Potential automorphy of $\mathrm{GSpin}_{2n+1}$-valued Galois representations

Abstract: We prove a potentially automorphy theorem for suitable Galois representations $\Gamma_{F^+} \to \mathrm{GSpin}{2n+1}(\overline{\mathbb{F}}_p)$ and $\Gamma{F^+} \to \mathrm{GSpin}{2n+1}(\overline{\mathbb{Q}}_p)$, where $\Gamma{F^+}$ is the absolute Galois group of a totally real field $F^+$. We also prove results on solvable descent for $\mathrm{GSp}{2n}(\mathbb{A}{F^+})$ and use these to put representations $\Gamma_{F^+} \to \mathrm{GSpin}{2n+1}(\overline{\mathbb{Q}}_p)$ into compatible systems of $\mathrm{GSpin}{2n+1}(\overline{\mathbb{Q}}_{\ell})$-valued representations.