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

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 ${\rm GL}_3$.