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
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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     author = {Kimball Martin},
     title = {Modularity of hypertetrahedral representations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {99--102},
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     number = {2},
     year = {2004},
     doi = {10.1016/j.crma.2004.05.003},
     language = {en},
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.003/

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