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Compatibility of local and global Langlands correspondences

Richard Taylor·Teruyoshi Yoshida
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Abstract

We prove the compatibility of local and global Langlands correspondences for GL_n, which was proved up to semisimplification by Harris-Taylor. More precisely, for the n-dimensional l-adic representation R_l(Π) of the Galois group of a CM-field L attached to a conjugate self-dual regular algebraic cuspidal automorphic representation Π, which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of R_l(Π) to the decomposition group of a prime v of L not dividing l corresponds to Π_v by the local Langlands correspondence. As a corollary, we show the irreducibility of R_l(Π) when Π_v is square integrable for some finite place v outside l. We also obtain conditional results in the case v|l.

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Richard Taylor, Teruyoshi Yoshida. 2005-04-07. Compatibility of local and global Langlands correspondences. https://arxiv.org/abs/math/0412357

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