Motivic Galois representations valued in Spin groups
Shiang Tang1
1 1409 West Green Street Urbana, IL 61801, United States
Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 197-221.

Soit m un entier tel que m≥7 et m≡0,1,7mod8. Nous construisons des systèmes strictement compatibles de repré- sentations l-adiques Γ ℚ →Spin m (ℚ ¯ l )→ spinGL N (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 .

Let m be an integer such that m≥7 and m≡0,1,7mod8. We construct strictly compatible systems of representations of Γ ℚ →Spin m (ℚ ¯ l )→ spinGL N (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 .

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1157
Classification : 11F80
Mots clés : Galois representations, spin groups, inverse Galois problem, automorphic representations
Shiang Tang 1

1 1409 West Green Street Urbana, IL 61801, United States
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Shiang Tang. Motivic Galois representations valued in Spin groups. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 1, pp. 197-221. doi : 10.5802/jtnb.1157. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1157/

[1] Jeffrey Adams; Xuhua He Lifting of elements of Weyl groups, J. Algebra, Volume 485 (2017), pp. 142-165 | DOI | MR | Zbl

[2] James Arthur The Endoscopic classification of representations orthogonal and symplectic groups, Colloquium Publications, 61, American Mathematical Society, 2013 | Zbl

[3] Thomas Barnet-Lamb; Toby Gee; David Geraghty; Richard Taylor Potential automorphy and change of weight, Ann. Math., Volume 179 (2014) no. 2, pp. 501-609 | DOI | MR | Zbl

[4] Rebecca Bellovin; Toby Gee G-valued local deformation rings and global lifts, Algebra Number Theory, Volume 13 (2019) no. 2, pp. 333-378 | DOI | MR | Zbl

[5] Armand Borel; Nolan Wallach Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Mathematical Surveys and Monographs, 67, American Mathematical Society, 2000 | MR

[6] Nicolas Bourbaki Lie groups and Lie algebras, Chapters 4-6, Springer, 2002 | Zbl

[7] George Boxer; Frank Calegari; Matthew Emerton; Brandon Levin; Keerthi Madapusi Pera; Stefan Patrikis Compatible systems of Galois representations associated to the exceptional group E 6 , Forum Math. Sigma, Volume 7 (2019), e4, 29 pages | MR | Zbl

[8] Kevin Buzzard; Toby Gee The conjectural connections between automorphic representations and Galois representations, Automorphic forms and Galois representations (Durham, 2011) (London Mathematical Society Lecture Note Series), Volume 414, London Mathematical Society, 2011, pp. 135-187 | Zbl

[9] Frank Calegari Even Galois representations and the Fontaine–Mazur conjecture. II, J. Am. Math. Soc., Volume 25 (2012) no. 2, pp. 533-554 | DOI | MR | Zbl

[10] Ana Caraiani Local–global compatibility and the action of monodromy on nearby cycles, Duke Math. J., Volume 161 (2012) no. 12, pp. 2311-2413 | MR | Zbl

[11] Laurent Clozel On limit multiplicities of discrete series representations in spaces of automorphic forms, Invent. Math., Volume 83 (1986) no. 2, pp. 265-284 | DOI | MR | Zbl

[12] Laurent Clozel Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and L-functions. Vol. I (Ann Arbor, 1988) (Perspectives in Mathematics), Volume 10, Academic Press Inc., 1990, pp. 77-159 | Zbl

[13] Pierre Deligne; James S. Milne; Arthur Ogus; Kuang-yen Shih Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, 900, Springer, 1982 | MR | Zbl

[14] Najmuddin Fakhruddin; Chandrashekhar Khare; Stefan Patrikis Relative deformation theory and lifting irreducible Galois representations (2019) (https://arxiv.org/abs/1904.02374)

[15] William Fulton; Joe Harris Representation theory: a first course, Graduate Texts in Mathematics, 129, Springer, 2013 | Zbl

[16] Chandrashekhar Khare; Jean-Pierre Wintenberger Serre’s modularity conjecture (I), Invent. Math., Volume 178 (2009) no. 3, pp. 485-504 | DOI | MR | Zbl

[17] Mark Kisin Potentially semi-stable deformation rings, J. Am. Math. Soc., Volume 28 (2008) no. 2, pp. 513-546 | MR | Zbl

[18] Arno Kret; Sug Woo Shin Galois representations for general symplectic groups (2016) (https://arxiv.org/abs/1609.04223, to appear in J. Eur. Math. Soc.)

[19] Arno Kret; Sug Woo Shin Galois representations for even general special orthogonal groups (2020) (https://arxiv.org/abs/2010.08408)

[20] Michael Larsen Maximality of Galois actions for compatible systems, Duke Math. J., Volume 80 (1995) no. 3, pp. 601-630 | MR | Zbl

[21] Michael Larsen; Richard Pink Determining representations from invariant dimensions, Invent. Math., Volume 102 (1990) no. 1, pp. 377-398 | DOI | MR | Zbl

[22] Michael Larsen; Richard Pink On ℓ–independence of algebraic monodromy groups in compatible systems of representations, Invent. Math., Volume 107 (1992) no. 3, pp. 603-636 | DOI | MR

[23] Jürgen Neukirch; Alexander Schmidt; Kay Wingberg Cohomology of number fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2013 | Zbl

[24] Stefan Patrikis Deformations of Galois representations and exceptional monodromy, Invent. Math., Volume 205 (2016) no. 2, pp. 269-336 | DOI | MR | Zbl

[25] Stefan Patrikis; Shiang Tang Potential automorphy of GSpin 2n+1 –valued Galois representations (2019) (https://arxiv.org/abs/1910.03164)

[26] Ravi Ramakrishna Deforming Galois representations and the conjectures of Serre and Fontaine–Mazur, Ann. Math., Volume 1566 (2002) no. 1, pp. 115-154 | DOI | MR | Zbl

[27] Jan Saxl; Gary M. Seitz Subgroups of algebraic groups containing regular unipotent elements, J. Lond. Math. Soc., Volume 55 (1997) no. 2, pp. 370-386 | DOI | MR | Zbl

[28] Jean-Pierre Serre Propriétés conjecturales des groupes de Galois motiviques et des représentations ℓ-adiques, Motives (Seattle, 1991) (Proceedings of Symposia in Pure Mathematics), Volume 55, American Mathematical Society, 1991, pp. 377-400

[29] Jean-Pierre Serre Sur la semi-simplicité des produits tensoriels de représentations de groupes, Invent. Math., Volume 116 (1994) no. 1, pp. 513-530 | DOI | Zbl

[30] Jean-Pierre Serre Topics in Galois theory, CRC Press, 2016

[31] Sug Woo Shin Galois representations arising from some compact Shimura varieties, Ann. Math., Volume 173 (2011) no. 3, pp. 1645-1741 | DOI | MR | Zbl

[32] Zhiwei Yun Motives with exceptional Galois groups and the inverse Galois problem, Invent. Math., Volume 196 (2014) no. 2, pp. 267-337 | MR | Zbl

[33] David Zywina The inverse Galois problem for orthogonal groups (2014) (https://arxiv.org/abs/1409.1151)

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