Automorphy of some residually S$_5$ Galois representations

Abstract

Let $\textit{F}$ be a totally real field and $\textit{p}$ an odd prime. We prove an automorphy lifting theorem for geometric representations $\rho$ : $\textit{G}_F$ → GL$_2$($\bar{\Bbb Q}_p$) which lift irreducible residual representations $\bar \rho$ that arise from Hilbert modular forms. The new result is that we allow the case $\textit{p}$ = 5, $\bar \rho$ has projective image S$_5$ $\cong$ PGL$_2$($\Bbb F_5$) and the fixed field of the kernel of the projective representation contains ζ$_5$. The usual Taylor–Wiles method does not work in this case as there are elements of dual Selmer that cannot be killed by allowing ramification at Taylor–Wiles primes. These elements arise from our hypothesis and the non-vanishing of H$^1$ (PGL$_2$($\Bbb F_5$), Ad(1)) where Ad(1) is the adjoint of the natural representation of GL$_2$($\Bbb F_5$) twisted by the quadratic character of PGL$_2$($\Bbb F_5$).