Motivic Galois representations valued in Spin groups

TL;DR

This paper constructs compatible Galois representations valued in Spin groups that are potentially automorphic and motivic, and applies these to the inverse Galois problem, extending understanding of automorphic Galois representations in this context.

Contribution

It introduces new constructions of motivic Galois representations in Spin groups for specific dimensions, linking them to automorphic forms and solving cases of the inverse Galois problem.

Findings

Constructed compatible systems of Spin-valued Galois representations.

Proved instances of the inverse Galois problem for Spin groups.

Compared new examples with existing automorphic Galois representations.

Abstract

Let $m$ be an integer such that $m \geq 7$ and $m \equiv 0,1,7 \mod 8$. We construct strictly compatible systems of representations of $\Gamma_{\mathbb Q} \to \mathrm{Spin}_m(\overline{\mathbb Q}_l) \xrightarrow{\mathrm{spin}} \mathrm{GL}_N(\overline{\mathbb Q}_l)$ 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 $\mathrm{Spin}_m$.