# Automorphic tensor products and cuspidal cohomology of the ${\rm GL}_{4}$

Chandrasheel Bhagwat & Sudipa Mondal Indian Institute of Science Education and
Research, Dr. Homi Bhabha Road, Pashan, Pune 411008, INDIA.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### 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.

###### 2020 Mathematics Subject Classification:

11F41, 11F75

##  1\. Introduction

For a connected reductive algebraic group over a number field ${\mathbb{F}}$,
one studies the notion of cohomological representations of
$G({\mathbb{A}}_{\mathbb{F}})$ in terms of the sheaves on the associated
adélic symmetric spaces, defined by the highest weight representations of $G$.

It is interesting to ask how much of the cohomology of $G$ is captured by the
automorphic representations of $G({\mathbb{A}}_{\mathbb{F}})$. In the case of
${\rm GL}_{n}$, such representations are often obtained by functorial transfer
from products of lower rank ${\rm GL}_{n^{\prime}}$ subgroups. In [4], authors
of this manuscript proved an estimate of the contribution of symmetric cube
transfer from ${\rm GL}_{2}$ to the cohomology of ${\rm GL}_{4}$.

The other possible functorial transfers for ${\rm
GL}_{4}({\mathbb{A}}_{\mathbb{Q}})$ are automorphic tensor product from ${\rm
GL}_{2}\times{\rm GL}_{2}$ to ${\rm GL}_{4}$ and Asai transfer from ${{\rm
GL}_{2}}|_{{\mathbb{K}}}$ for some imaginary quadratic extension
${\mathbb{K}}$ over ${\mathbb{Q}}$. In this manuscript, we prove the following
estimate on the contribution of automorphic tensor products from ${\rm
GL}_{2}\times{\rm GL}_{2}$ to cohomology of the ${\rm GL}_{4}$.

Let $k_{1}>k_{2}\geq 2$ be two positive integers. Suppose they are of
different parity. Without loss of generality, assume $k_{1}$ to be odd and
$k_{2}$ even. Define $\mathcal{T}_{k_{1},k_{2}}(N)$ to be the set of all
cuspidal automorphic representations of ${\rm
GL}_{4}(\mathbb{A}_{{\mathbb{Q}}})$ corresponding to the level structure
$K_{f}^{4}(N)$ that are obtained by automorphic tensor product of two cuspidal
automorphic representations of ${\rm GL}_{2}(\mathbb{A}_{{\mathbb{Q}}})$ with
highest weights $\left(\dfrac{k_{1}-1}{2},~{}\dfrac{1-k_{1}}{2}\right)$ and
$\left(\dfrac{k_{2}}{2}-1,~{}1-\dfrac{k_{2}}{2}\right)$, respectively. (If
$k_{1},~{}k_{2}$ are of same parity, we consider the half-integral twist of
the automorphic tensor product of two representations $\pi_{1},\pi_{2}$ of
${\rm GL}_{2}(\mathbb{A}_{{\mathbb{Q}}})$ with suitable highest weight
depending on $k_{1},k_{2}$ to be even or odd). See section 5 for the
notations.

######  Theorem 1.0.1.

Let $k_{1}>k_{2}\geq 2$ be two integers. Then for any integer $N>24$,

$|\mathcal{T}_{k_{1},~{}k_{2}}(p^{N})|\gg_{k_{1},~{}k_{2}}~{}p^{2\lfloor
N/4\rfloor}\text{ as the prime }p\longrightarrow\infty.$

We briefly describe the structure of this manuscript. After setting up
preliminaries about cusp forms in Sec. 2, we discuss the automorphic tensor
product representations in Sec 3 including a calculation on the conductor of
these representations. In Sec. 4, we discuss briefly about the cohomological
representations of ${\rm GL}_{4}({\mathbb{A}}_{\mathbb{Q}})$ and examples of
cohomological automorphic tensor product representations. In Sec. 5, we give
the proof of Thm. 1.0.1 using the construction and criteria for cuspidality of
automorphic tensor product for ${\rm GL}_{2}\times{\rm GL}_{2}$ proved by D.
Ramakrishnan in [13] and [14]. In Sec. 6, we give a proof of the fact 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. This
result may be possibly known to the experts and may be unsurprising. However,
we add the details of the calculations and proofs for the purpose of
completion.

##  2\. Preliminaries

###  2.1. Dimension formulae for spaces of cusp forms

We discuss some well-known and relevant results in this section. Let $N$ be an
integer. Consider the congruence subgroup $\Gamma_{1}(N)$ of
$\text{SL}_{2}(\mathbb{Z})$ defined by

$\displaystyle\Gamma_{1}(N)$
$\displaystyle=\left\\{A\in\text{SL}_{2}(\mathbb{Z}):A\equiv\begin{bmatrix}1&*\\\
0&1\end{bmatrix}(\text{mod }N)\right\\}.$

For an integer $k$, let $S_{k}(\Gamma_{1}(N))$ and
$S_{k}^{\text{new}}(\Gamma_{1}(N))$ denote the spaces of cusp forms of weight
$k$ and newforms of weight $k$, respectively for the congruence subgroup
$\Gamma_{1}(N)$. The following well-known results will be important in the
proof of the main theorem (see [15] for the details and [1] for the
formulation of these results as given below).

######  Lemma 2.1.1.

An estimate for the dimension of the space of cusp forms:

$\frac{\rm{dim}_{\mathbb{C}}(S_{k}(\Gamma_{1}(N)))}{N^{2}}=\frac{k-1}{4\pi^{2}}+o(1)\text{
as }N\longrightarrow\infty.$

######  Lemma 2.1.2.

A dimension formula for newforms for $\Gamma_{1}(N)$: For $n\geq 2$ and a
prime $p\geq 2$, we have

$\frac{{\rm
dim}_{\mathbb{C}}~{}(S_{k}^{\text{new}}(\Gamma_{1}(p^{n})))}{p^{2n}}=\left(\frac{k-1}{4\pi^{2}}\right)\left(1+\frac{1}{p^{2}}-\frac{2}{p^{4}}\right)+o(1)\quad\text{
as }~{}p\longrightarrow\infty.$

###### Proof.

Let $M\in\mathbb{N}$. For a prime $p$, let $v_{p}(M)$ denote the highest power
of $p$ that divides $M$. For $M\in{\mathbb{N}}$, let $a_{M}$ be the the number
of prime factors of $M$ such that $v_{p}(M)=2$. We define a function
$\bar{\mu}$ on ${\mathbb{N}}$ by,

$\bar{\mu}(M)=\begin{cases}0&\text{ if }v_{p}(M)\geq 3\text{ for some prime
}p\\\ (-2)^{a_{M}}&\text{ otherwise }\\\ \end{cases}.$

From ([15, Prop. 6.6]), we know that

${\rm
dim}_{\mathbb{C}}(S_{k}^{\text{new}}(\Gamma_{1}(M)))=\sum_{d|M}\bar{\mu}(M/d)\cdot{\rm
dim}_{\mathbb{C}}(S_{k}(\Gamma_{1}(d))).$

For $M=p^{n},n\geq 2$, there are at most three terms such that
$\bar{\mu}(M/d)\neq 0$ in the above sum. Adding these three terms gives

$\displaystyle{\rm dim}_{\mathbb{C}}(S_{k}^{\text{new}}(\Gamma_{1}(M)))={\rm
dim}_{\mathbb{C}}(S_{k}(\Gamma_{1}(p^{n})))+{\rm
dim}_{\mathbb{C}}(S_{k}(\Gamma_{1}(p^{n-1})))-2\text{
dim}_{\mathbb{C}}(S_{k}(\Gamma_{1}(p^{n-2}))).$

Using Lem. 2.1.1 and fixing $n$ while varying $p$, we get

$\displaystyle\frac{{\rm
dim}_{\mathbb{C}}(S_{k}^{\text{new}}(\Gamma_{1}(M)))}{p^{2n}}=\frac{k-1}{4\pi^{2}}\left(1+\frac{1}{p^{2}}-\frac{2}{p^{4}}\right)+o(1)\text{
as }p\longrightarrow\infty.$

∎

###  2.2. An upper bound for normalised eigenforms

Let $C_{k}(N)$ denote the set of normalised cusp eigenforms of Hecke operators
of $\Gamma_{1}(N)$ of weight $k$ obtained by automorphic induction of
größencharacters of all possible imaginary quadratic extensions of
${\mathbb{Q}}$. From the work of Ambi [1], we have an upper bound on the
cardinality of this set as described below.

Notation: Let $f$ and $g$ be two non-negative real valued functions on
$\mathbb{N}$. We write $f\ll g$ (or equivalently $g\gg f$) if there exists a
constant $C$ and $n_{0}\in\mathbb{N}$ such that

$f(n)\leq Cg(n)\quad\forall~{}n\geq n_{0}.$

We write $f\sim g$ if both $f\ll g$ and $g\ll f$ hold.

######  Theorem 2.2.1.

([1]) For $k,N\geq 1$ and $\epsilon\in(0,1),$ let
$N^{\prime}:=\prod\limits_{p~{}\text{prime},~{}p|N}p$. Then

$|C_{k}(N)|~{}\ll_{k,\epsilon}N\cdot{N^{\prime}}^{1+\epsilon}\text{ as
}N\longrightarrow\infty$

If $N=p^{n},n\in\mathbb{N}$, then more is true.

######  Theorem 2.2.2.

Let $k\geq 2$ and $n\in{\mathbb{N}}$. We have,

$|C_{k}(p^{n})|\ll_{\epsilon}p^{n+\frac{1}{2}+\epsilon}\text{ as
}\epsilon\longrightarrow 0,~{}p\longrightarrow\infty.$

The following auxiliary results are discussed in [1] and [4] respectively. We
mention them here for a reference.

######  Lemma 2.2.3.

Suppose $p$ is an odd prime.

  1. (1)

If $p\equiv 1\mod 4$, then

$|C_{k}(p^{n})|=0\quad\forall~{}\text{ }k\geq 2\text{ and }\forall\text{
}n\geq 1.$

  2. (2)

If $p\equiv 3\mod 4$, $\mathbb{E}={\mathbb{Q}}(\sqrt{-p})$ and
$\mathfrak{f}\subseteq\mathcal{O}_{\mathbb{E}}$ such that
$N_{\mathbb{E}/\mathbb{F}}(\mathfrak{f})\cdot\allowbreak|D_{\mathbb{E}/\mathbb{Q}}~{}||~{}p^{n}$
for some $n\geq 1$, then $\mathfrak{f}~{}|~{}p^{m}\mathcal{O}_{\mathbb{E}}$
where $m=\lfloor\dfrac{n}{2}\rfloor$.

We now prove Thm. 2.2.2. We need to only consider the case $p\equiv 3\mod 4$
case in view of Lem. 2.2.3.

###### Proof.

Let, for an ideal $\mathfrak{J}\subset\mathcal{O}_{\mathbb{E}}$,
$h_{\mathbb{E}}^{0}(\mathfrak{J})$ and $h_{\mathbb{E}}^{\prime}(\mathfrak{J})$
denote the number of Hecke characters with conductor exactly $\mathfrak{J}$
and the number of characters of the narrow ray class group modulo
$\mathfrak{J}$ respectively. Hence
$\sum\limits_{\mathfrak{f}~{}|~{}\mathfrak{J}}h_{\mathbb{E}}^{0}(\mathfrak{f})=h_{\mathbb{E}}^{\prime}(\mathfrak{J})$.
Using (Theorem 12.5, [8]) we have,

$|C_{k}(p^{n})|\leq\sum\limits_{\begin{subarray}{c}\mathbb{E}=\mathbb{Q}(\sqrt{-p}),\\\
N_{\mathbb{E}/\mathbb{Q}}(\mathfrak{f})\cdot|D_{\mathbb{E}/\mathbb{Q}}~{}||~{}p^{n}\end{subarray}}h_{\mathbb{E}}^{0}(\mathfrak{f}).$

By part (2) of Lem. 2.2.3,

$|C_{k}(p^{n})|\leq\sum\limits_{\begin{subarray}{c}\mathfrak{f}~{}|~{}p^{m}\mathcal{O}_{\mathbb{E}}\\\
\mathbb{E}=\mathbb{Q}(\sqrt{-p})\end{subarray}}h_{\mathbb{E}}^{0}(\mathfrak{f})=h^{\prime}_{\mathbb{E}}(p^{m}\mathcal{O}_{\mathbb{E}})\leq
h_{\mathbb{E}}\Phi_{\mathbb{E}}(p^{m}\mathcal{O}_{\mathbb{E}}),$

where $h_{\mathbb{E}}=$ the class number of $\mathbb{E}$ and
$\Phi_{\mathbb{E}}(\mathcal{I})=N_{\mathbb{E}/\mathbb{Q}}(\mathcal{I})\prod\limits_{\begin{subarray}{c}\mathfrak{p}|\mathcal{I},\\\
\mathfrak{p}\text{ prime
}\end{subarray}}(1-N_{\mathbb{E}/\mathbb{Q}}(\mathfrak{p})^{-1})$ with
$\Phi_{\mathbb{E}}(\mathcal{O}_{\mathbb{E}})=1$.

The last inequality follows from [11, Theorem 3.25(i)]. Hence,

$|C_{k}(p^{n})|\leq h_{\mathbb{E}}\cdot p^{2m}.$

We use the following asymptotic formula for the class number as in [11,
Corollary 1 of Theorem 8.14].

$\begin{split}&\quad{\log}~{}h_{{\mathbb{Q}}({\sqrt{-d}})}=\left(\frac{1}{2}+o(1)\right)\log|d|\quad\text{
as }~{}d\longrightarrow\infty\\\ \implies&\quad
h_{{\mathbb{Q}}({\sqrt{-d}})}\ll_{\epsilon}d^{{\frac{1}{2}}+\epsilon}\quad\text{
as }\epsilon\longrightarrow 0,~{}d\longrightarrow\infty\\\
\implies&\quad|C_{k}(p^{n})|\ll_{\epsilon}p^{2m+\frac{1}{2}+1}\text{ as
}\epsilon\longrightarrow 0,~{}p\longrightarrow\infty\\\
\implies&\quad|C_{k}(p^{n})|\ll_{\epsilon}p^{n+\frac{1}{2}+\epsilon}\text{ as
}\epsilon\longrightarrow 0,~{}p\longrightarrow\infty.\end{split}$

∎

###  2.3. The Level Structure :

Consider the algebraic group ${\rm GL}_{m}$. We very briefly recall the
definition of a level structure for ${\rm GL}_{m}(\mathbb{A}_{\mathbb{Q}})$.
For a finite prime $p$ of ${\mathbb{Q}}$, let ${\mathbb{Z}}_{p}$ be the ring
of $p$-adic integers. For each integer $n\geq 0$, define

$K_{p}^{m}(n):=\\{x=(x_{i,j})_{m\times m}\in{\rm
GL}_{m}(\mathbb{Z}_{p}):x_{m,k}\in p^{n}\mathbb{Z}_{p},~{}\text{ }1\leq
k<m,~{}x_{m,m}-1\in p^{n}{\mathbb{Z}}_{p}\\}.$

Let $\mathbb{A}_{f}$ denote the finite part of adèles over ${\mathbb{Q}}$. Let
$N$ be a positive integer with prime factorisation $N=\prod\limits_{1\leq
i\leq r,p_{i}|N}p_{i}^{n_{i}}$. Define a compact open subgroup
$K_{f}^{m}(N)=\prod\limits_{p}K_{p}$ of ${\rm GL}_{m}(\mathbb{A}_{f})$ where

$K_{p}=\begin{cases}&K_{p_{i}}^{m}{(n_{i})}\quad\text{if}~{}p\mid
N~{}\text{i.e.},~{}p=p_{i}\\\ &{\rm
GL}_{m}(\mathbb{Z}_{p})\quad\text{if}~{}p\nmid N\end{cases}.$

For each $N\geq 1$, the compact open subgroup $K_{f}^{m}(N)\subseteq{\rm
GL}_{m}(\mathbb{A}_{f})$ is called the level structure corresponding to $N$.

##  3\. Automorphic Tensor product

###  3.1. Definition

We recall the definition of automorphic tensor product transfer from
$\rm{GL}_{2}\times\rm{GL}_{2}$ to $\rm{GL}_{4}$ over a number field
${\mathbb{F}}$. Let $\pi_{1}=\bigotimes\limits_{v}\pi_{1,v}$ and
$\pi_{2}=\bigotimes\limits_{v}\pi_{2,v}$ be two cuspidal automorphic
representations of $\rm{GL}_{2}({\mathbb{A}}_{\mathbb{F}})$. By local
Langlands correspondence for each place $v$, the local constituents
$\pi_{1,v}$ and $\pi_{2,v}$ correspond to two dimensional representations of
the local Weil-Deligne group.

D. Ramakrishnan in [13, Thm. M] proved the existence of an automorphic
representation (automorphic tensor product) $\pi_{1}\boxtimes\pi_{2}$ of
$\rm{GL}_{4}({\mathbb{A}}_{\mathbb{F}})$ that satisfies:

  * •

Equality of local $L$-functions:
$L(s,(\pi_{1}\boxtimes\pi_{2})_{v})=L(s,\pi_{1,v}\times\pi_{2,v})$ for all
places $v$,

  * •

Equality of local $\epsilon$-factors:
$\varepsilon(s,(\pi_{1}\boxtimes\pi_{2})_{v})=\varepsilon(s,\pi_{1,v}\times\pi_{2,v})$
for all finite places $v$.

###  3.2. The cuspidality criterion for automorphic tensor product

We describe the cuspidality criterion for the automorphic tensor product
(proved by D. Ramakrishnan in [14]) (we need it only for the case of base
field = ${\mathbb{Q}}$).

######  Theorem 3.2.1.

Let $\pi_{1},~{}\pi_{2}$ be two cuspidal automorphic representations of ${\rm
GL}_{2}(\mathbb{A}_{{\mathbb{Q}}})$. Then $\pi_{1}\boxtimes\pi_{2}$ is an
isobaric automorphic representation of ${\rm
GL}_{4}(\mathbb{A}_{{\mathbb{Q}}})$. Further more $\pi_{1}\boxtimes\pi_{2}$ is
cuspidal under the following conditions.

  1. (1)

If exactly one of them is dihedral (i.e. automorphically induced from
imaginary quadratic extensions of ${\mathbb{Q}}$), then
$\pi_{1}\boxtimes\pi_{2}$ is cuspidal.

  2. (2)

If both of them are non-dihedral, then $\pi_{1}\boxtimes\pi_{2}$ is cuspidal
if and only if $\pi_{2}$ is not of the form $\pi_{1}\otimes\chi$ for any idéle
class character $\chi$ of ${\mathbb{Q}}$.

  3. (3)

Suppose $\pi_{1},~{}\pi_{2}$ are both dihedral, say
$\pi_{i}=I^{{\mathbb{Q}}}_{{\mathbb{K}}_{i}}$ for $i=1,~{}2$, where
${\mathbb{K}}_{i}$’s are quadratic extensions of ${\mathbb{Q}}$ and $\mu_{1}$
(resp., $\mu_{2}$) is an idéle class character of ${\mathbb{K}}_{1}$ (resp.,
$\mathbb{K}_{2}$). Then $\pi_{1}\boxtimes\pi_{2}$ is cuspidal if and only if
${\mathbb{K}}_{1}\neq{\mathbb{K}}_{2}$.

###  3.3. Conductors for automorphic tensor products

Conductor of an automorphic representation: We briefly recall the notion of
the conductor of an automorphic representation here.

Let
$(\rho,\mathcal{H})=(\bigotimes\limits_{p\leq\infty}\rho_{p},\bigotimes\limits_{p\leq\infty}\mathcal{H}_{p})$
be an irreducible automorphic representation of ${\rm
GL}_{m}(\mathbb{A}_{\mathbb{Q}})$.

######  Definition 3.3.1.

For each finite prime $p$, the conductor of $\rho_{p}$ is defined to be the
smallest integer $c(\rho_{p})\geq 0$ such that the space
$\mathcal{H}_{p}^{K_{p}^{m}({{c(\rho_{p})}})}$ consisting of all
$K_{p}^{m}({{c(\rho_{p})}})$-fixed vectors of $\mathcal{H}_{p}$ is non-zero.
Thus $c(\rho_{p})=0$ for almost all primes. The conductor of $\rho$ is defined
as

$N_{\rho}=\prod_{p<\infty}p^{c(\rho_{p})}.$

Conductor of a local character: Let $\mathcal{K}$ be a non-archimedean local
field of characteristic zero. Let $\mathcal{O}_{\mathcal{K}}$, $\mathfrak{p}$,
$\kappa$ be its ring of integers, the maximal ideal and the residue field
respectively.

######  Definition 3.3.2.

The conductor $c(\zeta)$ of a character $\zeta$ of $\mathcal{K}^{\times}$ is
the smallest positive integer $n$ such that
${\zeta}_{|_{1+\mathfrak{p}^{n}}}=1$.

A relation between level and conductor: For a supercuspidal representation
$\pi$ of ${\rm GL}_{n}({\mathbb{Q}}_{p})$, if $c(\pi)$ is the conductor of
$\pi$, we have $c(\pi)=n\left(1+\dfrac{m}{e}\right)$, where $m$ is the level
of $\pi$ and $e$ is the ramification index (see [2, p.149]). We will consider
only the case when $n=2$. Note that, $c(\pi)\geq 2$.

Since, $1\leq e\leq 2$ , we have

(1) $\begin{split}&\quad\dfrac{m}{2}\leq\dfrac{m}{e}\leq m\\\
\implies&\quad\frac{m}{2}\leq\dfrac{c(\pi)}{2}-1\leq m\\\ \implies&\quad
m+2\leq c(\pi)\leq 2m+2.\end{split}$

######  Lemma 3.3.3.

Let $p$ be an odd prime and $\pi_{1},~{}\pi_{2}$ be two supercuspidal
representations of $\rm{GL}_{2}({\mathbb{Q}}_{p})$. Let
$c(\pi_{1}),~{}c(\pi_{2})\text{ and }c(\pi_{1}\times\pi_{2})$ be the
conductors of $\pi_{1},~{}\pi_{2}\text{ and }\pi_{1}\times\pi_{2}$
respectively. Then we have,

$2\leq c(\pi_{1}\times\pi_{2})\leq 4c(\pi_{1})+4c(\pi_{2})-12.$

###### Proof.

Using [5, Thm. 6.5 ] and the calculation before the Lem. 3.3.3, we get the
desired inequality.

Take $e={\rm l.c.m.}(e_{1},e_{2})$, and $m_{1}$, $m_{2}$ be the levels of
$\pi_{1}$ and $\pi_{2}$, respectively. Let $m$ be defined by

$m=\max\left\\{\frac{em_{1}}{e_{1}},\frac{em_{2}}{e_{2}}\right\\}.$

As in the notation in [5], we have

$n_{1}=n_{2}=2,\quad 1\leq e_{1},e_{2}\leq 2,$ $1\leq d\leq 2,\quad 0\leq
c(\beta),c(\gamma),\quad l\leq m.$

Let $\check{\pi_{1}}$ denote the contragredient representation of $\pi$. Using
[5, Thm. 6.5 ], the inequality (1), and the fact $c(\pi)=c(\check{\pi})$ (see
[6, Prop., p.23]), we conclude that

$2\leq c(\pi_{1}\times\pi_{2})\leq 4c(\pi_{1})+4c(\pi_{2})-12.$

∎

##  4\. Cohomology of the ${\rm GL}_{m}$

###  4.1. Cohomological representations of ${{\rm
GL}}_{m}({\mathbb{A}}_{\mathbb{Q}})$

Let $\pi$ be a cuspidal automorphic representation of ${{\rm
GL}}_{m}({\mathbb{A}}_{\mathbb{Q}})$ with finite and infinite parts denoted by
$\pi_{f}$ and $\pi_{\infty}$ respectively. Let $\mu$ be a dominant integral
weight corresponding to the standard Borel subgroup of ${{\rm GL}}_{m}$ and
$\mathcal{M}_{\mu,\mathbb{C}}$ be the underlying vector space of the finite
dimensional irreducible highest weight representation corresponding to $\mu$.
For a given level structure $K_{f}^{m}\subset{\rm GL}_{m}(\mathbb{A}_{f})$,
let $\pi^{K^{m}_{f}}_{f}$ denote the space of $K_{f}^{m}$-fixed vectors of
$\pi_{f}$.

Define ${\rm Coh}({{\rm GL}}_{m},\mu,K_{f}^{m})$ = the set of all automorphic
representations $\pi$ such that its relative Lie algebra cohomology with
coefficients in $\mathcal{M}_{\mu,\mathbb{C}}$ is non zero in some degree
i.e.,

$H^{*}(\mathfrak{gl}_{m},\mathbb{R}_{+}^{\times}\cdot{\rm
SO}_{m}(\mathbb{R}),\pi_{\infty}\bigotimes\mathcal{M}_{\mu,\mathbb{C}})\bigotimes\pi_{f}^{K_{f}^{m}}\neq
0.$

We say that $\pi$ is cohomological if $\pi\in{\rm Coh}({{\rm
GL}}_{m},\mu,K_{f}^{m})$ for some highest weight $\mu$ and level structure
$K_{f}^{m}$. Now we state a necessary condition for $\pi$ to be cohomological.
Let $\mathcal{L}_{0}^{+}({\rm GL}_{m})$ be the set of all pairs $(w,\ell)$
with $\ell=(\ell_{1},\ell_{2},\dots,\ell_{m})\in{\mathbb{Z}}^{m}$ such that
$\ell_{1}>\ell_{2}>\dots>\ell_{m}>0$ and $\ell_{i}=-\ell_{m-i+1}$ and
$w\in{\mathbb{Z}}$ such that for all $i$,

$w+\ell_{i}\equiv\begin{cases}1\text{ if }m\text{ is even }\\\ 0\text{ if
}m\text{ is odd }\end{cases}.$

For $(w,\ell)\in\mathcal{L}_{0}^{+}({\rm GL}_{m})$, let $D_{\ell_{i}}$ be the
discrete series representation of $\rm{GL}_{2}({\mathbb{R}})$ of lowest weight
$\ell_{i}+1$, $1\leq i\leq m$. Define,

$J(w,\ell)=\begin{cases}{\rm Ind}_{P_{2,2,\dots,2}}^{{\rm
GL}_{m}}\left((D_{\ell_{1}}\otimes|.|^{w/2}_{{\mathbb{R}}})\otimes\dots\otimes(D_{\ell_{m/2}}\otimes|.|^{w/2}_{{\mathbb{R}}})\right)&\text{
if }m\text{ is even}\\\ {\rm Ind}_{P_{2,2,\dots,2,1}}^{{\rm
GL}_{m}}\left((D_{\ell_{1}}\otimes|.|^{w/2}_{{\mathbb{R}}})\otimes\dots\otimes(D_{\ell_{(m-1)/2}}\otimes|.|^{w/2}_{{\mathbb{R}}})\otimes|.|^{w/2}_{{\mathbb{R}}}\right)&\text{
if }m\text{ is odd}\end{cases}.$

If $\pi$ is cuspidal cohomological representation of ${\rm
GL}_{m}(\mathbb{A}_{{\mathbb{Q}}})$, then the representation $\pi_{\infty}$
must be equal to $J(w,\ell)$ for some $(w,\ell)\in\mathcal{L}_{0}^{+}({\rm
GL}_{m})$. See ([12], section 5) for more details.

###  4.2. Examples

We are interested in studying the contribution to the ${\rm
GL}_{4}$-cohomology of the representations that are obtained via automorphic
tensor products of two cusp forms on ${\rm GL}_{2}$. For that, consider the
following examples.

######  Example 4.2.1.

(see [12]) Let $k_{1}>k_{2}\geq 2$ be positive integers. Let $\phi_{1}\text{
and }\phi_{2}$ be two cusp forms of weight $k_{1},k_{2}$ respectively and
$\pi_{\phi_{1}}$ and $\pi_{\phi_{2}}$ be the corresponding automorphic
cuspidal representations of ${\rm GL}_{2}({\mathbb{A}}_{\mathbb{Q}})$. Let
$\pi:=\pi_{\phi_{1}}\boxtimes\pi_{\phi_{2}}$ denote their automorphic tensor
product. $\pi_{\infty}$ = the representation at infinity as a representation
of ${\rm GL}_{4}({\mathbb{R}})$, can be computed using a well-known recipe for
Langlands parameters as

$\pi_{\infty}={\rm{Ind}}_{P_{2,2}}^{{\rm GL}_{4}}(D_{k_{1}+k_{2}-2}\otimes
D_{k_{1}-k_{2}}).$

We have to consider two cases based on the parity of $k_{1}+k_{2}$.

Case (1):  Suppose $k_{1}+k_{2}$ is odd. In this case we see that
$\pi_{\infty}=J(0,\ell),$ where:

$\ell=(k_{1}+k_{2}-2,~{}k_{1}-k_{2},~{}k_{2}-k_{1},~{}-k_{1}-k_{2}+2).$

Hence $\pi$ is cohomological with respect to the dominant integral weight

$\mu_{\pi}=\nu_{k_{1},k_{2}}=\left(\frac{k_{1}+k_{2}-5}{2},~{}\frac{k_{1}-k_{2}-1}{2},~{}\frac{k_{2}-k_{1}+1}{2},~{}\frac{-k_{1}-k_{2}+5}{2}\right).$

Case (2):  Suppose $k_{1}+k_{2}$ is even. In this case we see that $\pi$ is
not cohomological but its half-integral Tate-twist $\pi\otimes|\cdot|^{1/2}$
is cohomological. Moreover, $(\pi\otimes|\cdot|^{1/2})_{\infty}=J(1,\ell),$
where

$\ell=(k_{1}+k_{2}-2,~{}k_{1}-k_{2},~{}k_{2}-k_{1},~{}-k_{1}-k_{2}+2).$

Hence $\pi$ is cohomological with respect to the dominant integral weight

$\mu_{\pi}=\left(\frac{k_{1}+k_{2}-4}{2},~{}\frac{k_{1}-k_{2}}{2},~{}\frac{k_{2}-k_{1}+2}{2},~{}\frac{-(k_{1}+k_{2})+6}{2}\right).$

######  Example 4.2.2.

The following example from ([12], section 5) shows that we must consider two
cusp forms of different weight for their automorphic tensor product to be
cohomological (up to a half-integral Tate-twist).

Let $\phi_{1},\phi_{2}$ are two cusp forms of weight $k\in{\mathbb{Z}},k$ even
such that $\pi=\pi_{\phi_{1}}\boxtimes\pi_{\phi_{2}}$ is cuspidal. Then the
similar computation involving the Langlands parameters gives that the infinite
part of $\pi$ is given by

$\pi_{\infty}={\rm Ind}_{P_{2,1,1}}^{{\rm GL}_{4}}(D_{2(k-1)}\otimes
sgn\otimes 1).$

From the shape of $\pi_{\infty}$, it follows that neither $\pi_{\infty}$ nor
$(\pi\otimes|\cdot|^{1/2})_{\infty}$ are of the type $J(w,\ell)$ for any
$(w,\ell)\in\mathcal{L}_{0}^{+}({\rm GL}_{m})$ and hence neither $\pi$ nor
$\pi\otimes|\cdot|^{1/2}$ are cohomological.

##  5\. An estimate of the contribution of automorphic tensor products to
cohomology of the ${\rm GL}_{4}$

###  5.1. Notations

We develop a few required notations for the statement of the main theorem in
this subsection.

  * •

Let $k_{1}>k_{2}\geq 2$ be two positive integers. Suppose they are of
different parity. Without loss of generality, assume that $k_{1}$ is odd and
$k_{2}$ is even. Define $\mathcal{T}_{k_{1},k_{2}}(N)$ to be the set of all
cuspidal automorphic representations of ${\rm
GL}_{4}(\mathbb{A}_{{\mathbb{Q}}})$ corresponding to the level structure
$K_{f}^{4}(N)$ that are obtained by automorphic tensor product of two cuspidal
automorphic representations of ${\rm GL}_{2}(\mathbb{A}_{{\mathbb{Q}}})$ with
highest weights $\left(\dfrac{k_{1}-1}{2},~{}\dfrac{1-k_{1}}{2}\right)$ and
$\left(\dfrac{k_{2}}{2}-1,~{}1-\dfrac{k_{2}}{2}\right)$, respectively. (If
$k_{1},~{}k_{2}$ are of same parity, we consider the half-integral twist of
the automorphic tensor product of two representations $\pi_{1},\pi_{2}$ of
${\rm GL}_{2}(\mathbb{A}_{{\mathbb{Q}}})$ with suitable highest weight
depending on $k_{1},k_{2}$ are even or odd).

  * •

Let $C^{0}_{k_{1},~{}k_{2}}(N_{1},N_{2})$ denote the set of ordered pairs
$(f_{1},f_{2})$ of all normalized cusp eigenforms $f_{1},f_{2}$ of Hecke
operators of $\Gamma_{1}(N_{1})$ of weight $k_{1}$ and $\Gamma_{1}(N_{2})$ of
weight $k_{2}$ respectively such that both $f_{1}$ and $f_{2}$ are dihedral,
and are obtained by automorphic induction from Hecke characters of (same)
imaginary quadratic extension of ${\mathbb{Q}}$.

Note that, $C^{0}_{k_{1},~{}k_{2}}(N_{1},N_{2})\subseteq
C_{k_{1}}(N_{1})\times C_{k_{2}}(N_{2})$ (see the Sec. 2.2 for notations).
Hence, if $N_{1}=p^{n_{1}}\text{ and }N_{2}=p^{n_{2}}$, using Thm. 2.2.2 we
have,

$|C^{0}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})|\ll_{\epsilon}p^{n_{1}+n_{2}+2+2\epsilon}\text{
as }p\longrightarrow\infty.$

  * •

Let $C^{\prime}_{k_{1},~{}k_{2}}(N_{1},N_{2})$ denote the set of ordered pairs
$(g_{1},g_{2})$ of normalized cusp eigenforms $g_{1},~{}g_{2}$ of Hecke
operators of $\Gamma_{1}(N_{1})$ of weight $k_{1}$ and $\Gamma_{1}(N_{2})$ of
weight $k_{2}$ respectively such that $g_{1},~{}g_{2}$ are both non-dihedral
and one of them is the twist of the other by some idéle class character of
${\mathbb{Q}}$.

Note that under the assumption that $k_{1}>k_{2},$ the set
$C^{\prime}_{k_{1},~{}k_{2}}(N_{1},N_{2})$ is indeed an empty set.

###  5.2. The main theorem

######  Theorem 5.2.1.

Let $k_{1}>k_{2}\geq 2$ be two integers. Then for any integer $N>24$,

$|\mathcal{T}_{k_{1},~{}k_{2}}(p^{N})|\gg_{k_{1},~{}k_{2}}~{}p^{2\lfloor
N/4\rfloor}\text{ as the prime }p\longrightarrow\infty.$

###### Proof.

We first consider the different parity case where $k_{1}+k_{2}$ is odd,
$k_{1}$ is odd and $k_{2}$ is even. Let $p$ be an odd prime. Suppose
$\pi_{1},\pi_{2}$ are cuspidal automorphic representations of ${\rm
GL}_{2}(\mathbb{A}_{{\mathbb{Q}}})$ corresponding to the level structure
$K_{f}^{2}(p^{n_{1}})$ and $K_{f}^{2}(p^{n_{2}})$, respectively, with
$n_{1},n_{2}\geq 1$ and of weights
$\left(\dfrac{k_{1}-1}{2},\dfrac{1-k_{1}}{2}\right)$ and
$\left(\dfrac{k_{2}}{2}-1,~{}1-\dfrac{k_{2}}{2}\right)$, respectively.

Using Thm. 3.2.1, we conclude that exactly one of the following holds:

  * •

$(\pi_{1},\pi_{2})\in C^{\prime}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})$

  * •

$(\pi_{1},\pi_{2})\in C^{0}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})$

  * •

$\pi_{1}\boxtimes\pi_{2}\in\bigcup\limits_{n\geq
1}\mathcal{T}_{k_{1},~{}k_{2}}(p^{n})$.

Let $\pi_{1}=\bigotimes_{q\leq\infty}\pi_{1,q}$ and
$\pi_{2}=\bigotimes_{q\leq\infty}\pi_{2,q}$. We know that $\pi_{1,p}$ (resp
$\pi_{2,p}$) corresponds to a unique newform in
$S_{k_{1}}^{\rm{new}}(\Gamma_{1}(p^{s_{1}}))$ (resp.
$S_{k_{2}}^{\rm{new}}(\Gamma_{1}(p^{s_{2}}))$) with $1\leq s_{1}\leq n_{1}$
(resp $1\leq s_{2}\leq n_{2}$). Note that, $c(\pi_{1,p})=s_{1}$ and
$c(\pi_{2,p})=s_{2}$.

We want to count the contributions of supercuspidal representations
$\pi_{i,p},i=1,2$ to $\bigcup\limits_{n\geq
1}\mathcal{T}_{k_{1},~{}k_{2}}(p^{n})$. Using Lem. 3.3.3, we can conclude that
two newforms in $S_{k_{1}}^{\rm{new}}(\Gamma_{1}(p^{s_{1}}))$ and
$S_{k_{2}}^{\rm{new}}(\Gamma_{1}(p^{s_{2}}))$ that are obtained from the
supercuspidal representations $\pi_{1,p}$ and $\pi_{2,p}$ respectively, can
contribute to $\mathcal{T}_{k_{1},~{}k_{2}}(p^{n})$ only if

$2\leq n\leq 4c(\pi_{1,p})+4c(\pi_{2,p})-12.$

Since $1\leq s_{i}\leq n_{i},i=1,2$, we have,

$2\leq n\leq 4n_{1}+4n_{2}-12.$

Note that, $n_{i}\geq 2$ since $s_{i}\geq 2$ as $\pi_{i,p}$ is supercuspidal,
$i=1,2$. Hence, we conclude that, automorphic tensor product of any pair of
supercuspidal representations in the set

$\bigoplus\limits_{1\leq i\leq
n_{1}}S_{k_{1}}^{\text{new}}(\Gamma_{1}(p^{i}))\times\bigoplus\limits_{1\leq
i\leq
n_{2}}S_{k_{2}}^{\text{new}}(\Gamma_{1}(p^{i}))~{}\setminus~{}\\{C^{0}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})\cup
C^{\prime}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})\\}$ is also in the set
$\mathcal{T}_{k_{1},~{}k_{2}}(p^{4n_{1}+4n_{2}-12})$.

Since we have the following bound

$\sum\limits_{1\leq i\leq n_{1}}{\rm
dim}_{\mathbb{C}}S_{k}^{\text{new}}(\Gamma_{1}(p^{i}))\gg_{k}p^{2n_{1}}$

and we also have

$|C^{0}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})\cup
C^{\prime}_{k_{1},~{}k_{2}}(p^{n_{1}},p^{n_{2}})|\ll_{k_{1},~{}k_{2}}p^{n_{1}+n_{2}+4},$

which is negligible compared to $p^{2n_{1}+2n_{2}}$ for
$p\longrightarrow\infty$ and $n_{1}+n_{2}>4$, we conclude that

$p^{2n_{1}+2n_{2}}\ll_{k_{1},~{}k_{2}}|\mathcal{T}_{k_{1},~{}k_{2}}(p^{4n_{1}+4n_{2}-12})|\text{
as }p\longrightarrow\infty\text{ and }n_{1}+n_{2}>4.$

In particular, taking $N=4(n_{1}+n_{2})$ with $N>24$, we have,

$|\mathcal{T}_{k_{1},~{}k_{2}}(p^{N})|\gg_{k_{1},~{}k_{2}}p^{2\lfloor
N/4\rfloor}\text{ as }p\longrightarrow\infty.$

The same parity case where $k_{1}+k_{2}$ is even can be handled in an exactly
same manner by considering $(\pi_{1}\boxtimes\pi_{2})\otimes|\cdot|^{1/2}$
instead of $\pi_{1}\boxtimes\pi_{2}$. ∎

Using Lem. 2.1.1 we have an upper bound as stated below.

######  Corollary 5.2.2.

Let $N\in\mathbb{N}$. Then,

$|\mathcal{T}_{k_{1},~{}k_{2}}(p^{N})|\ll_{k_{1},~{}k_{2}}~{}p^{2N}\text{ as
}p\longrightarrow\infty.$

##  6\. Possible Overlap between symmetric cube and automorphic tensor product
representations of ${\rm GL}_{4}$

It is natural to investigate the overlap between the two types of
contributions to the cohomology of ${\rm GL}_{4}$, the symmetric cube of a
representation of ${\rm GL}_{2}$ and the automorphic tensor product of two
representations of ${\rm GL}_{2}$.

###  6.1. Comparison of representations at infinity

We first describe the representations at infinity. Now suppose
$k_{1},k_{2},k_{3}$ are integers $\geq 2$, $k_{1}>k_{2}$ and they are weights
of cusp forms corresponding to three automorphic representations
$\pi_{1},\pi_{2},\pi_{3}$ of ${\rm GL}_{2}$, respectively.

From the discussion in Sec. 4.2, we conclude that for a half-integer $s$,

(2)
$((\pi_{1}\boxtimes\pi_{2})\otimes|\cdot|^{s})_{\infty}=\begin{cases}J(0,\ell)\otimes|\cdot|_{\infty}^{s}\quad&\text{if}~{}k_{1}+k_{2}~{}\text{is
odd}\\\
J(1,\ell)\otimes|\cdot|_{\infty}^{s-1/2}&\text{if}~{}k_{1}+k_{2}~{}\text{is
even}\end{cases}$

where in both cases,
$\ell=(k_{1}+k_{2}-2,~{}k_{1}-k_{2},~{}k_{2}-k_{1},~{}-k_{1}-k_{2}+2).$

On the other hand, $({\rm Sym^{3}}(\pi_{3})\otimes|\cdot|^{s})_{\infty}$ is
given by [12, Thm 5.5]

(3) $({\rm
Sym^{3}}(\pi_{3})\otimes|\cdot|^{s})_{\infty}=\begin{cases}J(1,\ell^{\prime})\otimes|\cdot|_{\infty}^{s-1/2}\quad&\text{if}~{}k_{3}~{}\text{is
odd}\\\
J(0,\ell^{\prime})\otimes|\cdot|_{\infty}^{s}&\text{if}~{}k_{3}~{}\text{is
even}\end{cases}$

where in both cases,
$\ell^{\prime}=(3k_{3}-3,~{}k_{3}-1,~{}1-k_{3},~{}3-3k_{3}).$

Refer to Sec. 4.1 for the discussion of cohomological representations
$J(w,\ell)$. It follows that for a half-integer $s$ (i.e.
$2s\in{\mathbb{Z}}$), the twisted representation
$J(w,\ell)\otimes|\cdot|^{s}_{\infty}$ is cohomological iff $s\in{\mathbb{Z}}$
(i.e. $2s$ is even) and in that case,

(4) $J(w,\ell)\otimes|\cdot|^{s}_{\infty}=J(w+2s,\ell)$

We analyze all possible scenarios as follows.

  1. (a)

Suppose $k_{1}+k_{2}$ and $k_{3}$ are even. Let $s$ be a half-integer (i.e.
$2s\in{\mathbb{Z}}$). Then from (2) and (4), we conclude that
$(\pi_{1}\boxtimes\pi_{2})\otimes|\cdot|^{s}$ is cohomoloigcal if and only if
$2s$ is odd, and in that case it equals $J(2s,\ell)$. Since ${\rm
Sym^{3}}(\pi_{3})_{\infty}=J(0,\ell^{\prime})$, we conclude that ${\rm
Sym^{3}}(\pi_{3})\otimes|\cdot|^{s}\neq\pi_{1}\boxtimes\pi_{2}$ for any half
integer $s$.

  2. (b)

Suppose $k_{1}+k_{2}$ and $k_{3}$ are odd. Let $s$ be a half-integer (i.e.
$2s\in{\mathbb{Z}}$). Then from (3) and (4) we conclude that ${\rm
Sym^{3}}(\pi_{3})\otimes|\cdot|^{s}$ is cohomoloigcal if and only if $2s$ is
odd, and in that case it equals $J(2s,\ell^{\prime})$. Since
$(\pi_{1}\boxtimes\pi_{2})_{\infty}=J(0,\ell)$, we conclude that ${\rm
Sym^{3}}(\pi_{3})\otimes|\cdot|^{s}\neq\pi_{1}\boxtimes\pi_{2}$ for any half
integer $s$.

  3. (c)

Suppose $k_{1}+k_{2}$ and $k_{3}$ are of different parity and $k_{1}$ is even.
Thus $k_{2},k_{3}$ have different parity. From (2) and (3), we conclude that:

$((\pi_{1}\boxtimes\pi_{2})\otimes|\cdot|^{s})_{\infty}={\rm
Sym^{3}}(\pi_{3})_{\infty}$ for some half integer $s$ (i.e.
$2s\in{\mathbb{Z}}$) if and only if $s=0$ and $\ell=\ell^{\prime}$.

If $\ell=\ell^{\prime}$, then we have

$\frac{k_{1}-1}{2}=k_{2}=k_{3}$

This is impossible since we assumed that $k_{1}$ is even and $k_{2},k_{3}$ are
of different parity. So we conclude that ${\rm
Sym^{3}}(\pi_{3})\otimes|\cdot|^{s}\neq\pi_{1}\boxtimes\pi_{2}$ for any half
integer $s$.

  4. (d)

Suppose $k_{1}+k_{2}$ and $k_{3}$ are of different parity and $k_{1}$ is odd.
Thus $k_{2},k_{3}$ are of same parity. Similar to the previous case, we have
the following necessary conditions for ${\rm
Sym^{3}}(\pi_{3})\otimes|\cdot|^{s}=\pi_{1}\boxtimes\pi_{2}$ for some half
integer $s$ (i.e. $2s\in{\mathbb{Z}}$):

     * •

$\ell=\ell^{\prime}$, i.e. $\dfrac{k_{1}-1}{2}=k_{2}=k_{3}$

     * •

$s=0$

This case is inconclusive and we can not rule out the possibility of an
overlap using the method of comparing the representations at infinity.

###  6.2. Proof of ‘lack of overlap’

Now we prove that the overlap is in fact impossible up to any twist by a Hecke
character of ${\rm GL}_{1}({\mathbb{A}}_{\mathbb{Q}})$ under suitable
conditions on the representations being cuspidal and cohomological.

######  Proposition 6.2.1.

Let $\pi_{1},\pi_{2},\pi_{3}$ be three cuspidal automorphic representations of
${{\rm GL}}_{2}{(\mathbb{A}_{{\mathbb{Q}}})}$ such that they are given by cusp
forms of weights $k_{1},k_{2},k_{3}\geq 2$ and $\pi_{1}\boxtimes\pi_{2}$ and
${{\rm Sym}}^{3}{(\pi_{3})}$ are cuspidal automorphic representations of
${{\rm GL}}_{4}{(\mathbb{A}_{{\mathbb{Q}}})}$.

Then there is no Hecke character of ${\rm GL}_{1}({\mathbb{A}}_{\mathbb{Q}})$
such that $\pi_{1}\boxtimes\pi_{2}={\rm Sym}^{3}{(\pi_{3})}\otimes\chi$.

###### Proof.

Suppose $\pi_{1}\boxtimes\pi_{2}={\rm Sym}^{3}{(\pi_{3})}\otimes\chi$ for some
Hecke character $\chi$. This gives

(5) $\Lambda^{2}(\pi_{1}\boxtimes\pi_{2})=\Lambda^{2}({\rm
Sym}^{3}{(\pi_{3})}\otimes\chi)$

Let $\omega_{\pi_{i}}$ denote the central character of $\pi_{i}$ for
$i=1,2,3$. From ([3], Proposition 3.1), [10], we have the following isobaric
decompositions of the exterior square of representations.

(6) $\begin{split}\Lambda^{2}(\pi_{1}\boxtimes\pi_{2})&~{}=~{}({\rm
Sym}^{2}{(\pi_{1})}\otimes\omega_{\pi_{2}})\text{ }\boxplus\text{ }({\rm
Sym}^{2}{(\pi_{2})}\otimes\omega_{\pi_{1}})\\\ \Lambda^{2}({\rm
Sym}^{3}{(\pi_{3})}\otimes\chi)&~{}=~{}({\rm
Sym}^{4}{(\pi_{3})}\otimes\chi^{2}\omega_{\pi_{3}})\text{ }\boxplus\text{
}\chi^{2}\omega_{\pi_{3}}^{3}\end{split}$

Combining the two equations (5) and (6), we get

(7) $~{}{\rm
Sym}^{2}{(\pi_{1})}\otimes\omega_{\pi_{2}}\otimes(\chi^{2}\omega_{\pi_{3}}^{3})^{-1}\text{
}\boxplus\text{ }{\rm
Sym}^{2}{(\pi_{2})}\otimes\omega_{\pi_{1}}\otimes(\chi^{2}\omega_{\pi_{3}}^{3})^{-1}~{}=~{}{\rm
Sym}^{4}{(\pi_{3})}\otimes\omega_{\pi_{3}}^{-2}\text{ }\boxplus\text{ }1$

We assumed that ${\rm Sym}^{3}(\pi_{3})$ is cuspidal. If ${\rm
Sym}^{4}(\pi_{3})$ is not cuspidal, then using Langlands-Tunnel theorem (see
[7, Thm.1.3, Gelbart’s article]), we conclude that $\pi_{3}$ must have come
from a cusp form of weight one and this already contradicts the assumption on
the weight for $\pi_{3}$ i.e., $k_{3}\geq 2$. Thus ${\rm Sym}^{4}(\pi_{3})$ is
cuspidal.

Thus the first term ${\rm Sym}^{4}{(\pi_{3})}\otimes\omega_{\pi_{3}}^{-2}$ on
RHS in (7) is a representation of ${\rm GL}_{5}({\mathbb{A}}_{\mathbb{Q}})$
and it will not split further into an isobaric sum with respect to lower
dimensional ${\rm GL}_{n^{\prime}}$. We get a contradiction as there is no
${\rm GL}_{5}$-isobaric term on the LHS of Eq. (7). ∎

######  Remark 6.2.2.

The authors of this manuscript established lower bound estimates on the
contribution of symmetric cube transfer representations to the cohomology of
the ${\rm GL}_{4}$ in [4]. Prop. 6.2.1 establishes ‘lack of overlap between
symmetric cube and automorphic tensor product’ and hence the estimates
obtained in this manuscript improve the lower bound on the total contribution
of cuspidal automorphic representations to the cohomology of the ${\rm
GL}_{4}$. The case of Asai transfer from ${{\rm GL}_{2}}|_{{\mathbb{K}}}$ for
some imaginary quadratic extension ${\mathbb{K}}$ over ${\mathbb{Q}}$ is
remaining to be explored. One can ask these questions for the ${\rm GL}_{4}$
over a general number field and it will be interesting to know whether the
similar estimates hold in that case also.

Acknowledgements: Both the authors are thankful to A. Raghuram for valuable
discussions regarding various aspects of this manuscript, in particular about
the proof of ‘lack of overlap between symmetric cube and automorphic tensor
product’. Chandrasheel Bhagwat would like to acknowledge the support of
MATRICS research grant MTR/2018/000102 from the Science and Engineering
Research Board, Department of Science and Technology, Government of India
during this work. Sudipa Mondal would like to acknowledge the support of CSIR
PhD fellowship during this work.

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