOCATION:MR5
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