Mathematical Research Letters

Volume 23 (2016)

Number 4

Eigenvarieties for classical groups and complex conjugations in Galois representations

Pages: 1167 – 1220

DOI: http://dx.doi.org/10.4310/MRL.2016.v23.n4.a10

Author

Olivier Taïbi (Department of Mathematics, Imperial College London, United Kingdom)

Abstract

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\mathrm{GL}_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\mathrm{GL}_{2n/F}$ whose multiplicative character is “odd”. We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are “many” points corresponding to (quasi-)irreducible Galois representations. Recent work of James Arthur describing the automorphic spectrum for these groups is used to define these Galois representations, and also to transfer self-dual automorphic representations of the general linear group to these classical groups.

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