Modularity of hypertetrahedral representations

Abstract

Let F be a number field, G F its absolute Galois group, and ρ : G F → GL 4 ( C ) an irreducible continuous Galois representation. Let G ¯ denote the projective image of ρ in PGL 4 ( C ) . We say that ρ is hypertetrahedral if 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. To cite this article: K. Martin, C. R. Acad. Sci. Paris, Ser. I 339 (2004).