Modularity of hypertetrahedral representations

Kimball Martin

doi:10.1016/j.crma.2004.05.003

Group Theory/Number Theory

Modularity of hypertetrahedral representations

Presented by: Hervé Jacquet

Kimball Martin ¹

¹ Caltech 253-37, Pasadena, CA 91125, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 99-102

Abstract (Original language)
Résumé

Let F be a number field, $G_{F}$ its absolute Galois group, and $ρ : G_{F} \to {GL}_{4} (C)$ an irreducible continuous Galois representation. Let $\bar{G}$ denote the projective image of ρ in ${PGL}_{4} (C)$ . We say that ρ is hypertetrahedral if $\bar{G}$ is an extension of $A_{4}$ by the Klein group $V_{4}$ . In this case, we show that ρ is modular, i.e., ρ corresponds to an automorphic representation π of ${GL}_{4} (A_{F})$ such that their L-functions are equal. This gives new examples of irreducible 4-dimensional monomial representations which are modular, but are not induced from normal extensions and are not essentially self-dual.

Soient F un corps de nombres, $G_{F} = Gal (\bar{F} / F)$ et $ρ : G_{F} \to {GL}_{4} (C)$ une représentation irréductible et continue. Soit $\bar{G}$ l'image projective ρ. Nous appellerons une telle représentation hypertétraèdrale si $\bar{G}$ est une extension de $A_{4}$ par le groupe de Klein $V_{4}$ . Nous démontrons qu'une représentation hypertétraèdrale est modulaire, i.e., il existe une représentation cuspidale π de ${GL}_{4} (A_{F})$ tel que $L (s, ρ) = L (s, π)$ . Ceci donne de nouveaux exemples de représentations modulaires qui ne sont pas induites par des extensions normales et ne sont pas essentiellement auto-duales.

Received: 2004-01-15
Accepted: 2004-05-11
Published online: 2004-07-15

DOI: 10.1016/j.crma.2004.05.003

Author's affiliations:

Kimball Martin ¹

¹ Caltech 253-37, Pasadena, CA 91125, USA

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     title = {Modularity of hypertetrahedral representations},
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Kimball Martin. Modularity of hypertetrahedral representations. Comptes Rendus. Mathématique, Volume 339 (2004) no. 2, pp. 99-102. doi: 10.1016/j.crma.2004.05.003

References
Cited by

[1] J. Arthur; L. Clozel Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Ann. of Math. Stud., vol. 120, Princeton University Press, 1999

[2] S. Gelbart; H. Jacquet A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup., Volume 11 (1979), pp. 471-542

[3] H. Kim Functoriality for the exterior square of ${GL}_{4}$ and the symmetric fourth of ${GL}_{2}$ , J. Amer. Math. Soc., Volume 16 (2003), pp. 139-183

[4] H. Kim; F. Shahidi Functorial products for ${GL}_{2} \times {GL}_{3}$ and functorial symmetric cube for ${GL}_{2}$ , C. R. Acad. Sci. Paris Sér. I Math., Volume 331 (2000), pp. 599-604

[5] R.P. Langlands Base Change for GL(2), Ann. of Math. Stud., vol. 96, Princeton University Press, 1980

[6] K. Martin A symplectic case of Artin's conjecture, Math. Res. Let., Volume 10 (2003), pp. 483-492

[7] J. Neukirch; A. Schmidt; K. Wingberg Cohomology of Number Fields, Springer-Verlag, 2000

[8] D. Ramakrishnan Modularity of solvable Artin representations of GO(4)-type, Int. Math. Res. Not. (2002), pp. 1-54

[9] J. Tunnell Artin's conjecture for representation of octahedral type, Bull. Amer. Math. Soc., Volume 5 (1981), pp. 173-175

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