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Automorphy and irreducibility of some l-adic representations

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  24 October 2014

Stefan Patrikis  and
Richard Taylor
Stefan Patrikis
Affiliation:
Department of Mathematics, MIT, Building E18, 77 Massachusetts Avenue, Cambridge, MA 02139, USA email patrikis@math.mit.edu
Richard Taylor
Affiliation:
School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA email rtaylor@ias.edu
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Abstract

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In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz’s theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if $F$ is a CM or totally real field and if ${\it\pi}$ is a polarizable, regular algebraic, cuspidal automorphic representation of $\text{GL}_{n}(\mathbb{A}_{F})$, then for a positive Dirichlet density set of rational primes $l$, the $l$-adic representations $r_{l,\imath }({\it\pi})$ associated to ${\it\pi}$ are irreducible.

MSC classification

Primary: 11F80: Galois representations 11R39: Langlands-Weil conjectures, nonabelian class field theory
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 2 , February 2015 , pp. 207 - 229
Copyright
© The Author(s) 2014 

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