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

Abstract

In this thesis, we estimate the contribution of symmetric cube transfer and tensor product transfer to the cuspidal cohomology of ${\rm{GL}_4}$. Let $\mathbb{E}=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic extension of $\mathbb{Q}$. Let $\chi$ be a Hecke character of the group of ideles over $\mathbb{E}$. Using Langlands functoriality, it gives an automorphic cusp form of ${\rm{GL}}_2(\mathbb{A}_\mathbb{Q})$ by automorphic induction. Consider a cuspidal automorphic representation of ${\rm{GL}}_2(\mathbb{A}_\mathbb{Q})$. Due to Kim and Shahidi, symmetric cube of this representation gives an automorphic representation of ${\rm{GL}}_4(\mathbb{A}_\mathbb{Q})$ which is further cuspidal if the representation is not dihedral, that is, if the representation of ${\rm{GL}}_2(\mathbb{A}_\mathbb{Q})$ is not obtained by automorphic induction. We provide an estimate of the number of cuspidal automorphic representations of ${\rm{GL}}_4(\mathbb{A}_\mathbb{Q})$ obtained from ${\rm{GL}}_2(\mathbb{A}_\mathbb{Q})$ by symmetric cube transfer corresponding to a specific level structure.

Similarly, we consider another transfer called tensor product transfer or automorphic tensor product. If we start with two cuspidal representations $\pi_1$ and $\pi_2$ of ${\rm{GL}}_2$, then the automorphic tensor product $\pi_1 \boxtimes \pi_2$ gives a representation of ${\rm{GL}}_4$. Using the cuspidality criterion by Dinakar Ramakrishnan, we estimate the cuspidal cohomology of ${\rm{GL}}_4(\mathbb{A}_{\mathbb{Q}})$ obtained from ${\rm{GL}_2}\times {\rm{GL}_2}$ by tensor product transfer.

We have also shown that there is no overlap between these two procedures, i.e., there does not exist any cuspidal representation of ${\rm{GL}}_4({\mathbb{A}_\mathbb{Q}})$ which is obtained at the same time as the symmetric cube of a representation of ${\rm{GL}}_2({\mathbb{A}_\mathbb{Q}})$ and as the automorphic tensor product of two representations of ${\rm{GL}}_2({\mathbb{A}_\mathbb{Q}})$.