Hypothesis H and the prime number theorem for automorphic representations

Abstract

For any unitary cuspidal representations $\pi_n$ of $GL_n(\mathbb{Q}_\mathbb{A})$, $n=2,3,4$, respectively, consider two automorphic representations $\Pi$ and $\Pi'$ of $GL_6(\mathbb{Q}_\mathbb{A})$, where $\Pi_p\cong\wedge^2\pi_{4,p}$ for $p\neq 2,3$ and $\pi_{4,p}$ not supercuspidal ($\pi_{4, p}$ denotes the local component of $\pi_4$), and $\Pi'=\pi_2\boxtimes\pi_3$. First, Hypothesis H for $\Pi$ and $\Pi'$ is proved. Then contributions from prime powers are removed from the prime number theorem for cuspidal representations $\pi$ and $\pi'$ of $GL_m(\mathbb{Q}_\mathbb{A})$ and $GL_{m'}(\mathbb{Q}_\mathbb{A})$, respectively. The resulting prime number theorem is unconditional when $m,m'\leq 4$ and is under Hypothesis H otherwise.