Artin representations for $G{L}_n$

Abstract

Let $\pi$ be a cuspidal automorphic representation of $G{L}_n({\mathbb{A}}_{\mathbb{Q}})$ which satisfies certain reasonable assumptions such as integrality of Hecke polynomials, the existence of mod $\ell$ Galois representations attached to $\pi$. Under Langlands functoriality of exterior $m$-th power ${\wedge }^m(\pi )$, $m=2,...,[\frac{n}{2}]$, we will construct a unique Artin representation associated to $\pi$. As a corollary, we obtain that such a cuspidal representation of $G{L}_n({\mathbb{A}}_{\mathbb{Q}})$ satisfies the Ramanujan conjecture. We also revisit our previous work on Artin representations associated to non-holomorphic Siegel cusp forms of weight (2,1), and show that we can associate non-holomorphic Siegel modular forms of weight $(2,1)$ to Maass forms for $G{L}_2/\mathbb{Q}$ and cuspidal representations of $G{L}_2$ over imaginary quadratic fields.