Motivic Galois representations valued in Spin groups

Shiang Tang

doi:10.5802/jtnb.1157

Shiang Tang ¹

¹ 1409 West Green Street Urbana, IL 61801, United States

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

Abstract
Résumé

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 $Γ_{ℚ} \to {Spin}_{m} ({\bar{ℚ}}_{l}) \overset{spin}{\to} {GL}_{N} ({\bar{ℚ}}_{l})$ 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 ${GSpin}_{m}$ .

Soit $m$ un entier tel que $m \geq 7$ et $m \equiv 0, 1, 7 mod 8$ . Nous construisons des systèmes strictement compatibles de repré- sentations $l$ -adiques $Γ_{ℚ} \to {Spin}_{m} ({\bar{ℚ}}_{l}) \overset{spin}{\to} {GL}_{N} ({\bar{ℚ}}_{l})$ 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 ${GSpin}_{m}$ .

Received: 2020-06-12
Accepted: 2020-12-13
Published online: 2021-05-21

DOI: 10.5802/jtnb.1157

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

Author's affiliations:

Shiang Tang ¹

¹ 1409 West Green Street Urbana, IL 61801, United States

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Shiang Tang},
     title = {Motivic {Galois} representations valued in {Spin} groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {197--221},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
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     doi = {10.5802/jtnb.1157},
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Shiang Tang. Motivic Galois representations valued in Spin groups. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 197-221. doi : 10.5802/jtnb.1157. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1157/

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