Automorphic tensor products and cuspidal cohomology of the ${\rm GL}_4$

Abstract: In this article, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\rm GL}4(\mathbb A{\mathbb Q})$ to cuspidal cohomology of the ${\rm GL}4$ which are obtained from automorphic tensor product of two automorphic representations of ${\rm GL}_2(\mathbb A{\mathbb Q})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\rm GL}_2$ and the automorphic tensor product of two representations of ${\rm GL}_2$ can not be equal (up to a twist by a character of ${\rm GL}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.