Geodesic
Local-global compatibility for regular algebraic cuspidal automorphic representations when $\ell \neq p$
Forum of Mathematics, Sigma, Tome 12 (2024)

Voir la notice de l'article provenant de la source Cambridge University Press

We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$. We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$, the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$. 
@article{10_1017_fms_2024_7,
     author = {Ila Varma},
     title = {Local-global compatibility for regular algebraic cuspidal automorphic representations when $\ell \neq p$},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {12},
     year = {2024},
     doi = {10.1017/fms.2024.7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2024.7/}
}
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Ila Varma. Local-global compatibility for regular algebraic cuspidal automorphic representations when $\ell \neq p$. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.7

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