Local-global compatibility for l=p, I

TL;DR

This paper proves the compatibility of local and global Langlands correspondences at places dividing l for certain automorphic representations over CM fields, under specific conditions on the representations.

Contribution

It establishes the local-global compatibility for l-adic Galois representations associated to regular algebraic conjugate self-dual automorphic representations over imaginary CM fields, assuming Iwahori-fixed vectors and Shin-regular weight.

Findings

Proves local-global compatibility at places dividing l.

Validates compatibility under Iwahori-fixed and Shin-regular conditions.

Advances understanding of Galois representations in the Langlands program.

Abstract

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GL_n over an imaginary CM field, under the assumption that the automorphic representations have Iwahori-fixed vectors at places dividing l and have Shin-regular weight.