On the Rigid Cohomology of Certain Shimura Varieties

TL;DR

This paper constructs compatible systems of l-adic representations for automorphic representations of GL_n over CM or totally real fields, establishing local-global compatibility without requiring self-duality.

Contribution

It introduces a method to build l-adic representations for automorphic forms without the self-duality assumption, expanding the scope of known compatibility results.

Findings

Constructed compatible l-adic systems for a broad class of automorphic representations.

Verified local-global compatibility away from l and ramified primes.

Extended the theory to cases without self-duality constraints.

Abstract

We construct the compatible system of $l$-adic representations associated to a regular algebraic cuspidal automorphic representation of $GL_n$ over a CM (or totally real) field and check local-global compatibility for the $l$-adic representation away from $l$ and finite number of rational primes above which the CM field or the automorphic representation ramify. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.