Motivic Galois representations valued in Spin groups

Abstract

Let $m$ be an integer such that $m\ge 7$ and $m\equiv 0,1,7\mathrm{mod}8$. We construct strictly compatible systems of representations of ${\Gamma }_{\mathbb{Q}}\to {\mathrm{Spin}}_m({\overline{\mathbb{Q}}}_l)\stackrel{\mathrm{spin}}{\to }{\mathrm{GL}}_N({\overline{\mathbb{Q}}}_l)$ that is potentially automorphic and motivic. As an application, we prove instances of the inverse Galois problem for the ${\mathbb{F}}_p$--points of the spin groups. For odd $m$, we compare our examples with the work of A. Kret and S. W. Shin, which studies automorphic Galois representations valued in ${\mathrm{Spin}}_m$.

21 pages. Comments are welcome!