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.

Comments welcome! In this version, the descent arguments of section 5 are reworked