On the growth of cuspidal cohomology of ${\rm GL}_4$

TL;DR

This paper provides an asymptotic estimate for the number of cuspidal automorphic representations of GL_4 that arise from symmetric cube transfer of GL_2 representations, extending previous work on GL_3.

Contribution

It establishes a new asymptotic estimate for GL_4 cuspidal cohomology contributions from symmetric cube transfers, generalizing prior results for GL_3.

Findings

Derived an asymptotic estimate for the growth of certain automorphic representations.

Extended previous results from GL_3 to GL_4.

Analyzed automorphic representations with varying level structures.

Abstract

In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of ${\rm GL}_4(\mathbb A_{\mathbb Q})$ which contribute to the cuspidal cohomology of ${\rm GL}_4$ and are obtained from symmetric cube transfer of automorphic representations of ${\rm GL}_2(\mathbb A_{\mathbb Q})$ of a given weight and with varying level structure. This generalises the recent work of C. Ambi [2020] about the similar problem for ${\rm GL}_3$.