We prove the compatibility at places dividing l of the local and global Langlands correspondences for the l-adic Galois representations associated to regular algebraic essentially (conjugate) self-dual cuspidal automorphic representations of GL_n over an imaginary CM or totally real field. We prove this compatibility up to semisimplification in all cases, and up to Frobenius semisimplification in the case of Shin-regular weight.
We generalize the local-global compatibility result in arXiv:1506.04022 to higher dimensional cases, by examining the relation between Scholze's functor and cohomology of Kottwitz-Harris-Taylor type Shimura varieties. Along the way we prove a cuspidality criterion from type theory. We also deal with compatibility for torsion classes in the case of semisimple mod $p$ Galois representations with distinct irreducible components under certain flatness hypotheses.
Compatibility of local and global Langlands correspondences
本文证明了GL_n的局部和整体Langlands对应的相容性，Harris-Taylor半简化地证明了这一点。更精确地说，对于CM域L的伽罗瓦群的n维l-adic表示R_l（R_l），它与共轭自对偶正则代数尖形自守表示R_l（R_l）相连，R_l（R_l）在有限处平方可积〔的〕，我们证明了将R_l（R_l）限制到L的素数v的不整除l的分解群的Frobenius半简化对应于局部Langlands对应.作为推论，我们证明了当n_v对l外的某个有限位置v是平方可积〔的〕时，R_l（n）的不可约性.我们也得到了条件结果的情况下v| L.
We remove the semisimple condition in the mod-$p$ local-global compatibility result of arXiv:2106.10674. Namely, assuming flatness of $π_{\mathfrak{m}}^\vee$ and $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ being multiplicity free, we prove that $H^{n-1}_{\text{ét}}(\mathbb{P}_{\mathbb{C}_p}^{n-1}, \mathcal{F}_{π_{\mathfrak{m}}[\mathfrak{m}]})$ determines $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ uniquely. We give remarks on our assumptions at the end of this note.
Sur la compatibilite entre les correspondances de Langlands locale et globale pour U(3). (On the compatibility between local and global Langlands correspondances for U(3))
On mod $p$ local-global compatibility for $\mathrm{GL}_n(\mathbf{Q}_p)$ in the ordinary case
令$p$为素数，$n>2$为整数，$F$为其中$p$完全分裂的CM字段。假设连续的自同构伽罗瓦表示$\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$是上三角的，并且在$p$上的一个位置$w$处满足特定的泛型条件，并且$>2$的$\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$的每个子商都是Fontaine-Laffaille泛型的。本文证明了$\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$的同构类是由$\mathrm{GL}_n(F_w)$ -作用在模$p$代数自守形式空间上确定的，该模$p$代数自守形式空间是由与$\overline{r}$相关联的Hecke代数的极大理想切割出来的，并假设一个权消去结果，这是Bao V. Le Hung在他即将发表的论文~ \cite{LeH}中的一个定理.特别是，我们表明，广泛分歧的$\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$的一部分是由雅可比和运营商（视为元素的$\mathbf{F}_…
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties of the refined local Langlands correspondence.
证明了Lubin-Tate塔的p-adic上同调的一个有限性结果。对于任意n>=1且p-adic域F，这给出了从GL_n（F）的容许p-adic表示到Gal_F x D^* 的容许p-adic表示的一个典范函子，其中Gal_F是F的绝对伽罗瓦群，D/F是不变量1/n的中心除代数.此外，我们验证了这种对应关系的局部-全局-兼容性声明，以及与Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin的修补构造的兼容性。
