Motivic Galois representations valued in Spin groups

Tang, Shiang

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 197-221.

Soit $m$ un entier tel que $m\ge 7$ et $m\equiv 0,1,7mod8$. Nous construisons des systèmes strictement compatibles de repré- sentations $l$-adiques ${\Gamma }_{\mathbb{Q}}\to {\mathrm{Spin}}_m\left({\overline{\mathbb{Q}}}_l\right)\stackrel{\mathrm{spin}}{\to }{\mathrm{GL}}_N\left({\overline{\mathbb{Q}}}_l\right)$ qui sont potentiellement automorphes et motiviques. Comme application, dans certains cas nous donnons une réponse positive au problème de Galois inverse pour les groupes spinoriels sur ${𝔽}_p$. Pour $m$ impair, nous comparons nos exemples avec le travail de A. Kret et S. W. Shin ([18]), qui étudie les représentations galoisiennes automorphes à valeurs dans ${\mathrm{GSpin}}_m$.

Let $m$ be an integer such that $m\ge 7$ and $m\equiv 0,1,7mod8$. We construct strictly compatible systems of representations of ${\Gamma }_{\mathbb{Q}}\to {\mathrm{Spin}}_m\left({\overline{\mathbb{Q}}}_l\right)\stackrel{\mathrm{spin}}{\to }{\mathrm{GL}}_N\left({\overline{\mathbb{Q}}}_l\right)$ that are potentially automorphic and motivic. As an application, we prove instances of the inverse Galois problem for the ${𝔽}_p$–points of the spin groups. For odd $m$, we compare our examples with the work of A. Kret and S. W. Shin ([18]), which studies automorphic Galois representations valued in ${\mathrm{GSpin}}_m$.

Classification : 11F80
Mots clés : Galois representations, spin groups, inverse Galois problem, automorphic representations

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