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CATEGORIES:Number Theory Seminar
SUMMARY:S_5 Galois extensions of totally real fields and a
utomorphy - Shekhar Khare (UCLA)
DTSTART;TZID=Europe/London:20161109T143000
DTEND;TZID=Europe/London:20161109T153000
UID:TALK67456AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/67456
DESCRIPTION:We consider S_5 extensions of totally real fields
F that are totally odd. These arise as splitting f
ields of quintic polynomials over F not all of who
se roots are real.\n\nNoting the isomorphism S_5=P
GL_2(F_5)\, one can ask if these arise as splittin
g fields of the 5-torsion of an elliptic curve def
ined over F\, or more generally from the 5-torsion
of an abelian variety defined over F with real mu
ltiplication. One can also ask if such S_5 extensi
ons arise from Hilbert modular forms. The case whe
n the images of complex conjugations are even perm
utations (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 interestin
g from the point of view of automorphy lifting res
ults of Wiles\, Taylor-Wiles et al as when the fix
ed field of PSL_2(F_5) is given by F(zeta_5) this
falls in a blind spot of the Taylor-Wiles patching
method. We will describe joint work with Jack Tho
rne which combines patching with an argument using
p-adic approximations to overcome this blind spot
.
LOCATION:MR5
CONTACT:Jack Thorne
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