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

TL;DR

This paper estimates the number of cuspidal automorphic representations of GL(n+1) arising from symmetric power lifts of GL(2) representations, contributing to the cuspidal cohomology, with fixed weight and varying level.

Contribution

It provides an asymptotic estimate for the count of such representations for general n, extending previous results for GL(3) and GL(4).

Findings

Established asymptotic estimates for the number of relevant automorphic representations.

Generalized previous results from GL(3) and GL(4) to higher n.

Connected symmetric power lifts to cuspidal cohomology contributions.

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 ${\mathrm {GL}}_4$.