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

We prove a potential automorphy theorem for suitable Galois representations $$\Gamma _{F^+} \rightarrow \mathrm {GSpin}_{2n+1}(\overline{\mathbb {F}}_p)$$ and $$\Gamma _{F^+} \rightarrow \mathrm {GSpin}_{2n+1}({\overline{{\mathbb {Q}}}}_p)$$ , 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 $$\Gamma _{F^+} \rightarrow \mathrm {GSpin}_{2n+1}({\overline{{\mathbb {Q}}}}_p)$$ into compatible systems of $$\mathrm {GSpin}_{2n+1}({\overline{{\mathbb {Q}}}}_l)$$ -valued representations.