Abstract

We consider S_5 extensions of totally real fields F that are totally odd. These arise as splitting fields of quintic polynomials over F not all of whose roots are real.

Noting the isomorphism S_5=PGL_2(F_5), one can ask if these arise as splitting fields of the 5-torsion of an elliptic curve defined over F, or more generally from the 5-torsion of an abelian variety defined over F with real multiplication. One can also ask if 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.

The 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 the Taylor-Wiles patching method. We will describe joint work with Jack Thorne which combines patching with an argument using p-adic approximations to overcome this blind spot.