Contribution of symmetric power transfers to the cuspidal cohomology of ${\rm GL_n}$

Abstract

Let $\pi$ be a cuspidal automorphic representation of ${\mathrm {GL}}_2(\mathbb{A}_\mathbb{Q})$. Newton and Thorne have proved that for every $n\geq 1$, the symmetric power lifting ${{\mathrm {sym}}^n(\pi)}$ is automorphic if $\pi$ is attached to a non-CM Hecke eigenform. In this article, we establish an asymptotic estimate of the number of cuspidal automorphic representations of ${\mathrm {GL}}_{n+1}(\mathbb{A}_\mathbb{Q})$ which contribute to the cuspidal cohomology of ${\mathrm {GL}}_{n+1}$ and are obtained by symmetric $n$th transfer of cuspidal representations of ${\mathrm {GL}}_2(\mathbb{A}_\mathbb{Q})$. Here we fix the weight and vary the level. This generalises the previous works done for ${\mathrm {GL}}_3$ and ${\mathrm {GL}}_4$.