abconsymbol=c

# Zeros of Rankin–Selberg $L$-functions in families

Peter Humphries  Department of Mathematics, University of Virginia,
Charlottesville, VA 22904, USA<EMAIL_ADDRESS>https://sites.google.com/view/peterhumphries/ and  Jesse Thorner  Department
of Mathematics, University of Illinois, Urbana, IL 61801, USA
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathfrak{F}_{n}$ be the set of all cuspidal automorphic representations
$\pi$ of $\mathrm{GL}_{n}$ with unitary central character over a number field
$F$. We prove the first unconditional zero density estimate for the set
$\mathcal{S}=\\{L(s,\pi\times\pi^{\prime})\colon\pi\in\mathfrak{F}_{n}\\}$ of
Rankin–Selberg $L$-functions, where $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$
is fixed. We use this density estimate to prove:

  1. (i)

a strong average form of effective multiplicity one for $\mathrm{GL}_{n}$;

  2. (ii)

that given $\pi\in\mathfrak{F}_{n}$ defined over $\mathbb{Q}$, the convolution
$\pi\times\widetilde{\pi}$ has a positive level of distribution in the sense
of Bombieri–Vinogradov;

  3. (iii)

that almost all $L(s,\pi\times\pi^{\prime})\in\mathcal{S}$ have a hybrid-
aspect subconvexity bound on $\textup{Re}(s)=\frac{1}{2}$;

  4. (iv)

a hybrid-aspect power-saving upper bound for the variance in the discrepancy
of the measures $|\varphi(x+iy)|^{2}y^{-2}dxdy$ associated to
$\mathrm{GL}_{2}$ Hecke–Maaß newforms $\varphi$ with trivial nebentypus,
extending work of Luo and Sarnak for level 1 cusp forms; and

  5. (v)

a nonsplit analogue of quantum ergodicity: almost all restrictions of Hilbert
Hecke–Maaß newforms to the modular surface dissipate as their Laplace
eigenvalues grow.

##  1\. Introduction and statement of the main result

Let $\mathbb{A}_{F}$ be the ring of adèles over a number field $F$ with
absolute norm $\mathrm{N}=\mathrm{N}_{F/\mathbb{Q}}$ and absolute discriminant
$D_{F}$. Let $\mathfrak{F}_{n}$ be the set of cuspidal automorphic
representations $\pi=\bigotimes_{v}\pi_{v}$ of
$\mathrm{GL}_{n}(\mathbb{A}_{F})$, where the (restricted) tensor product runs
over all places of $F$ and $\pi$ is normalized so that its central character
is trivial on the diagonally embedded copy of the positive reals. Let
$\mathfrak{q}_{\pi}$ be the arithmetic conductor of $\pi$, $C(\pi)\geq 1$ the
analytic conductor of $\pi$ (see (3.3)), and
$\mathfrak{F}_{n}(Q)=\\{\pi\in\mathfrak{F}_{n}\colon C(\pi)\leq Q\\}$. The
analytic conductor $C(\pi)$ is a useful measure for the arithmetic and
spectral complexity of $\pi$. Our normalization for the central characters
ensures that $|\mathfrak{F}_{n}(Q)|$ is finite.

Given $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$,
let $L(s,\pi\times\pi^{\prime})$ be the associated Rankin–Selberg
$L$-function, and let $\widetilde{\pi}\in\mathfrak{F}_{n}$ and
$\widetilde{\pi}^{\prime}\in\mathfrak{F}_{n^{\prime}}$ be the contragredient
representations. When
$\pi^{\prime}\in\\{\widetilde{\pi},\widetilde{\pi}^{\prime}\\}$, work of
Brumley [29, Appendix] and the authors [30] shows that there exists an
absolute and effectively computable constant
$\Cl[abcon]{ZFR_{s}tandard_{i}ntro}>0$ such that $L(s,\pi\times\pi^{\prime})$
has a “standard” zero-free region of the shape

(1.1) $\textup{Re}(s)\geq
1-\frac{\Cr{ZFR_{s}tandard_{i}ntro}}{(n+n^{\prime})^{3}\log(C(\pi)C(\pi^{\prime})(|\textup{Im}(s)|+3)^{n[F:\mathbb{Q}]})}$

apart from at most one real simple zero. This is comparable to the classical
zero-free region for Dirichlet $L$-functions. Brumley ([8] and [41, Appendix])
proved a much narrower zero-free region for all choices of $\pi$ and
$\pi^{\prime}$. The generalized Riemann hypothesis (GRH) asserts that
$L(s,\pi\times\pi^{\prime})\neq 0$ for $\textup{Re}(s)>\frac{1}{2}$. Zeros
that lie close to the line $\textup{Re}(s)=1$ typically cause the most damage
in applications, and one observes that a zero-free region of the shape
$\textup{Re}(s)\geq 1-\delta$ for some constant $\delta>0$ is often times
sufficient to obtain arithmetic consequences that are commensurate with what
GRH predicts.

Since strong zero-free regions for Rankin–Selberg $L$-functions remain out of
reach, it is useful to show that zeros near the line $\textup{Re}(s)=1$ must
be sparse. A suitable quantitative formulation can serve as a proxy for a
zero-free region of the shape $\textup{Re}(s)\geq 1-\delta$. Famous
consequences of this philosophy include Hoheisel’s proof [27] that
$p_{n+1}-p_{n}\ll p_{n}^{1-1/33000}$ (where $p_{n}$ is the $n$-th prime) and
Linnik’s proof [44] that there exists an absolute and effectively computable
constant $B>0$ and a prime $p\leq q^{B}$ such that $p\equiv
a\,(\mathrm{mod}\,\,q)$.

To quantify our notion of “sparse”, we define for $\sigma\geq 0$ and $T\geq 1$
the quantity

$N_{\pi\times\pi^{\prime}}(\sigma,T)=|\\{\rho=\beta+i\gamma\colon
L(\rho,\pi\times\pi^{\prime})=0,~{}\beta\geq\sigma,~{}|\gamma|\leq T\\}|.$

Note that $N_{\pi\times\pi^{\prime}}(\frac{1}{2},T)$ is roughly of size
$T\log(C(\pi)C(\pi^{\prime})T)$ via the argument principle and the functional
equation, and GRH can be restated as $N_{\pi\times\pi^{\prime}}(\sigma,T)=0$
for all $\sigma>\frac{1}{2}$. The zero density estimate

$N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{n,n^{\prime},[F:\mathbb{Q}]}(C(\pi)C(\pi^{\prime})T^{[F:\mathbb{Q}]})^{10^{7}(n^{\prime}n)^{4}(1-\sigma)}.$

follows from work of Soundararajan and Thorner [65, Corollary 2.6]. Therefore,
while an arbitrary Rankin–Selberg $L$-function $L(s,\pi\times\pi^{\prime})$
does not yet have a standard zero-free region, the proportion of zeros near
the line $\textup{Re}(s)=1$ is vanishingly small. It follows from work of
Brumley, Thorner, and Zaman [10, Theorem 1.3] that if
$\max\\{n,n^{\prime}\\}\leq 4$, then111Both [10] and [65] assumed that
$F=\mathbb{Q}$. The proofs can be made uniform over other number fields $F$
with minor modifications.

(1.2)
$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{n,n^{\prime},[F:\mathbb{Q}]}(C(\pi^{\prime})QT^{[F:\mathbb{Q}]})^{10^{7}(n^{\prime}n)^{4}(1-\sigma)}.$

If $\max\\{n,n^{\prime}\\}\geq 5$, then (1.2) holds under certain unproven
progress towards the generalized Ramanujan conjecture (GRC) (see (3.2)). When
$n\geq 1$ and $\pi^{\prime}\in\mathfrak{F}_{1}$ is the trivial representation,
Thorner and Zaman [66, Theorem 1.2] unconditionally proved (1.2). Their
methods are strong enough to prove (1.2) when $n\geq 1$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$ satisfies GRC.

We prove the first unconditional zero density estimate for any family of
Rankin–Selberg $L$-functions of the shape
$\\{L(s,\pi\times\pi^{\prime})\colon\pi\in\mathfrak{F}_{n}(Q)\\}$.

######  Theorem 1.1.

Let $n,n^{\prime}\geq 1$ and $\varepsilon>0$. If $\frac{1}{2}\leq\sigma\leq
1$, $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$, and $Q,T\geq 1$, then

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{n,n^{\prime},[F:\mathbb{Q}],\varepsilon}(C(\pi^{\prime})QT^{[F:\mathbb{Q}]})^{9(n+1)\max\\{n,n^{\prime}\\}(1-\sigma)+\varepsilon}.$

######  Remark.

Let $S\subseteq\mathfrak{F}_{n}$ (for example, a family of automorphic
representations in the sense of Sarnak, Shin, and Templier [63]) and
$S(Q)\coloneqq\\{\pi\in S\colon C(\pi)\leq Q\\}$. Theorem 1.1 immediately
implies that

(1.3) $\sum_{\pi\in
S(Q)}N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{n,n^{\prime},[F:\mathbb{Q}],\varepsilon}(C(\pi^{\prime})QT^{[F:\mathbb{Q}]})^{9(n+1)\max\\{n,n^{\prime}\\}(1-\sigma)+\varepsilon}.$

If there exists a constant $\delta=\delta_{n,[F:\mathbb{Q}]}>0$ such that
$|S(Q)|\gg_{n,F}Q^{\delta}$, then (1.3) is nontrivial when
$\sigma>1-\frac{\delta}{9(n+1)\max\\{n,n^{\prime}\\}}$. Brumley and Milićević
[9] proved (among other things) that $|\mathfrak{F}_{n}(Q)|\gg_{n,F}Q^{n+1}$,
but the flexibility of Theorem 1.1 for subfamilies follows from the upper
bound

(1.4)
$|\mathfrak{F}_{n}(Q)|\ll_{n,[F:\mathbb{Q}],\varepsilon}D_{F}^{-n^{2}}Q^{2n+\varepsilon},$

proved by Brumley, Thorner, and Zaman [10, Theorem A.1].222In [10, Appendix
A], the convention is that the analytic conductor does not include
$D_{F}$-dependence. Here, we include $D_{F}$ dependence in our definition of
the analytic conductor. This is why (1.4) visibly differs from in [10, Theorem
A.1].

The main new tool required to prove Theorem 1.1 is a large sieve inequality.
To describe this, let $\mathfrak{n}$ denote an integral ideal in the ring of
integers $\mathcal{O}_{F}$ of $F$, and we let $\mu_{F}(\mathfrak{n})$ be the
Möbius function on the integral ideals of $\mathcal{O}_{F}$. Let
$\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$. Let
$\lambda_{\pi}(\mathfrak{n})$ denote the $\mathfrak{n}$-th Hecke eigenvalue of
$\pi$, let $\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})$ denote the
$\mathfrak{n}$-th Dirichlet coefficient of $L(s,\pi\times\pi^{\prime})$, and
let $\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$ denote the $\mathfrak{n}$-th
Dirichlet coefficient of $L(s,\pi\times\pi^{\prime})^{-1}$.

We detect zeros of $L(s,\pi\times\pi^{\prime})M_{X}(s,\pi\times\pi^{\prime})$
instead of $L(s,\pi\times\pi^{\prime})$, where

(1.5) $M_{X}(s,\pi\times\pi^{\prime})=\sum_{\mathrm{N}\mathfrak{n}\leq
X}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{s}}$

approximates $L(s,\pi\times\pi^{\prime})^{-1}$. The zero detection process
pioneered by Montgomery [52] naturally leads to us bound an integral roughly
of the shape

(1.6)
$\int_{-T}^{T}\Big{|}\sum_{A<\mathrm{N}\mathfrak{n}<B}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+it}}\Big{|}^{2}dt.$

In order to provide a strong bound for (1.6) as one averages over
$\pi\in\mathfrak{F}_{n}(Q)$, one is led to prove a large sieve inequality
roughly of the form

(1.7)
$\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{x<\mathrm{N}\mathfrak{n}\leq
ex}\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})b(\mathfrak{n})\Big{|}^{2}\ll_{n,[F:\mathbb{Q}],\varepsilon}x^{1+\varepsilon}\sum_{x<\mathrm{N}\mathfrak{n}\leq
ex}|b(\mathfrak{n})|^{2},\qquad x>Q^{C},$

where $b$ is an arbitrary function and $C$ is a suitable constant.

If $\mathfrak{n}$ is squarefree and coprime to the product
$\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$ of the conductors of $\pi$ and
$\pi^{\prime}$, then
$\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})=\mu_{F}(\mathfrak{n})\lambda_{\pi}(\mathfrak{n})\lambda_{\pi^{\prime}}(\mathfrak{n})$.
Therefore, assuming that $L(s,\pi\times\pi^{\prime})$ is well-approximated by
its “unramified squarefree part” in the sense that we have

(1.8)
$L(s,\pi\times\pi^{\prime})\approx\prod_{\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}\Big{(}1+\frac{\lambda_{\pi}(\mathfrak{p})\lambda_{\pi^{\prime}}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}^{s}}\Big{)},\qquad\textup{Re}(s)\geq
1-\delta$

for some fixed $\delta=\delta_{n,n^{\prime},[F:\mathbb{Q}]}\in(0,1)$, where
the product is over prime ideals $\mathfrak{p}$ of $\mathcal{O}_{F}$ not
dividing $\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$, the sums in (1.5)
and (1.6) could be replaced by sums over squarefree integral ideals coprime to
$\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$. One might then appeal to
recent work of Thorner and Zaman [66, Theorem 1.1] and use their large sieve
inequality

(1.9)
$\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
x\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}\lambda_{\pi}(\mathfrak{n})b(\mathfrak{n})\Big{|}^{2}\ll_{n,[F:\mathbb{Q}],\varepsilon}x^{1+\varepsilon}\sum_{x<\mathrm{N}\mathfrak{n}\leq
ex}|b(\mathfrak{n})|^{2},\quad x>Q^{n^{2}+n}|\mathfrak{F}_{n}(Q)|.$

We could choose
$b(\mathfrak{n})=\lambda_{\pi^{\prime}}(\mathfrak{n})\mu_{F}(\mathfrak{n})a(\mathfrak{n})$,
where $a$ is another arbitrary function, in which case
$\lambda_{\pi}(\mathfrak{n})b(\mathfrak{n})=\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})a(\mathfrak{n})$.
Thus, for $x>Q^{n^{2}+n}|\mathfrak{F}_{n}(Q)|$, we would have that

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
ex\\\ \textup{$\mathfrak{n}$ squarefree}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})a(\mathfrak{n})\Big{|}^{2}\ll_{n,[F:\mathbb{Q}],\varepsilon}x^{1+\varepsilon}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
ex\\\ \textup{$\mathfrak{n}$
squarefree}\end{subarray}}\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})|a(\mathfrak{n})|^{2}.$

This would provide us with a version of (1.7) that is perfectly suited for our
purposes.

However, without substantial progress333The uniform bounds
$\theta_{n},\theta_{n^{\prime}}\leq\frac{1}{4}-\delta$ would suffice. See
(3.2). toward GRC beyond what is known unconditionally, the approximation
(1.8) is not yet known hold in a region of the shape $\textup{Re}(s)\geq
1-\delta$ when $n$ and $n^{\prime}$ are large. Therefore, we require a large
sieve inequality for $\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$ without the
highly convenient restriction to squarefree $\mathfrak{n}$. When
$\mathfrak{n}$ is not squarefree, the coefficients
$\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$ do not exhibit a clean
relationship with
$\lambda_{\pi}(\mathfrak{n})\lambda_{\pi^{\prime}}(\mathfrak{n})$ that we can
exploit. This renders (1.9) useless for our purposes. Despite this setback, we
prove the following result.

######  Theorem 1.2.

Let $b$ be a complex-valued function supported on the integral ideals of
$\mathcal{O}_{F}$. Let $n$ and $n^{\prime}$ be positive integers, and let
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$. Let $Q,T\geq 1$, $\varepsilon>0$,
and $x\geq 1$. Both

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})b(\mathfrak{n})\Big{|}^{2}\qquad\text{and}\qquad\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})b(\mathfrak{n})\Big{|}^{2}$

are

$\ll_{n,[F:\mathbb{Q}],\varepsilon}Q^{\varepsilon}\Big{(}\frac{x}{T}+Q^{n^{2}+n}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}|\mathfrak{F}_{n}(Q)|\Big{)}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})|b(\mathfrak{n})|^{2}.$

######  Remark.

Theorem 1.2 recovers (1.9) when $\pi^{\prime}=\mathbbm{1}$ and $T=1$.

Theorem 1.2 provides the first nontrivial unconditional mean value estimates
for the Dirichlet coefficients $\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})$
or $\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$ where $n$ and $n^{\prime}$ are
allowed to be arbitrary. Theorem 1.2 follows from a more general large sieve
inequality (Proposition 4.1) for sequences of products of Schur polynomials
evaluated on the set $A_{\pi}(\mathfrak{p})$ of Satake parameters of $\pi$. We
use explicit relationships amongst $A_{\pi}(\mathfrak{p})$,
$A_{\pi^{\prime}}(\mathfrak{p})$,
$\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})$, and
$\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$ arising from Cauchy’s identities
to circumvent the need for an approximation like (1.8). The large sieve
inequalities in Theorem 1.2 are not sharp in the $Q$-aspect, but they are
sharp in the length-aspect once $x$ is a suitably large power of $Q$ (this
uses (1.4)) while $T$ is a suitably small power of $x$. This suffices to prove
Theorem 1.1.

##  2\. Applications

We will now describe some applications of Theorem 1.1.

###  2.1. Notation

In what follows, we write $f\ll_{\nu}g$, $f=O_{\nu}(g)$, and $f\gg_{\nu}g$ to
denote that there exists a constant $c>0$ such that $|f|\leq c|g|$ in the
stated range. The implied constant $c$, which is effectively computable unless
otherwise stated, will depend at most on $\nu$, $n$, $n^{\prime}$, and
$[F:\mathbb{Q}]$. The expression $f\asymp_{\nu}g$ means that $f\ll_{\nu}g$ and
$g\ll_{\nu}f$. We use $\varepsilon>0$ to denote an arbitrarily small quantity
that depends at most on $n$, $n^{\prime}$, and $[F:\mathbb{Q}]$.

###  2.2. Strong zero-free region

Let $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$.
If $\pi^{\prime}\in\\{\widetilde{\pi},\widetilde{\pi}^{\prime}\\}$, then
$L(s,\pi\times\pi^{\prime})$ has the standard zero-free region (1.1) apart
from a possible Landau–Siegel zero, which must be both real and simple. In all
other cases, it follows from work of Brumley (see [8], [29, Section 1], and
[41, Appendix]) that there exists an absolute and effectively computable
constant $\Cl[abcon]{Brumley}>0$ such that $L(s,\pi\times\pi^{\prime})\neq 0$
in the much narrower region

(2.1) $\textup{Re}(s)\geq
1-\frac{\Cr{Brumley}}{(C(\pi)C(\pi^{\prime})(3+|\textup{Im}(s)|)^{[F:\mathbb{Q}]})^{4n^{\prime}n}}.$

Theorem 1.1 implies that aside from a few exceptional
$\pi\in\mathfrak{F}_{n}(Q)$, a much stronger zero-free region for
$L(s,\pi\times\pi^{\prime})$ holds, regardless of whether
$\pi^{\prime}\in\\{\widetilde{\pi},\widetilde{\pi}^{\prime}\\}$.

######  Theorem 2.1.

Let $\varepsilon>0$, $A>0$, $n,n^{\prime}\geq 1$, and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$. For all except
$O_{A,\varepsilon}((C(\pi^{\prime})Q)^{\varepsilon})$ of the
$\pi\in\mathfrak{F}_{n}(Q)$, the $L$-function $L(s,\pi\times\pi^{\prime})$ is
nonzero in the region

$1-\textup{Re}(s)\leq\begin{cases}\displaystyle\frac{\varepsilon\log
C(\pi)}{19n\max\\{n,n^{\prime}\\}\log(C(\pi)C(\pi^{\prime})(3+|\textup{Im}(s)|)^{[F:\mathbb{Q}]})}&\mbox{if
$|\textup{Im}(s)|\leq C(\pi)^{A}$,}\vspace{0.1in}\\\
\displaystyle\frac{\Cr{Brumley}}{(C(\pi)C(\pi^{\prime})(3+|\textup{Im}(s)|)^{[F:\mathbb{Q}]})^{4n^{\prime}n}}&\mbox{if
$|\textup{Im}(s)|>C(\pi)^{A}$.}\end{cases}$

######  Remark.

The same conclusion holds when $\mathfrak{F}_{n}(Q)$ is replaced by a
subfamily $S(Q)$ that grows polynomially in $Q$.

###  2.3. Effective multiplicity one

Let $\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)$ have (restricted) tensor product
representations $\pi=\bigotimes_{v}\pi_{v}$ and
$\pi^{\prime}=\bigotimes_{v}\pi_{v}^{\prime}$. Assume GRH for
$L(s,\pi\times\widetilde{\pi})$ and $L(s,\pi\times\pi^{\prime})$ and that
$\pi_{\mathfrak{p}}$ and $\pi_{\mathfrak{p}}^{\prime}$ are tempered for all
prime ideals $\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$. If
$Y=(\log Q)^{2}$ and $\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}^{\prime}$
$\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$ with
$\mathrm{N}\mathfrak{p}\ll Y$, then $\pi=\pi^{\prime}$. Graham and Ringrose
[24] proved that primitive Dirichlet characters to modulus at most $Q$ are not
determined by their values at primes $p\leq(\log Q)\log\log\log Q$. This is
can be extended to $\mathrm{GL}_{2}$ via work of Humphries [28, Section 6.2]
and a twisting argument. Therefore, we expect $Y\ll_{\varepsilon}(\log
Q)^{1+\varepsilon}$ to suffice.

Brumley [8], improving on work of Moreno [53], showed that there exists a
constant $B_{n}>0$ such that one may take $Y=Q^{B_{n}}$ with no assumptions of
GRH or temperedness. This result makes effective the multiplicity one theorems
of Jacquet and Shalika [35, Theorem 4.8] and Piatetski-Shapiro [60].
Currently, any fixed $B_{n}>2n$ suffices [45].

When $n=2$, we have an average result that is commensurate with what GRH
predicts. Specifically, let $\mathfrak{F}_{2}^{\flat}$ be the subset of
$\pi\in\mathfrak{F}_{2}$ with squarefree conductor and trivial central
character, and let $\pi^{\prime}\in\mathfrak{F}_{2}^{\flat}$. For all
$\varepsilon>0$, there exists an effectively computable constant
$N_{\varepsilon}>0$, depending at most on $\varepsilon$ and $[F:\mathbb{Q}]$,
such that

(2.2)
$|\\{\pi\in\mathfrak{F}_{2}^{\flat}(Q)\colon\textup{$\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}^{\prime}$
for all $\mathfrak{p}\nmid\mathfrak{q}_{\pi}$ with
$\mathrm{N}\mathfrak{p}\leq(\log
Q)^{N_{\varepsilon}}$}\\}|\ll_{\varepsilon}Q^{\varepsilon}.$

In particular, the implied constant does not depend on $\pi^{\prime}$. This
was proved by Duke and Kowalski [17, Theorem 3] when $F=\mathbb{Q}$ in a
stronger form under the assumption of GRC. See Brumley’s Ph.D. thesis [7,
Corollary 5.2.2] for a proof that does not use GRC.

If $\pi\in\mathfrak{F}_{2}$ and $\widetilde{\pi}\in\mathfrak{F}_{2}$ is the
contragredient representation, then $\pi\otimes\widetilde{\pi}$ exhibits an
isobaric decomposition $\mathbbm{1}\boxplus\mathrm{Ad}^{2}\pi$, where
$\mathrm{Ad}^{2}\pi$ is the adjoint square lift of $\pi$ from
$\mathrm{GL}_{2}$ to $\mathrm{GL}_{3}$. The arguments in [7, 17] rely on two
key properties of $\mathrm{Ad}^{2}\pi$ for $\pi\in\mathfrak{F}_{2}$:

  1. (1)

Gelbart and Jacquet [20] proved that $\mathrm{Ad}^{2}\pi$ is an isobaric
automorphic representation of $\mathrm{GL}_{3}(\mathbb{A}_{F})$, complete with
a comprehensive criterion by which one can determine whether
$\mathrm{Ad}^{2}\pi$ is cuspidal (and hence lies in $\mathfrak{F}_{3}$).

  2. (2)

If $\mathfrak{F}_{2}^{\flat}(Q)\subseteq\mathfrak{F}_{2}(Q)$ is the subset of
nondihedral $\pi$, then the preimage of the map
$\mathrm{Ad}^{2}\colon\mathfrak{F}_{2}^{\flat}(Q)\to\mathfrak{F}_{3}(Q^{2})$
has cardinality $O_{\varepsilon}(Q^{1/2+\varepsilon})$ for all
$\varepsilon>0$.

In generalizing the strategy of Duke and Kowalski to $\mathrm{GL}_{n}$ for
$n\geq 3$, one encounters some deep open problems. If
$\pi\in\mathfrak{F}_{n}$, then for $\textup{Re}(s)$ sufficiently large,
$L(s,\pi\times\widetilde{\pi})$ factors as
$\zeta_{F}(s)L(s,\pi,\mathrm{Ad}^{2})$, where $\mathrm{Ad}^{2}$ is the adjoint
square lift from $\mathrm{GL}_{n}$ to $\mathrm{GL}_{n^{2}-1}$ and
$\zeta_{F}(s)$ is the Dedekind zeta function of $F$. Apart from some special
cases, the following obstacles arise:

  1. (1)

The adjoint square lift is not yet known to be automorphic for $n\geq 3$, and
if it were, there is no known criterion for cuspidality.

  2. (2)

It is not known when the preimage of the lifting
$\mathrm{Ad}^{2}\colon\mathfrak{F}_{n}\to\mathfrak{F}_{n^{2}-1}$ is small.

Despite these setbacks, we can use Theorem 2.1 to prove a $\mathrm{GL}_{n}$
analogue of (2.2) when $\pi^{\prime}$ is fixed. In fact, using recent work of
the authors [30], we can allow $C(\pi^{\prime})$ to grow mildly with $Q$ while
keeping all implied constants effective.

######  Theorem 2.2.

Let $\varepsilon>0$, $n\geq 1$, and $\pi^{\prime}\in\mathfrak{F}_{n}$. Let $Q$
be sufficiently large with respect to $\varepsilon$. There exists an absolute
and effectively computable constant $\Cl[abcon]{multone_{1}}>0$ such that if
$C(\pi^{\prime})\ll_{\varepsilon}(\log
Q)^{\Cr{multone_{1}}/(n^{4}[F:\mathbb{Q}]^{2})}$, then

$|\\{\pi\in\mathfrak{F}_{n}(Q)\colon\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}^{\prime}~{}\textup{for
all}~{}\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}~{}\textup{with}~{}\mathrm{N}\mathfrak{p}\leq(\log
Q)^{41n^{2}/\varepsilon}\\}|\ll_{\varepsilon}Q^{\varepsilon}.$

######  Remark.

The same conclusion holds when $\mathfrak{F}_{n}(Q)$ is replaced by a
subfamily $S(Q)$ that grows polynomially in $Q$.

The proof is very flexible. For instance, if $\varepsilon>0$ and
$C(\pi^{\prime})\ll_{\varepsilon}Q^{\Cr{multone_{1}}\varepsilon^{2}/(n^{4}[F:\mathbb{Q}]^{2})}$,
then the same proof with minor changes in choices of parameters produces the
bound

$|\\{\pi\in\mathfrak{F}_{n}(Q)\colon\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}^{\prime}~{}\textup{for
all}~{}\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}~{}\textup{with}~{}\mathrm{N}\mathfrak{p}\leq
Q^{\varepsilon}\\}|\ll_{\varepsilon}Q^{\varepsilon}.$

While the number of $\mathfrak{p}$ for which one needs to check that
$\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}^{\prime}$ is larger, the range of
$C(\pi^{\prime})$ is greatly extended, and the threshold
$\mathrm{N}\mathfrak{p}\leq Q^{\varepsilon}$ (reminiscent of Vinogradov’s
conjecture on the size of the least quadratic nonresidue) still greatly
improves on $\mathrm{N}\mathfrak{p}\ll_{\varepsilon}Q^{2n+\varepsilon}$ from
[45].

###  2.4. Automorphic level of distribution

Let $\Lambda(m)$ be the von Mangoldt function, equal to $\log p$ if $m$ is a
power of a prime $p$ and zero otherwise. The celebrated Bombieri–Vinogradov
theorem states that if $\theta<\frac{1}{2}$ is fixed, then for all $A>0$, we
have

(2.3) $\sum_{q\leq x^{\theta}}~{}\max_{\gcd(a,q)=1}\max_{y\leq
x}\Big{|}\sum_{\begin{subarray}{c}m\leq y\\\ m\equiv
a\,(\mathrm{mod}\,\,q)\end{subarray}}\Lambda(m)-\frac{y}{\varphi(q)}\Big{|}\ll_{A}\frac{x}{(\log
x)^{A}}.$

This may be viewed as an average form of GRH for Dirichlet $L$-functions. As
part of his proof [6], Bombieri proved a strong form of the zero density
estimate in Theorem 1.1 for Dirichlet $L$-functions. We call any $\theta$ for
which (2.3) holds a level of distribution for the primes. Elliott and
Halberstam conjectured that any fixed $\theta<1$ is a level of distribution
for the primes.

Number theorists have proved several interesting extensions and variations of
(2.3). For example, Murty and Murty [55] proved that primes in the Chebotarev
density theorem have a positive level of distribution. To describe a different
direction for automorphic representations over $\mathbb{Q}$, we let $m\geq 2$
and consider $\pi\in\mathfrak{F}_{n}$ with conductor $q_{\pi}$. Let
$\Lambda(m)$ be the von Mangoldt function, and define the numbers $a_{\pi}(m)$
by

$\sum_{n=1}^{\infty}\frac{a_{\pi}(m)\Lambda(m)}{m^{s}}=-\frac{L^{\prime}}{L}(s,\pi)=\sum_{p}\sum_{k=1}^{\infty}\frac{\sum_{j=1}^{n}\alpha_{j,\pi}(p)^{k}\log
p}{p^{ks}},\qquad\textup{Re}(s)>1.$

Note that $a_{\pi}(p)=\lambda_{\pi}(p)$. For fixed $\theta<\frac{1}{n^{2}-2}$,
Wong [71, Theorem 9] proved that if $\pi$ satisfies GRC and
$L(s,\pi\times(\widetilde{\pi}\otimes\chi))$ has no Landau–Siegel zero for all
primitive Dirichlet characters $\chi$, then for any $A>0$,

(2.4) $\sum_{q\leq x^{\theta}}~{}\max_{\gcd(a,q)=1}\max_{y\leq
x}\Big{|}\sum_{\begin{subarray}{c}m\leq y\\\ \gcd(m,q_{\pi})=1\\\ m\equiv
a\,(\mathrm{mod}\,\,q)\end{subarray}}|a_{\pi}(m)|^{2}\Lambda(m)-\frac{y}{\varphi(q)}\Big{|}\ll_{A,\pi}\frac{x}{(\log
x)^{A}}.$

This conditionally endows $\pi\times\widetilde{\pi}$ with a positive level of
distribution $\theta$. In particular, (2.4) holds unconditionally (with
$\theta<\frac{1}{2}$) for $\pi$ associated to nondihedral holomorphic cuspidal
Hecke eigenforms on congruence subgroups of $\mathrm{SL}_{2}(\mathbb{Z})$.

Let $\pi\in\mathfrak{F}_{n}$. Using Theorem 1.1, we unconditionally endow
$\pi\times\widetilde{\pi}$ with a notion of positive level of distribution. In
particular, we avoid recourse to unproven progress toward GRC or the absence
of Landau–Siegel zeros.

######  Theorem 2.3.

Let $F=\mathbb{Q}$ and $\pi\in\mathfrak{F}_{n}$. Fix $\theta<1/(9n^{2}(n+1))$.
If $A>0$, then

$\sum_{\begin{subarray}{c}q\leq x^{\theta}\\\
\gcd(q,q_{\pi})=1\end{subarray}}\max_{\gcd(a,q)=1}\max_{y\leq
x}\Big{|}\sum_{\begin{subarray}{c}m\leq y\\\ m\equiv
a\,(\mathrm{mod}\,\,q)\end{subarray}}|a_{\pi}(m)|^{2}\Lambda(m)-\frac{y}{\varphi(q)}\Big{|}\ll_{A,\pi}\frac{x}{(\log
x)^{A}}.$

If $n\leq 4$, then for the same range of $\theta$, we have

$\sum_{\begin{subarray}{c}q\leq x^{\theta}\\\
\gcd(q,q_{\pi})=1\end{subarray}}\max_{\gcd(a,q)=1}\max_{y\leq
x}\Big{|}\sum_{\begin{subarray}{c}p\leq y\\\ p\equiv
a\,(\mathrm{mod}\,\,q)\end{subarray}}|\lambda_{\pi}(p)|^{2}\log
p-\frac{y}{\varphi(q)}\Big{|}\ll_{A,\pi}\frac{x}{(\log x)^{A}}.$

The implied constants are ineffective.

######  Remark.

The theorem remains the same when $|a_{\pi}(m)|^{2}$ (respectively
$|\lambda_{\pi}(p)|^{2}$) is replaced by $a_{\pi\times\widetilde{\pi}}(m)$
(respectively $\lambda_{\pi\times\widetilde{\pi}}(p)$). Also, the theorem
remains the same when the sum over $m$ (respectively $p$) has the additional
constraint that $m$ (respectively $p$) is coprime to $q_{\pi}$. With
additional work, the condition that $\gcd(q,q_{\pi})=1$ can be removed under
the additional hypothesis that $\pi\neq\pi\otimes\chi$ for all primitive
quadratic Dirichlet characters.

In addition to Theorem 1.1, we also require a standard zero-free region and
some understanding of possible Landau–Siegel zeros associated to the twists
$L(s,\pi\times(\widetilde{\pi}\otimes\chi))$ as $\chi$ varies among primitive
Dirichlet characters. A standard zero-free region for
$L(s,\pi\times\widetilde{\pi})$ (apart from a possible Landau–Siegel zero) has
been known for a long time when $\pi=\widetilde{\pi}$ (see [53], for example),
but such a zero-free region when $\pi\neq\widetilde{\pi}$ was only recently
proved by the authors [30]. Informed by the ideas therein, we prove the
following result, which appears to be new to the literature and might be of
independent interest.

######  Theorem 2.4.

Let $Q\geq 3$ and $\pi\in\mathfrak{F}_{n}$. There exists an effectively
computable constant
$\Cl[abcon]{ZFR_{B}V_{f}inal}=\Cr{ZFR_{B}V_{f}inal}(\pi)>0$ such that for all
primitive Dirichlet characters $\chi\,(\mathrm{mod}\,\,q)$ with $q\leq Q$ and
$\gcd(q,q_{\pi})=1$ apart from at most one exception, the $L$-function
$L(s,\pi\times(\widetilde{\pi}\otimes\chi))$ does not vanish in the region

$\textup{Re}(s)\geq
1-\frac{\Cr{ZFR_{B}V_{f}inal}}{\log(Q(|\textup{Im}(s)|+3))}.$

If the exceptional character $\chi_{1}\,(\mathrm{mod}\,\,q_{1})$ exists, then
$L(s,\pi\times(\widetilde{\pi}\otimes\chi_{1}))$ has at most one zero, say
$\beta_{1}$, in this region; $\beta_{1}$ is both real and simple; and
$\chi_{1}$ must be quadratic. Moreover, for all $\varepsilon>0$, there exists
an ineffective constant $c_{\pi}(\varepsilon)>0$ such that $\beta_{1}\leq
1-c_{\pi}(\varepsilon)q_{1}^{-\varepsilon}$.

######  Remark.

Theorem 2.4 implies that if $\pi\in\mathfrak{F}_{n}$ satisfies
$\widetilde{\pi}\otimes\psi\neq\pi$ for all primitive characters $\psi$ of
order at most 2, then each $L$-function in the family
$\\{L(s,\pi\times(\widetilde{\pi}\otimes\chi))\colon\textup{$\chi$
quadratic}\\}$ has a standard zero-free region and is of the form
$L(s,\pi_{1}\times\pi_{2})$ with
$\pi_{2}\notin\\{\widetilde{\pi}_{1},\widetilde{\pi}_{2}\\}$. This appears to
be the first such infinite family of $\mathrm{GL}_{n}\times\mathrm{GL}_{n}$
Rankin–Selberg $L$-functions when $n\geq 3$.

###  2.5. Bounds for Rankin–Selberg $L$-functions

It is a classical problem for Dirichlet $L$-functions to find strong bounds on
the critical line $\textup{Re}(s)=\frac{1}{2}$. The Phragmén–Lindelöf
convexity principle shows that if $q_{\chi}$ is the conductor of a primitive
Dirichlet character $\chi$, then $|L(\tfrac{1}{2},\chi)|\ll q^{1/4}$;
improving this bound by replacing $1/4$ with a smaller exponent is known as a
_subconvex_ bound. The multiplicative version of the classical large sieve
inequality combined with an approximate functional equation shows that for
almost all $\chi$, we have the bound
$|L(\frac{1}{2},\chi)|\ll_{\varepsilon}q_{\chi}^{\varepsilon}$ for all
$\varepsilon>0$, consistent with the generalized Lindelöf hypothesis (GLH). Up
to the explication of the $\varepsilon$, the bound
$|L(\frac{1}{2},\chi)|\ll_{\varepsilon}q_{\chi}^{\varepsilon}$ is optimal and
follows from GRH.

For $\pi\in\mathfrak{F}_{2}$, GLH predicts that
$|L(\tfrac{1}{2},\pi)|\ll_{\varepsilon}C(\pi)^{\varepsilon}$. Michel and
Venkatesh [49] proved that there exists a fixed positive $\delta>0$ such that
$|L(\frac{1}{2},\pi)|\ll_{F}C(\pi)^{1/4-\delta}$. When $F=\mathbb{Q}$, a sharp
mean value estimate for Hecke eigenvalues proved by Deshouillers and Iwaniec
[16], in conjunction with the approximate functional equation, implies the
bound $|L(\frac{1}{2},\pi)|\ll_{\varepsilon}(qT)^{\varepsilon}$ for almost all
$\pi\in\mathfrak{F}_{2}$ of arithmetic conductor $q$, trivial central
character, and whose archimedean part $C(\pi)/q$ of the analytic conductor
$C(\pi)$ lies in the dyadic interval $[T,2T]$.

For $\pi\in\mathfrak{F}_{n}$ with $n\geq 3$, the best uniform result towards
the bound $|L(\frac{1}{2},\pi)|\ll_{\varepsilon}C(\pi)^{\varepsilon}$
predicted by GLH is that of Thorner and Soundararajan [65, Corollary 2.7],
namely

(2.5) $|L(\tfrac{1}{2},\pi)|\ll\frac{C(\pi)^{\frac{1}{4}}}{(\log
C(\pi))^{1/(10^{17}n^{3})}}.$

We mention three results that improve on (2.5) in an average sense, each
having complementary strengths. Jana [36, Theorem 6] extended the GLH-on-
average bound of Deshouillers and Iwaniec to the family of cuspidal
automorphic representations of $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ of
arithmetic conductor 1. Blomer [4, Corollary 5] proved the corresponding
result for the family of cuspidal automorphic representations of
$\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ of a large given arithmetic
conductor $q$, trivial central character, and whose archimedean components are
principal series representations such that the associated Laplace eigenvalue
is bounded. Thorner and Zaman [66, Theorem 1.3] proved that there exists a
constant $\Cl[abcon]{TZ_{g}ln}=\Cr{TZ_{g}ln}(n,[F:\mathbb{Q}])>0$ such that if
$0\leq\delta\leq 1$, then

(2.6)
$|\\{\pi\in\mathfrak{F}_{n}(Q)\colon|L(\tfrac{1}{2},\pi)|\geq\Cr{TZ_{g}ln}C(\pi)^{\frac{1}{4}-\frac{\delta}{10^{16}n^{3}}}\\}|\ll|\mathfrak{F}_{n}(Q)|^{\delta}.$

Unlike the preceding results, (2.6) is uniform in both the arithmetic
conductor and spectral aspects, but the savings over (2.5) is not comparable
to GLH on average.

Given $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$,
the best bound for Rankin–Selberg $L$-functions on the critical line is due to
Soundararajan and Thorner [65, Corollary 2.7], who proved when $F=\mathbb{Q}$
that

(2.7)
$|L(\tfrac{1}{2},\pi\times\pi^{\prime})|\ll|L(\tfrac{3}{2},\pi\times\pi^{\prime})|^{2}\frac{C(\pi\times\pi^{\prime})^{\frac{1}{4}}}{(\log
C(\pi\times\pi^{\prime}))^{1/(10^{17}(n^{\prime}n)^{3})}}.$

The best general upper bound that is currently known for
$|L(\tfrac{3}{2},\pi\times\pi^{\prime})|^{2}$ is larger than any fixed power
of $\log C(\pi\times\pi^{\prime})$ [42, Theorem 2]. The factor of
$|L(\tfrac{3}{2},\pi\times\pi^{\prime})|^{2}$ can be removed under partial
progress toward GRC, which is available in some special cases. The bound
$|L(\frac{1}{2},\pi\times\pi^{\prime})|\ll_{\varepsilon}(C(\pi\times\pi^{\prime}))^{\varepsilon}$
is predicted by GLH.

In order to improve (2.7) on average with uniformity in $\pi$ and
$\pi^{\prime}$, one might first try to mimic the approach that worked well for
Dirichlet $L$-functions and $\mathrm{GL}_{2}$ $L$-functions using trace
formulae, approximate functional equations, the spectral large sieve, Voronoĭ
summation, etc. While such methods have seen great success for
$\mathrm{GL}_{n}\times\mathrm{GL}_{n^{\prime}}$ with
$n,n^{\prime}\in\\{1,2\\}$, suitably uniform and flexible versions of these
tools do not appear to be available yet in the general setting. Using a period
integral approach that completely avoids the aforementioned tools, Blomer [3,
Theorem 2] proved an optimal second moment bound for
$|L(\frac{1}{2},f\times\overline{g})|$ in the special case where $f$ and $g$
are extremely “spectrally close”. Here, $f$ is a tempered, spherical
Hecke–Maaß newform for the group $\mathrm{SL}_{n}(\mathbb{Z})$ with spectral
parameter $\vec{\nu}_{f}\in i\mathbb{R}^{n-1}$ and $g$ varies over an
orthogonal basis $\mathcal{B}$ of tempered Hecke–Maaß newforms $g$ for the
group $\mathrm{SL}_{n}(\mathbb{Z})$ with spectral parameters $\vec{\nu}_{g}\in
i\mathbb{R}^{n-1}$ satisfying $\|\vec{\nu}_{f}-\vec{\nu}_{g}\|\leq 1$; this
last condition ensures that $f$ and $g$ are spectrally close, so that the
analytic conductor of $f\times\overline{g}$ exhibits extreme conductor-
dropping. A consequence of Blomer’s result is that if $\delta>0$ and
$\varepsilon>0$, then

(2.8) $|\\{g\in\mathcal{B}\colon\|\vec{\nu}_{f}-\vec{\nu}_{g}\|\leq
1,~{}|L(\tfrac{1}{2},f\times
g)|\geq\|\vec{\nu}_{f}\|^{\delta}\\}|\ll_{\varepsilon}\|\vec{\nu}_{f}\|^{1-2\delta+\varepsilon}.$

Since $|\\{g\in\mathcal{B}\colon\|\vec{\nu}_{f}-\vec{\nu}_{g}\|\leq
1\\}|\asymp\|\vec{\nu}_{f}\|$ by Weyl’s law for $\mathrm{SL}_{n}$, the
generalized Lindelöf hypothesis for $|L(\frac{1}{2},f\times g)|$ holds on
average over the $g$ that are spectrally close to $f$.

Along the same lines as (2.6), we use Theorem 1.1 to prove the following
result.

######  Theorem 2.5.

Let $n,n^{\prime}\geq 1$ be integers, and let
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$ and $t\in\mathbb{R}$. Let
$C(\pi\times\pi^{\prime},t)$ be defined as in (3.8). For all $\varepsilon>0$,
there exists an effectively computable constant
$\Cl[abcon]{subcon_{c}onst}=\Cr{subcon_{c}onst}(n,n^{\prime},[F:\mathbb{Q}],\varepsilon)>0$
such that

$|\\{\pi\in\mathfrak{F}_{n}(Q)\colon|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|\geq\Cr{subcon_{c}onst}C(\pi\times\pi^{\prime},t)^{\frac{1}{4}-\frac{\varepsilon}{10^{11}n\max\\{n,n^{\prime}\\}}}\\}|\ll_{\varepsilon}(C(\pi^{\prime})(3+|t|)^{[F:\mathbb{Q}]}Q)^{\varepsilon}.$

######  Remark.

The same conclusion holds when $\mathfrak{F}_{n}(Q)$ is replaced by a
subfamily $S(Q)$ that grows polynomially in $Q$.

In contrast with Blomer’s work, Theorem 2.3 provides a much smaller power-
saving improvement over (2.7), but the improvement is uniform in the
arithmetic conductor and spectral aspects. The exceptional set in Theorem 2.5
is a much smaller than that of (2.8). Finally, Theorem 2.5 removes the
requirements that $\pi$ and $\pi^{\prime}$ be “spectrally close”,
$F=\mathbb{Q}$, $n=n^{\prime}$, and $q_{\pi}=q_{\pi^{\prime}}=1$.

###  2.6. Hybrid-aspect averaged discrepancy bounds

Shnirelman [67], Colin de Verdière [15], and Zelditch ([74] in the compact
case, [75] in the noncompact case) proved the quantum analogue of geodesic
flow on a finite volume Riemannian manifold $X$ being ergodic. To be more
specific, let $\Delta$ denote the Laplace–Beltrami operator on $X$, and let
$(\varphi_{k})_{k=1}^{\infty}$ be an orthonormal basis of $L^{2}$-integrable
eigenfunctions of $\Delta$. Let $\lambda_{k}$ be the Laplace eigenvalue of
$\varphi_{k}$; we may order $(\varphi_{k})_{k=1}^{\infty}$ so that
$(\lambda_{k})_{k=1}^{\infty}$ is monotonically nondecreasing. Consider the
probability measures $d\mu_{k}\coloneqq|\varphi_{k}|^{2}d\mu$, where $d\mu$ is
the volume form on $X$. If the geodesic flow on the unit cotangent bundle is
ergodic (which happens, for instance, when $X$ has negative curvature), then
there exists a density one subsequence $\varphi_{k_{n}}$ along which

$\lim_{n\to\infty}\mu_{k_{n}}(A)=\frac{\mu(A)}{\mu(X)}$

when $A$ is a “nice” set (e.g., compact sets with smooth boundary or geodesic
balls). This has been termed quantum ergodicity. Rudnick and Sarnak [62] have
conjectured that

$\lim_{k\to\infty}\mu_{k}(A)=\frac{\mu(A)}{\mu(X)}$

when $X$ has negative curvature, which would alleviate the need to pass to a
density one subsequence. This has been termed quantum unique ergodicity.

Let $\Gamma=\mathrm{SL}_{2}(\mathbb{Z})$. The orbifold
$\Gamma\backslash\mathbb{H}$ has attracted much attention because of its
arithmetic structure. The volume form $d\mu(z)$ is the measure
$y^{-2}\,dx\,dy$ (where $z=x+iy$), and
$\mu(\Gamma\backslash\mathbb{H})=\frac{\pi}{3}$. Let
$(\varphi_{k})_{k=1}^{\infty}$ denote an orthonormal basis of Maaß cusp forms
satisfying $\Delta\varphi_{k}(z)=\lambda_{\varphi}\varphi_{k}(z)$, where
$\Delta=-y^{2}(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}})$. We may diagonalize the space of
Maaß cusp forms so that we may take each $\varphi_{k}$ to be a newform.
Necessarily, each such $\varphi_{k}$ is an eigenfunction of all Hecke
operators and the involution
$\varphi_{k}(z)\mapsto\varphi_{k}(-\overline{z})$. We call such an
eigenfunction a Hecke–Maaß newform. It is expected that the cuspidal spectrum
of $\Gamma\backslash\mathbb{H}$ is simple, in which case every Maaß cusp form
is a Hecke–Maaß newform.

Let $\mathbf{1}_{B}$ be the indicator function for an injective geodesic ball
$B$ in $\Gamma\backslash\mathbb{H}$, and for a probability measure $\nu$ on
$\Gamma\backslash\mathbb{H}$, define the discrepancy

(2.9)
$D(\nu)\coloneqq\sup_{B\subseteq\Gamma\backslash\mathbb{H}}\Big{|}\nu(\mathbf{1}_{B})-\frac{3}{\pi}\mu(\mathbf{1}_{B})\Big{|}.$

If $D(\mu_{k})\to 0$ as $k\to\infty$, as predicted by the quantum unique
ergodicity conjecture, then the measures $d\mu_{k}$ converge in the weak-*
topology to $(3/\pi)d\mu$. The rate of decay of $D(\mu_{k})$ as $k\to\infty$
then quantifies the rate of convergence. Watson’s thesis [69] shows that GLH
for certain families of $\mathrm{GL}_{1}\times\mathrm{GL}_{3}$ and
$\mathrm{GL}_{2}\times\mathrm{GL}_{3}$ Rankin–Selberg $L$-functions is
essentially equivalent to

(2.10)
$\mu_{k}(\mathbf{1}_{B})=\frac{3}{\pi}\mu(\mathbf{1}_{B})+O_{B,\varepsilon}(\lambda_{k}^{-\frac{1}{4}+\varepsilon})$

for every fixed injective geodesic ball
$B\subseteq\Gamma\backslash\mathbb{H}$. Young refines this by explicating the
dependence on $B$ in the error term in (2.10) under the assumption of GLH [73,
Proposition 1.5]. In particular, he conditionally proves small scale quantum
unique ergodicity, namely that the error term in (2.10) is smaller than the
main term so long as $\mu(\mathbf{1}_{B})$ is a little larger than
$\lambda_{k}^{-1/6}$. Moreover, even if $\mu(\mathbf{1}_{B})$ is smaller than
$\lambda_{k}^{-1/6}$, he obtains bounds for the error term in (2.10) uniform
in $B$ that are strong enough to imply the optimal bound
$D(\mu_{k})\ll_{\varepsilon}\lambda_{k}^{-1/4+\varepsilon}$, thereby resolving
under GLH a conjecture of Luo and Sarnak [48, p. 210].

Unconditionally, there are no known individual bounds $D(\mu_{k})$ other than
the work of Lindenstrauss [43] and Soundararajan [64], which shows that

$\lim_{k\to\infty}D(\mu_{k})=0$

with an unspecified rate of convergence. On the other hand, unconditional
bounds for $D(\mu_{k})$ on average are quite strong. Luo and Sarnak [48,
Theorem 1.5] proved a strong quantitative version of Zelditch’s quantum
ergodicity result, namely

(2.11) $\frac{1}{|\\{\lambda_{k}\leq T\\}|}\sum_{\lambda_{k}\leq
T}D(\mu_{k})^{2}\ll_{\varepsilon}T^{-\frac{1}{21}+\varepsilon}.$

Chebyshev’s inequality then implies that for fixed positive real numbers
$\alpha$ and $\beta$ satisfying $2\alpha+\beta<1/21$, we have that
$D(\mu_{k})\leq\lambda_{k}^{-\alpha}$ for all $\varphi_{k}$ with
$\lambda_{k}\leq T$ outside of an exceptional set of density $T^{-\beta}$.

For an integer $q\geq 1$, consider the orbifold
$\Gamma_{0}(q)\backslash\mathbb{H}$ associated to the congruence subgroup
$\Gamma_{0}(q)$ of $\mathrm{SL}_{2}(\mathbb{Z})$. Let $\varphi$ be a weight
zero Hecke–Maaß newform of level $q_{\varphi}$ on
$\Gamma_{0}(q_{\varphi})\backslash\mathbb{H}$ with trivial nebentypus and
Laplace eigenvalue $\lambda_{\varphi}>0$. The pushforward to
$\Gamma\backslash\mathbb{H}$ of the $L^{2}$-mass of $\varphi$ (weighted by
$H$) is the finite measure on $\Gamma\backslash\mathbb{H}$ given by

$\mu_{\varphi}(H)\coloneqq\int_{\Gamma_{0}(q_{\varphi})\backslash\mathbb{H}}|\varphi(z)|^{2}H(z)d\mu(z)$

for each bounded measurable function $H$ on $\Gamma\backslash\mathbb{H}$. We
use a slightly weaker though more convenient variant of Theorem 2.5 (see
Corollary 6.1 below) to prove the following result.

######  Theorem 2.6.

Let $\varphi$ denote a Hecke–Maaß newform with trivial nebentypus, Laplace
eigenvalue $\lambda_{\varphi}$, and level $q_{\varphi}$. Define
$\mathpzc{F}(Q)\coloneqq\\{\varphi\colon\lambda_{\varphi}q_{\varphi}\leq
Q\\}$. We have

$\frac{1}{|\mathpzc{F}(Q)|}\sum_{\varphi\in\mathpzc{F}(Q)}D(\mu_{\varphi})^{2}\ll
Q^{-\frac{1}{10^{13}}}.$

The implied constant is ineffective.

######  Remark.

The implied constant is ineffective because in order to bound the contribution
from the dihedral $\varphi\in\mathpzc{F}(Q)$, we need Siegel’s ineffective
lower bound for $L$-functions at $s=1$ associated to primitive quadratic
Dirichlet characters.

The numerical quality of the power savings in (2.11) relies heavily on the
aforementioned mean value estimate for the (spectrally normalized) Fourier
coefficients of Maaß cusp forms on $\Gamma\backslash\mathbb{H}$ with
$\lambda_{\varphi}\leq Q$ proved by Deshouillers and Iwaniec [16]. Their
result is uniform in the level $q$, but the proof of Theorem 2.6 requires a
large sieve inequality that averages over a larger family, namely the weight
zero Hecke–Maaß newforms $\varphi$ with Laplace eigenvalue $\lambda_{\varphi}$
level $q_{\varphi}$ satisfying $\lambda_{\varphi}q_{\varphi}\leq Q$ (cf. [32,
Problem 7.25]). The only unconditional $\mathrm{GL}_{2}$ large sieve
inequality that exhibits such uniformity is (1.9); unfortunately, the
$Q$-dependence is not strong enough to prove Theorem 2.6 using the approach in
[48],444When $n=2$, the $Q$-dependence in (1.9) can be improved slightly using
progress toward GRC that is only available for $\pi\in\mathfrak{F}_{2}$. This
improvement still leaves much to be desired in the $Q$-aspect. which involves
a spectral expansion with respect to a basis of incomplete Eisenstein series
and incomplete Poincaré series. Our approach, which involves a spectral
expansion with respect to a basis of unitary Eisenstein series and Hecke–Maaß
newforms, is well-suited for an application of Theorem 2.5.

###  2.7. Nonsplit quantum ergodicity

Let $E=\mathbb{Q}(\sqrt{D})$ be a real quadratic field with narrow class
number $1$, and let $(\phi_{j})_{j=1}^{\infty}$ be an orthonormal basis of
Hilbert Maaß cusp forms on
$\mathrm{SL}_{2}(\mathcal{O}_{E})\backslash\mathbb{H}\times\mathbb{H}$.
Consider the signed measure $d\mu_{j}(z)\coloneqq\phi_{j}(z,z)\,d\mu(z)$ on
$\Gamma\backslash\mathbb{H}$ obtained by restricting the function
$\phi_{j}(z_{1},z_{2})$ on
$\mathrm{SL}_{2}(\mathcal{O}_{E})\backslash\mathbb{H}\times\mathbb{H}$ to the
diagonal embedding of $\Gamma\backslash\mathbb{H}$. Given $H\in
C_{b}(\Gamma\backslash\mathbb{H})$, we define

$\mathcal{D}_{j}(H)\coloneqq\int_{\Gamma\backslash\mathbb{H}}H(z)\,d\mu_{j}(z)-\frac{3}{\pi}\mu_{j}(\Gamma\backslash\mathbb{H})\int_{\Gamma\backslash\mathbb{H}}H(z)\,d\mu(z).$

Our final application pertains to the following conjecture regarding the
signed measures $d\mu_{j}$.

######  Conjecture 2.7.

For any fixed $H\in C_{b}(\Gamma\backslash\mathbb{H})$, we have that

(2.12) $\lim_{C(\operatorname{As}\phi_{j})\to\infty}\mathcal{D}_{j}(H)=0.$

Here $C(\operatorname{As}\phi_{j})$ denotes the archimedean part of the
analytic conductor of the Asai transfer $\operatorname{As}\phi_{j}$ of
$\phi_{j}$, as introduced in [1]. It is given by

(2.13)
$C(\operatorname{As}\phi_{j})\coloneqq(3+|t_{1,j}+t_{2,j}|)^{2}(3+|t_{1,j}-t_{2,j}|)^{2},$

where $t_{1,j},t_{2,j}$ are the two archimedean spectral parameters of
$\phi_{j}$.

This conjecture may be thought of as a nonsplit analogue of quantum unique
ergodicity in configuration space for $\Gamma\backslash\mathbb{H}$, where the
restriction of a Hilbert Maaß cusp form $\phi_{j}(z_{1},z_{2})$ on
$\mathrm{SL}_{2}(\mathcal{O}_{E})\backslash\mathbb{H}\times\mathbb{H}$ to the
diagonal embedding of $\Gamma\backslash\mathbb{H}$ replaces the restriction of
$\varphi_{k}(z_{1})\varphi_{k}(z_{2})$ on
$\Gamma\backslash\mathbb{H}\times\Gamma\backslash\mathbb{H}$ to the diagonal
embedding of $\Gamma\backslash\mathbb{H}$, where $\varphi_{k}$ is a Maaß cusp
form on $\Gamma\backslash\mathbb{H}$. As we show in Lemma 9.1, if $\phi_{j}$
is additionally an eigenfunction of all the Hecke operators, then
$\mu_{j}(\Gamma\backslash\mathbb{H})=0$ except on the rare occasion that
$\phi_{j}$ is the base change of a Hecke–Maaß newform on
$\Gamma_{0}(D)\backslash\mathbb{H}$ of level $D$ and nebentypus $\chi_{D}$,
the primitive quadratic character modulo $D$. Thus, Conjecture 2.7 may be
interpreted as stating that the restriction of a Hilbert Hecke–Maaß newform
$\phi_{j}$ to $\Gamma\backslash\mathbb{H}$ dissipates, rather than
equidistributes, as $C(\operatorname{As}\phi_{j})\to\infty$ apart from when
$\phi_{j}$ is a base change.

Conjecture 2.7 seems out of reach by current methods. When $\phi_{j}$ is the
base change of a cuspidal _holomorphic_ Hecke eigenform of even weight $k$ and
$H$ is a fixed Hecke–Maaß newform, Nelson has shown that the analogue of
(2.12) holds as $k\to\infty$ [58, Theorem B]. His method is also valid when
$\phi_{j}$ is the base change of a Hecke–Maaß newform provided one
additionally assumes GRC, but no longer applies when $\phi_{j}$ is not a base
change.

A natural weakening of Conjecture 2.7 is the conjecture that there exists a
density one subsequence $\phi_{j_{n}}$ for which (2.12) holds for all $H\in
C_{b}(\Gamma\backslash\mathbb{H})$; this in turn may be thought of as a
nonsplit analogue of quantum ergodicity, as described in Section 2.6. We use
Corollary 6.1 to prove the following result towards this, where we instead
first fix a nice test function $H$ and then show that, apart from a very small
number of exceptional Hilbert Hecke–Maaß newforms $\phi_{j}$,
$|\mathcal{D}_{j}(H)|$ decays polynomially in $C(\operatorname{As}\phi_{j})$.

######  Theorem 2.8.

Fix $H\in C_{c}^{\infty}(\Gamma\backslash\mathbb{H})$ and $\varepsilon>0$. Let
$(\phi_{j})_{j=1}^{\infty}$ be an orthonormal basis of Hilbert Hecke–Maaß
newforms, and for $Q>1$, let
$\mathpzc{F}_{\mathrm{As}}(Q)\coloneqq\\{\phi_{j}\colon Q\leq
C(\operatorname{As}\phi_{j})\leq 2Q\\}$. Then for all
$0<\delta<\tfrac{1}{2}\cdot 10^{-12}\varepsilon$, we have that

$\big{|}\big{\\{}\phi_{j}\in\mathpzc{F}_{\mathrm{As}}(Q)\colon\big{|}\mathcal{D}_{j}(H)\big{|}>C(\operatorname{As}\phi_{j})^{-\delta}\big{\\}}\big{|}\ll_{D,H,\varepsilon}Q^{\varepsilon}.$

As we point out in Lemma 9.7, Theorem 2.8 is nontrivial once
$\varepsilon<\tfrac{1}{2}$. We emphasize that in Theorem 2.8, the density one
sequence is dependent on the choice of test function $H\in
C_{c}^{\infty}(\Gamma\backslash\mathbb{H})$, in contrast to classical quantum
ergodicity results. One would like to overcome this via a diagonalization and
approximation argument, as in [74, Section 6]. Unfortunately, there is a major
hindrance in our setting: $\mu_{j}$ is a signed measure, rather than a
probability measure. A diagonalization and approximation argument would
require strong control over the total mass of the measure $|\mu_{j}|$, which
we presently lack.

There is also an analogue of Conjecture 2.7 when $D<0$, so that $E$ is an
imaginary quadratic field and $\phi_{j}$ is a Bianchi Maaß cusp form. We
explain in Section 9.4 how the same methods allow one to prove a nonsplit
quantum ergodicity result in the imaginary quadratic case akin to Theorem 2.8.

### Overview of the paper

In Section 3, we recall basic properties of standard $L$-functions and
Rankin–Selberg $L$-functions that we will use in our proofs. In Section 4, we
prove Theorem 1.2 along with a corollary on mean values of Dirichlet
polynomials, which is the crux of our proof of Theorem 1.1 in Section 5. We
then prove Theorems 2.1 and 2.5 in Section 6, Theorem 2.2 in Section 7,
Theorem 2.6 in Section 8, Theorem 2.8 in Section 9, and Theorems 2.3 and 2.4
in Section 10.

### Acknowledgements

We thank Yao Cheng, Gergely Harcos, Subhajit Jana, and Rizwanur Khan for
helpful comments.

##  3\. Properties of $L$-functions

We recall some standard facts about $L$-functions arising from automorphic
representations and their Rankin-Selberg convolutions. See [8, 21, 34, 50,
65].

###  3.1. Standard $L$-functions

Given $\pi\in\mathfrak{F}_{n}$, let $\widetilde{\pi}\in\mathfrak{F}_{n}$ be
the contragredient representation and $\mathfrak{q}_{\pi}$ be the conductor of
$\pi$. We express $\pi$ as a restricted tensor product $\bigotimes_{v}\pi_{v}$
of smooth admissible representations of $\mathrm{GL}_{n}(F_{v})$, where $v$
varies over places of $F$. When $v$ is nonarchimedean and the prime ideal
$\mathfrak{p}$ corresponds with $v$, then the local $L$-function
$L(s,\pi_{\mathfrak{p}})$ is defined in terms of the Satake parameters
$A_{\pi}(\mathfrak{p})=\\{\alpha_{1,\pi}(\mathfrak{p}),\ldots,\alpha_{n,\pi}(\mathfrak{p})\\}$
by

(3.1)
$L(s,\pi_{\mathfrak{p}})=\prod_{j=1}^{n}(1-\alpha_{j,\pi}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s})^{-1}=\sum_{k=0}^{\infty}\frac{\lambda_{\pi}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}.$

We have $\alpha_{j,\pi}(\mathfrak{p})\neq 0$ for all $j$ whenever
$\mathfrak{p}\nmid\mathfrak{q}_{\pi}$, and when
$\mathfrak{p}|\mathfrak{q}_{\pi}$, it might be the case that there exist $j$
such that $\alpha_{j,\pi}(\mathfrak{p})=0$. The standard $L$-function
$L(s,\pi)$ associated to $\pi$ is of the form

$L(s,\pi)=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}})=\sum_{\mathfrak{n}}\frac{\lambda_{\pi}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{s}}.$

The Euler product and Dirichlet series converge absolutely when
$\textup{Re}(s)>1$.

At each archimedean place $v$ of $F$, there are $n$ Langlands parameters
$\mu_{j,\pi}(v)\in\mathbb{C}$ from which we define

$L(s,\pi_{\infty})=\prod_{v|\infty}\prod_{j=1}^{n}\Gamma_{v}(s+\mu_{j,\pi}(v)),\qquad\Gamma_{v}(s)\coloneqq\begin{cases}\pi^{-s/2}\Gamma(s/2)&\mbox{if
$F_{v}=\mathbb{R}$,}\\\ 2(2\pi)^{-s}\Gamma(s)&\mbox{if
$F_{v}=\mathbb{C}$.}\end{cases}$

Luo, Rudnick, and Sarnak [47] and Mueller and Speh [54] proved that there
exists $\theta_{n}\in[0,\frac{1}{2}-1/(n^{2}+1)]$ such that we have the
uniform bounds

(3.2)
$|\alpha_{j,\pi}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}^{\theta_{n}}\qquad\textup{
and }\qquad\textup{Re}(\mu_{j,\pi}(v))\geq-\theta_{n},$

and GRC asserts that in (3.2), one may take $\theta_{n}=0$. We have
$\mathfrak{q}_{\pi}=\mathfrak{q}_{\widetilde{\pi}}$, and for each
$\mathfrak{p}$ and each $v$, we have the equalities of sets
$\\{\alpha_{j,\widetilde{\pi}}(\mathfrak{p})\\}=\\{\overline{\alpha_{j,\pi}(\mathfrak{p})}\\}$
and $\\{\mu_{j,\widetilde{\pi}}(v)\\}=\\{\overline{\mu_{j,\pi}(v)}\\}$.

Let $r_{\pi}$ be the order of the pole of $L(s,\pi)$ at $s=1$. The completed
$L$-function

$\Lambda(s,\pi)=(s(s-1))^{r_{\pi}}(D_{F}^{n}\mathrm{N}\mathfrak{q}_{\pi})^{s/2}L(s,\pi)L(s,\pi_{\infty})$

is entire of order 1, and there exists a complex number $W(\pi)$ of modulus 1
such that for all $s\in\mathbb{C}$, we have the functional equation

$\Lambda(s,\pi)=W(\pi)\Lambda(1-s,\widetilde{\pi}).$

Let $d(v)=1$ if $F_{v}=\mathbb{R}$ and $d(v)=2$ if $F_{v}=\mathbb{C}$. The
analytic conductor of $\pi$ [33] is given by

(3.3) $C(\pi,t)\coloneqq
D_{F}^{n}\mathrm{N}\mathfrak{q}_{\pi}\prod_{v|\infty}\prod_{j=1}^{n}(3+|it+\mu_{j,\pi}(v)|^{d(v)}),\qquad
C(\pi)\coloneqq C(\pi,0).$

###  3.2. Rankin–Selberg $L$-functions

Let $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$.
There exist suitable complex numbers
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$ such that

(3.4)
$L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}(1-\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s})^{-1}=\sum_{k=0}^{\infty}\frac{\lambda_{\pi\times\pi^{\prime}}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}.$

If $\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$, then we
have the equality of sets

(3.5)
$\\{\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})\\}=\\{\alpha_{j,\pi}(\mathfrak{p})\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p})\\}.$

The Rankin-Selberg $L$-function $L(s,\pi\times\pi^{\prime})$ associated to
$\pi$ and $\pi^{\prime}$ is of the form

$L(s,\pi\times\pi^{\prime})=\prod_{\mathfrak{p}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})=\sum_{\mathfrak{n}}\frac{\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{s}}.$

Let $\mathfrak{q}_{\pi\times\pi^{\prime}}$ be the arithmetic conductor of
$\pi\times\pi^{\prime}$. At an archimedean place $v$ of $F$, there are
$n^{\prime}n$ complex Langlands parameters
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$ from which we define

$L(s,\pi_{\infty}\times\pi_{\infty}^{\prime})=\prod_{v|\infty}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\Gamma_{v}(s+\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)).$

Using the explicit descriptions of
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})$ and
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$ in [29, 65], one sees that

(3.6)
$|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}^{\theta_{n}+\theta_{n^{\prime}}},\qquad\textup{Re}(\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v))\geq-\theta_{n}-\theta_{n^{\prime}}.$

Let

$r_{\pi\times\pi^{\prime}}=-\mathop{\mathrm{ord}}_{s=1}L(s,\pi\times\pi^{\prime}),\qquad\kappa_{\pi\times\pi^{\prime}}=\mathop{\mathrm{Res}}_{s=1}L(s,\pi\times\pi^{\prime})\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})^{-1}.$

By our normalization for the central characters of $\pi$ and $\pi^{\prime}$,
we have that $r_{\pi\times\pi^{\prime}}=0$ and
$\kappa_{\pi\times\pi^{\prime}}=0$ if and only if
$\pi\neq\widetilde{\pi}^{\prime}$. Otherwise, $r_{\pi\times\widetilde{\pi}}=1$
and $\kappa_{\pi\times\widetilde{\pi}}>0$. The function

(3.7)
$\Lambda(s,\pi\times\pi^{\prime})=(s(s-1))^{r_{\pi\times\pi^{\prime}}}(D_{F}^{n^{\prime}n}\mathrm{N}\mathfrak{q}_{\pi\times\pi^{\prime}})^{s/2}L(s,\pi\times\pi^{\prime})L(s,\pi_{\infty}\times\pi_{\infty}^{\prime})$

is entire of order 1, and there exists a complex number
$W(\pi\times\pi^{\prime})$ of modulus 1 such that
$\Lambda(s,\pi\times\pi^{\prime})$ satisfies the functional equation

$\Lambda(s,\pi\times\pi^{\prime})=W(\pi\times\pi^{\prime})\Lambda(1-s,\widetilde{\pi}\times\widetilde{\pi}^{\prime}).$

As with $L(s,\pi)$, the analytic conductor of $\pi\times\pi^{\prime}$ is given
by

(3.8) $\displaystyle C(\pi\times\pi^{\prime},t)$ $\displaystyle\coloneqq
D_{F}^{n^{\prime}n}\mathrm{N}\mathfrak{q}_{\pi\times\pi^{\prime}}\prod_{v|\infty}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}(3+|it+\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)|^{d(v)}),$
$\displaystyle C(\pi\times\pi^{\prime})$ $\displaystyle\coloneqq
C(\pi\times\pi^{\prime},0).$

The combined work of Bushnell and Henniart [12] and Brumley [29, Appendix]
yields

(3.9) $C(\pi\times\pi^{\prime},t)\ll
C(\pi\times\pi^{\prime})(3+|t|)^{[F:\mathbb{Q}]n^{\prime}n},\qquad
C(\pi\times\pi^{\prime})\ll C(\pi)^{n^{\prime}}C(\pi^{\prime})^{n}.$

It follows from work of Li [42, Theorem 2] (with minor adjustments when
$F\neq\mathbb{Q}$) that there exists an absolute and effectively computable
constant $\Cl[abcon]{Li}>0$, which we assume to be sufficiently large for
future convenience, such that

(3.10)
$\lim_{\sigma_{0}\to\sigma}(\sigma_{0}-1)^{r_{\pi\times\pi^{\prime}}}|L(\sigma_{0},\pi\times\pi^{\prime})|\ll\exp\Big{(}\Cr{Li}n^{\prime}n[F:\mathbb{Q}]\frac{\log
C(\pi\times\pi^{\prime})}{\log\log
C(\pi\times\pi^{\prime})}\Big{)},\quad\sigma\in[1,3].$

We can change $\pi^{\prime}$ to $\pi^{\prime}\otimes\left|\det\right|^{it}$;
at the archimedean places, this has the effect of adding $it$ to each
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$. We then apply functional
equation, the Phragmén–Lindelöf principle, and (3.9) to obtain for all
$\sigma\geq 0$

(3.11)
$\displaystyle\lim_{\sigma_{0}\to\sigma}|(\sigma_{0}-1)^{r_{\pi\times\pi^{\prime}}}L(\sigma_{0}+it,\pi\times\pi^{\prime})|$
$\displaystyle\ll_{\varepsilon}C(\pi\times\pi^{\prime},t)^{\frac{\max\\{1-\sigma,0\\}}{2}+\frac{\varepsilon}{n^{\prime}n[F:\mathbb{Q}]}}$
$\displaystyle\ll_{\varepsilon}(C(\pi)^{n^{\prime}}C(\pi^{\prime})^{n}(3+|t|)^{n^{\prime}n[F:\mathbb{Q}]})^{\frac{\max\\{1-\sigma,0\\}}{2}+\frac{\varepsilon}{n^{\prime}n[F:\mathbb{Q}]}}.$

######  Lemma 3.1.

If $\pi\in\mathfrak{F}_{n}$, $X\geq 3$, and $\varepsilon>0$, then
$\sum_{\mathrm{N}\mathfrak{n}\leq
X}\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})/\mathrm{N}\mathfrak{n}\ll_{\varepsilon}C(\pi)^{\varepsilon}\log
X$.

###### Proof.

Since $\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})\geq 0$ for all
$\mathfrak{n}$ [26, Lemma a], we observe by (3.10) that the sum is

$\ll\sum_{\mathfrak{n}}\frac{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{1}{\log
X}}}\ll(\log X)\mathop{\mathrm{Res}}_{s=1}L(s,\pi\times\widetilde{\pi}).$

The desired bounded now follows from (3.11) with $\sigma=1$, $t=0$, and
$\pi^{\prime}=\widetilde{\pi}$. ∎

Since
$|\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})|\leq\mathrm{N}\mathfrak{p}$,
we have the bound

$\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}|L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})|^{-1}=\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}\Big{|}1-\frac{\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(\mathfrak{p})}{\mathrm{N}\mathfrak{p}}\Big{|}\leq\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}2^{n^{\prime}n}.$

Since
$|\\{\mathfrak{p}\colon\mathfrak{p}|\mathfrak{n}\\}|\ll(\log\mathrm{N}\mathfrak{n})/\log\log\mathrm{N}\mathfrak{n}$
by the proof of [70, Lemma 1.13b], it follows that

(3.12)
$\kappa_{\pi\times\widetilde{\pi}}\ll_{\varepsilon}C(\pi)^{\varepsilon}$

###  3.3. Rankin–Selberg combinatorics

A partition $\mu=(\mu_{i})_{i=1}^{\infty}$ is a sequence of nonincreasing
nonnegative integers $\mu_{1}\geq\mu_{2}\geq\cdots$ with only finitely many
nonzero entries. For a partition $\mu$, let $\ell(\mu)$ be the number of
nonzero $\mu_{i}$, and let $|\mu|=\sum_{i=1}^{\infty}\mu_{i}$. For a set
$\\{\alpha_{1},\ldots,\alpha_{n}\\}$ of real numbers and a partition $\mu$
with $\ell(\mu)\leq n$, let $s_{\mu}(\\{\alpha_{1},\ldots,\alpha_{n}\\})$ be
the Schur polynomial
$\det[(\alpha_{i}^{\lambda(j)+n-j})_{ij}]/\det[(\alpha_{i}^{n-j})_{ij}]$
associated to $\mu$. If $|\mu|=0$, then
$s_{\mu}(\\{\alpha_{1},\ldots,\alpha_{n}\\})$ is identically one. By
convention, if $\ell(\mu)>n$, then
$s_{\mu}(\\{\alpha_{1},\ldots,\alpha_{n}\\})$ is identically zero.

Let $\pi\in\mathfrak{F}_{n}$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$.
By (3.4), (3.5), and Cauchy’s identity [11, Theorem 38.1], we have

$\sum_{k=0}^{\infty}\frac{\lambda_{\pi\times\pi^{\prime}}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}=L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})=\sum_{\lambda}\frac{s_{\mu}(A_{\pi}(\mathfrak{p}))s_{\mu}(A_{\pi^{\prime}}(\mathfrak{p}))}{\mathrm{N}\mathfrak{p}^{s|\lambda|}},\quad\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}.$

where the sum ranges over all partitions. This yields

$\lambda_{\pi\times\pi^{\prime}}(\mathfrak{p}^{k})=\sum_{|\lambda|=k}s_{\mu}(A_{\pi}(\mathfrak{p}))s_{\mu}(A_{\pi^{\prime}}(\mathfrak{p})),\qquad\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}.$

For an integral ideal $\mathfrak{n}$ with factorization
$\mathfrak{n}=\prod_{\mathfrak{p}}\mathfrak{p}^{\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})}$
(with $\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})=0$ for all but finitely many
$\mathfrak{p}$), the multiplicativity of
$\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})$ tells us that if
$\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}$,
then

(3.13)
$\displaystyle\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})=\prod_{\mathfrak{p}}\lambda_{\pi\times\pi^{\prime}}(\mathfrak{p}^{\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})})=\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))s_{\mu_{\mathfrak{p}}}(A_{\pi^{\prime}}(\mathfrak{p})),$

where $(\mu_{\mathfrak{p}})_{\mathfrak{p}}$ denotes a sequence of partitions
indexed by prime ideals and

(3.14)
$\underline{\mu}[\mathfrak{n}]\coloneqq\\{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\colon|\mu_{\mathfrak{p}}|=\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})\textup{
for all $\mathfrak{p}$}\\}.$

Define the numbers $\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$ on unramified
prime powers by

$\sum_{k=0}^{\infty}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}=L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})^{-1}=\prod_{j=1}^{n}\prod_{j^{\prime}=1}^{n^{\prime}}(1-\alpha_{j,\pi}(\mathfrak{p})\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p})\mathrm{N}\mathfrak{p}^{-s}),\qquad\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}.$

By multiplicativity, this defines $\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$
when
$\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}$.
For a partition $\mu=(\mu_{i})_{i=1}^{\infty}$, let
$\mu^{*}=(\mu_{i}^{*})_{i=1}^{\infty}$ be the dual partition defined by
$\mu_{i}^{*}=|\\{j\colon\mu_{j}\geq i\\}|$. It follows from the dual Cauchy
identity [11, Chapter 38] and (3.5) that

$\sum_{k=0}^{\infty}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{p}^{k})}{\mathrm{N}\mathfrak{p}^{ks}}=\sum_{\lambda}\frac{s_{\mu}(A_{\pi}(\mathfrak{p}))s_{\mu^{*}}(-A_{\pi^{\prime}}(\mathfrak{p}))}{\mathrm{N}\mathfrak{p}^{|\lambda|s}},\qquad\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}},$

where
$-A_{\pi^{\prime}}(\mathfrak{p})=\\{-\alpha_{1,\pi^{\prime}}(\mathfrak{p}),\ldots,-\alpha_{n,\pi^{\prime}}(\mathfrak{p})\\}$.
Hence we have

(3.15)
$\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})=\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))s_{\mu^{*}_{\mathfrak{p}}}(-A_{\pi}(\mathfrak{p})),\qquad\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}.$

######  Lemma 3.2.

If
$\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}$,
then we have
$|\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})|\leq\frac{1}{2}(\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})+\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n}))$.

###### Proof.

We apply the inequality of arithmetic and geometric means to (3.15):

$|\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})|\leq\frac{1}{2}\Big{(}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{|}^{2}+\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\prod_{\mathfrak{p}}s_{\mu^{*}_{\mathfrak{p}}}(-A_{\pi^{\prime}}(\mathfrak{p}))\Big{|}^{2}\Big{)}.$

The first sum equals $\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})$ by
(3.13). For the second sum, note that since $|\mu|=|\mu^{*}|$, we have
$(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]$ if and
only if
$(\mu_{\mathfrak{p}}^{*})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]$.
Hence by rearranging, we see that

(3.16)
$\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\prod_{\mathfrak{p}}s_{\mu^{*}_{\mathfrak{p}}}(-A_{\pi^{\prime}}(\mathfrak{p}))\Big{|}^{2}=\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(-A_{\pi^{\prime}}(\mathfrak{p}))\Big{|}^{2}=\lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n}),$

where the last equality holds because
$\alpha_{j,\pi^{\prime}}(\mathfrak{p})\overline{\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p})}=(-\alpha_{j,\pi^{\prime}}(\mathfrak{p}))\overline{(-\alpha_{j^{\prime},\pi^{\prime}}(\mathfrak{p}))}$.
∎

##  4\. Proof of Theorem 1.2

We begin with a mean value estimate for the Satake parameters of $\pi$ as
$\pi\in\mathfrak{F}_{n}$ varies. Throughout, we let

(4.1) $a:\bigcup_{x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}}\underline{\mu}[\mathfrak{n}]\to\mathbb{C},\qquad\alpha:\mathfrak{F}_{n}(Q)\to\mathbb{C}$

be functions that are not identically zero. Their $\ell^{2}$ norms $\|a\|_{2}$
and $\|\alpha\|_{2}$ are defined by

$\|a\|_{2}=\Big{(}\sum_{x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}}~{}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}|a((\mu_{\mathfrak{p}})_{\mathfrak{p}})|^{2}\Big{)}^{1/2},\qquad\|\alpha\|_{2}=\Big{(}\sum_{\pi\in\mathfrak{F}_{n}(Q)}|\alpha(\pi)|^{2}\Big{)}^{1/2}.$

For convenience, we define $\mathbf{1}_{(\mathfrak{n},\mathfrak{q})}$ to equal
one when $\gcd(\mathfrak{n},\mathfrak{q})=\mathcal{O}_{F}$ and zero otherwise.

######  Proposition 4.1.

Let $x\geq 1$ and $Q,T\geq 1$. Define

$C(Q,T,x)\coloneqq\sup_{\|a\|_{2}\neq
0}\cfrac{\displaystyle\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}~{}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{]}a((\mu_{\mathfrak{p}})_{\mathfrak{p}})\Big{|}^{2}}{\displaystyle\sum_{x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}}~{}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}|a((\mu_{\mathfrak{p}})_{\mathfrak{p}})|^{2}}.$

We have the bound

$\displaystyle
C(Q,T,x)\ll_{\varepsilon}Q^{\varepsilon}\Big{(}\frac{x}{T}+Q^{n^{2}+n}T^{n^{2}[F:\mathbb{Q}]+\varepsilon}|\mathfrak{F}_{n}(Q)|\Big{)}.$

###### Proof.

We observe that

(4.2) $\displaystyle
C(Q,T,x)=\sup_{\|a\|_{2}=1}\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}\end{subarray}}~{}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{]}\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi})}a((\mu_{\mathfrak{p}})_{\mathfrak{p}})\Big{|}^{2}.$

By the duality principle for bilinear forms, (4.2) is bounded by the supremum
over the functions $\alpha\colon\mathfrak{F}_{n}(Q)\to\mathbb{C}$ such that
$\|\alpha\|_{2}=1$ of

(4.3) $\sum_{x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}}~{}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{]}\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi})}\alpha(\pi)\Big{|}^{2}.$

Let $\phi$ be a fixed smooth test function, supported in a compact subset of
$[-2,2]$, such that $\phi(t)=1$ for $t\in[0,1]$ and $\phi(t)\in[0,1)$
otherwise. Then (4.3) is bounded by

(4.4)
$\sum_{\mathfrak{n}}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{]}\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi})}\alpha(\pi)\Big{|}^{2}\phi\Big{(}T\log\frac{\mathrm{N}\mathfrak{n}}{x}\Big{)}.$

We expand the square, interchange the order of summation, and apply (3.13) to
find that (4.4) equals

$\displaystyle\sum_{\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)}\alpha(\pi)\overline{\alpha(\pi^{\prime})}\sum_{\mathfrak{n}}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{]}\overline{\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi^{\prime}}(\mathfrak{p}))\Big{]}}\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi})}\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi^{\prime}})}\phi\Big{(}T\log\frac{\mathrm{N}\mathfrak{n}}{x}\Big{)}$
$\displaystyle=\sum_{\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)}\alpha(\pi)\overline{\alpha(\pi^{\prime})}\sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}}\lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\phi\Big{(}T\log\frac{\mathrm{N}\mathfrak{n}}{x}\Big{)}.$

Let $\widehat{\phi}(s)=\int_{\mathbb{R}}\phi(y)e^{sy}dy$. It follows from a
standard contour integral calculation using (3.6) and (3.11) that the above
display equals

(4.5)
$\sum_{\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)}\alpha(\pi)\overline{\alpha(\pi^{\prime})}\Big{(}\frac{1}{2\pi
iT}\int_{3-i\infty}^{3+i\infty}\frac{L(s,\pi\times\widetilde{\pi}^{\prime})}{\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}}^{\prime})}x^{s}\widehat{\phi}(s/T)ds\Big{)}\\\
=\sum_{\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)}\alpha(\pi)\overline{\alpha(\pi^{\prime})}\Big{(}\kappa_{\pi\times\widetilde{\pi}^{\prime}}x\frac{\widehat{\phi}(\frac{1}{T})}{T}+\frac{1}{2\pi
iT}\int_{\frac{1}{\log x}-i\infty}^{\frac{1}{\log
x}+i\infty}\frac{L(s,\pi\times\widetilde{\pi}^{\prime})x^{s}\widehat{\phi}(s/T)}{\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}}^{\prime})}ds\Big{)}\\\
=\sum_{\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)}\alpha(\pi)\overline{\alpha(\pi^{\prime})}\Big{(}\kappa_{\pi\times\widetilde{\pi}^{\prime}}x\frac{\widehat{\phi}(\frac{1}{T})}{T}+O_{\phi,\varepsilon}(Q^{n^{2}+n+\varepsilon}T^{[F:\mathbb{Q}]n^{2}+\varepsilon})\Big{)}.$

Recall that $\kappa_{\pi\times\widetilde{\pi}^{\prime}}>0$ when
$\pi=\pi^{\prime}$, and $\kappa_{\pi\times\widetilde{\pi}^{\prime}}=0$
otherwise. Since $\|\alpha\|_{2}=1$ and $\phi$ is fixed, it follows from the
inequality of arithmetic and geometric means that (4.5) equals

$\frac{\widehat{\phi}(\frac{1}{T})}{T}x\sum_{\pi\in\mathfrak{F}_{n}(Q)}|\alpha(\pi)|^{2}\kappa_{\pi\times\widetilde{\pi}^{\prime}}+O_{\phi}(Q^{n^{2}+n+\varepsilon}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}|\mathfrak{F}_{n}(Q)|)\\\
\ll_{\phi,\varepsilon}\frac{x}{T}\max_{\pi\in\mathfrak{F}_{n}(Q)}\kappa_{\pi\times\widetilde{\pi}^{\prime}}+Q^{n^{2}+n+\varepsilon}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}|\mathfrak{F}_{n}(Q)|.$

The desired result follows from (3.12). ∎

We use Proposition 4.1 to prove Theorem 1.2.

###### Proof of Theorem 1.2.

For the sum involving $\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})$, we apply
Proposition 4.1 with

$a((\mu_{\mathfrak{p}})_{\mathfrak{p}})=b\Big{(}\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|}\Big{)}\mathbf{1}_{(\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|},\mathfrak{q}_{\pi^{\prime}})}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}^{*}}(-A_{\pi^{\prime}}(\mathfrak{p})).$

If $(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]$, then
by (3.14), we have
$|\mu_{\mathfrak{p}}|=\mathrm{ord}_{\mathfrak{p}}(\mathfrak{n})$ and

$a((\mu_{\mathfrak{p}})_{\mathfrak{p}})=b(\mathfrak{n})\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi^{\prime}})}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}^{*}}(-A_{\pi^{\prime}}(\mathfrak{p})).$

By (3.15), the left hand side of Proposition 4.1 is then

$\displaystyle\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}~{}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{[}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))\Big{]}a((\mu_{\mathfrak{p}})_{\mathfrak{p}})\Big{|}^{2}$
$\displaystyle=\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi})=\mathcal{O}_{F}\end{subarray}}b(\mathfrak{n})\mathbf{1}_{(\mathfrak{n},\mathfrak{q}_{\pi^{\prime}})}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi}(\mathfrak{p}))s_{\mu_{\mathfrak{p}}^{*}}(-A_{\pi^{\prime}}(\mathfrak{p})\Big{|}^{2}$
$\displaystyle=\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq
xe^{1/T}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})b(\mathfrak{n})\Big{|}^{2}.$

Similarly, the right hand side of Proposition 4.1 becomes

$\displaystyle
Q^{\varepsilon}\Big{(}\frac{x}{T}+Q^{n^{2}+n}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}|\mathfrak{F}_{n}(Q)|\Big{)}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}|b(\mathfrak{n})|^{2}\sum_{(\mu_{\mathfrak{p}})_{\mathfrak{p}}\in\underline{\mu}[\mathfrak{n}]}\Big{|}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}^{*}}(-A_{\pi^{\prime}}(\mathfrak{p}))\Big{|}^{2}.$

The desired result now follows from (3.16). Apart from the new choice of

$a((\mu_{\mathfrak{p}})_{\mathfrak{p}})=b\Big{(}\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|}\Big{)}\mathbf{1}_{(\prod_{\mathfrak{p}}\mathrm{N}\mathfrak{p}^{|\mu_{\mathfrak{p}}|},\mathfrak{q}_{\pi^{\prime}})}\prod_{\mathfrak{p}}s_{\mu_{\mathfrak{p}}}(A_{\pi^{\prime}}(\mathfrak{p})),$

the result for the sum involving
$\lambda_{\pi\times\pi^{\prime}}(\mathfrak{n})$ is handled in the same manner.
∎

######  Corollary 4.2.

Let $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$ and $Q,T\geq 1$. If $X\geq
Q^{n^{2}+n}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}|\mathfrak{F}_{n}(Q)|$ and $Y$
satisfies $\log Y\asymp\log X$, then we have the bounds

$\displaystyle\sum_{\pi\in\mathfrak{F}_{n}(Q)}\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}X<\mathrm{N}\mathfrak{n}\leq
X^{\log Y}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{1}{\log
Y}+iv}}\Big{|}^{2}dv$
$\displaystyle\ll_{\varepsilon}C(\pi^{\prime})^{\varepsilon}Q^{\varepsilon}(\log
X)^{2},$
$\displaystyle\sum_{\pi\in\mathfrak{F}_{n}(Q)}\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leq
X\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{\frac{1}{2}+\frac{1}{\log
Y}+iv}}\Big{|}^{2}dv$
$\displaystyle\ll_{\varepsilon}C(\pi^{\prime})^{\varepsilon}Q^{\varepsilon}X(\log
X)^{2}.$

###### Proof.

We prove the first result; the second result is proved completely analogously.
A formal generalization of [19, Theorem 1] to number fields tells us that if
$c(\mathfrak{n})$ is a complex-valued function supported on the integral
ideals of $\mathcal{O}_{F}$ with
$\sum_{\mathfrak{n}}|c(\mathfrak{n})|<\infty$, then

$\int_{-T}^{T}\Big{|}\sum_{\mathfrak{n}}\frac{c(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{it}}\Big{|}^{2}dt\ll
T^{2}\int_{0}^{\infty}\Big{|}\sum_{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]}c(\mathfrak{n})\Big{|}^{2}\frac{dx}{x}.$

We choose $b(\mathfrak{n})=\mathrm{N}\mathfrak{n}^{-1-\frac{1}{\log Y}}$ if
$\mathrm{N}\mathfrak{n}\in(X,X^{\log Y}]$ and $b(\mathfrak{n})=0$ otherwise,
which leads to

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}X<\mathrm{N}\mathfrak{n}\leq
X^{\log Y}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{1}{\log
Y}+it}}\Big{|}^{2}dt\ll
T^{2}\int_{0}^{\infty}\sum_{\pi\in\mathfrak{F}_{n}(Q)}\Big{|}\sum_{\mathrm{N}\mathfrak{n}\in(x,xe^{1/T}]}\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})b(\mathfrak{n})\Big{|}^{2}\frac{dx}{x}.$

Theorem 1.2 and the fact that
$\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\geq 0$ for
all $\mathfrak{n}$ imply that the above display is

$\displaystyle\ll_{\varepsilon}Q^{\varepsilon}\sum_{\begin{subarray}{c}X<\mathrm{N}\mathfrak{n}\leq
X^{\log Y}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{2}{\log
Y}}}\Big{(}1+\frac{Q^{n^{2}+n}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}|\mathfrak{F}_{n}(Q)|}{\mathrm{N}\mathfrak{n}}\Big{)}\ll_{\varepsilon}Q^{\varepsilon}\sum_{X<\mathrm{N}\mathfrak{n}\leq
X^{\log
Y}}\frac{\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}}.$

The desired result now follows from Lemma 3.1 and our choices of $X$ and $Y$.
∎

##  5\. The zero density estimate: proof of Theorem 1.1

Let $n,n^{\prime}\geq 1$ be integers, let $\pi\in\mathfrak{F}_{n}(Q)$ and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$. Let $s=\sigma+it$. We assume that
$n^{\prime}n[F:\mathbb{Q}]\geq 2$, for otherwise we could appeal to the
stronger results in [52].

We have the bound $|\\{\rho=\beta+i\gamma\colon\beta\geq 0,~{}|\gamma-t|\leq
1\\}|\ll\log C(\pi\times\pi^{\prime},t)$ for all $t\in\mathbb{R}$ [32,
Proposition 5.7]. We decompose the rectangle $[\sigma,1]\times[-T,T]$ into
disjoint boxes of the shape $[\sigma,1]\times[y,y+2(\log X)^{2}]$, each of
which contains $\ll(\log C(\pi\times\pi^{\prime},T))^{3}$ zeros. If we write
$n_{\pi\times\pi^{\prime}}$ for the number of such boxes containing at least
one zero $\rho$ of $L(s,\pi\times\pi^{\prime})$, then

$N_{\pi\times\pi^{\prime}}(\sigma,T)\ll(\log(C(\pi^{\prime})QT))^{3}n_{\pi\times\pi^{\prime}}.$

We will use Montgomery’s ideas for detecting nontrivial zeros [52] to estimate
$n_{\pi\times\pi^{\prime}}$. Let

(5.1)
$X\coloneqq(Q^{n^{2}+n}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}+C(\pi^{\prime})^{n}Q^{2n^{\prime}n+n^{\prime}/2}T^{1+n^{\prime}n[F:\mathbb{Q}]/2})|\mathfrak{F}_{n}(Q)|,\quad
Y\coloneqq X^{\frac{2}{3-2\sigma}}.$

Let $\rho=\beta+i\gamma$ be a nontrivial zero of $L(s)\coloneqq
L(s,\pi\times\pi^{\prime})\prod_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}L(s,\pi_{\mathfrak{p}}\times\pi_{\mathfrak{p}}^{\prime})^{-1}$,
and let

$M_{X}(s)\coloneqq\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leq X\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{s}}.$

If $\rho=\beta+i\gamma$ is a nontrivial zero of $L(s)$ with
$\beta\geq\sigma>\frac{1}{2}$, then we compute

(5.2) $\displaystyle e^{-\frac{1}{Y}}$ $\displaystyle=\frac{1}{2\pi
i}\int_{1-\beta+\frac{1}{\log Y}-i\infty}^{1-\beta+\frac{1}{\log
Y}+i\infty}(1-L(\rho+w)M_{X}(\rho+w))\Gamma(w)Y^{w}dw$
$\displaystyle+\frac{1}{2\pi i}\int_{\frac{1}{2}-\beta+\frac{1}{\log
Y}-i\infty}^{\frac{1}{2}-\beta+\frac{1}{\log
Y}+i\infty}L(\rho+w)M_{X}(\rho+w)\Gamma(w)Y^{w}dw$
$\displaystyle+\kappa_{\pi\times\pi^{\prime}}M_{X}(1)\Gamma(1-\rho)Y^{1-\rho}.$

It follows from (3.11), Lemmata 3.1 and 3.2, and partial summation that if
$\textup{Re}(w)=1-\beta+(\log Y)^{-1}$, then

(5.3) $\displaystyle|1-L(\rho+w)M_{X}(\rho+w)|$
$\displaystyle=|L(\rho+w)|\cdot|L(\rho+w)^{-1}-M_{X}(\rho+w)|$
$\displaystyle\ll_{\varepsilon}C(\pi\times\pi^{\prime},|\gamma+\textup{Im}(w)|+1)^{\varepsilon}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}>X\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{1}{\log
Y}+i(\gamma+\textup{Im}(w))}}\Big{|}$
$\displaystyle\ll_{\varepsilon}C(\pi\times\pi^{\prime},|\gamma+\textup{Im}(w)|+1)^{\varepsilon}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}>X\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\lambda_{\pi\times\widetilde{\pi}}(\mathfrak{n})+\lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{1}{\log
Y}}}$
$\displaystyle\ll_{\varepsilon}C(\pi)^{\varepsilon}C(\pi^{\prime})^{\varepsilon}(|\gamma+\textup{Im}(w)|+1)^{\varepsilon}(\log
X)/X.$

We therefore deduce from Stirling’s formula that

$\Big{|}\frac{1}{2\pi i}\int_{1-\beta+\frac{1}{\log
Y}-i\infty}^{1-\beta+\frac{1}{\log
Y}+i\infty}(1-L(\rho+w)M_{X}(\rho+w))\Gamma(w)Y^{w}dw\\\ -\frac{1}{2\pi
i}\int_{1-\beta+\frac{1}{\log Y}-i(\log Y)^{2}}^{1-\beta+\frac{1}{\log
Y}+i(\log
Y)^{2}}(1-L(\rho+w)M_{X}(\rho+w))\Gamma(w)Y^{w}dw\Big{|}\ll\frac{1}{Y}.$

The other terms in (5.2) are handled similarly. Since $e^{-1/Y}=1+O(1/Y)$, it
follows that

(5.4) $\displaystyle 1+O\Big{(}\frac{1}{Y}\Big{)}$
$\displaystyle=\frac{1}{2\pi i}\int_{1-\beta+\frac{1}{\log Y}-i(\log
Y)^{2}}^{1-\beta+\frac{1}{\log Y}+i(\log
Y)^{2}}(1-L(\rho+w)M_{X}(\rho+w))\Gamma(w)Y^{w}dw$
$\displaystyle+\frac{1}{2\pi i}\int_{\frac{1}{2}-\beta+\frac{1}{\log Y}-i(\log
Y)^{2}}^{\frac{1}{2}-\beta+\frac{1}{\log Y}+i(\log
Y)^{2}}L(\rho+w)M_{X}(\rho+w)\Gamma(w)Y^{w}dw$
$\displaystyle+O_{\varepsilon}((C(\pi)^{\varepsilon}+C(\pi^{\prime})^{\varepsilon})Y^{1-\sigma}\log
X),$

where we have invoked (3.9). The Cauchy–Schwarz inequality and (5.4) imply
that

$1\ll_{\varepsilon}Y^{2(1-\sigma)}(\log
Y)^{2}\Big{(}(C(\pi)^{\varepsilon}+C(\pi^{\prime})^{\varepsilon}\\\
+\int_{1-\beta+\frac{1}{\log Y}-i(\log Y)^{2}}^{1-\beta+\frac{1}{\log
Y}+i(\log Y)^{2}}|1-L(\rho+w)M_{X}(\rho+w))|^{2}\cdot|dw|\Big{)}\\\
+Y^{\frac{1}{2}-\sigma}\int_{\frac{1}{2}-\beta+\frac{1}{\log Y}-i(\log
Y)^{2}}^{\frac{1}{2}-\beta+\frac{1}{\log Y}+i(\log
Y)^{2}}|L(\rho+w)M_{X}(\rho+w)|\cdot|dw|.$

Collecting the contributions from all boxes and applying the inequality of
arithmetic and geometric means, we find that

$\displaystyle n_{\pi\times\pi^{\prime}}$
$\displaystyle\ll_{\varepsilon}Y^{2(1-\sigma)}(\log
Y)^{2}C(\pi)^{\varepsilon}C(\pi^{\prime})^{\varepsilon}\int_{-T}^{T}|1-L(1+\tfrac{1}{\log
Y}+iv)M_{X}(1+\tfrac{1}{\log Y}+iv)|^{2}dv$
$\displaystyle+Y^{\frac{1}{2}-\sigma}\int_{-T}^{T}|L(\tfrac{1}{2}+\tfrac{1}{\log
Y}+iv)M_{X}(\tfrac{1}{2}+\tfrac{1}{\log Y}+iv)|dv$
$\displaystyle\ll_{\varepsilon}Y^{2(1-\sigma)}(\log
Y)^{2}C(\pi)^{\varepsilon}C(\pi^{\prime})^{\varepsilon}\int_{-T}^{T}|1-L(1+\tfrac{1}{\log
Y}+iv)M_{X}(1+\tfrac{1}{\log Y}+iv)|^{2}dv$
$\displaystyle+Y^{\frac{1}{2}-\sigma}\Big{(}\int_{-T}^{T}|L(\tfrac{1}{2}+\tfrac{1}{\log
Y}+iv)|^{2}dv+\int_{-T}^{T}|M_{X}(\tfrac{1}{2}+\tfrac{1}{\log
Y}+iv)|^{2}dv\Big{)}.$

The trivial bound

$\int_{-T}^{T}|L(\tfrac{1}{2}+\tfrac{1}{\log
Y}+iv)|^{2}dv\ll_{\varepsilon}C(\pi^{\prime})^{\frac{n}{2}+\varepsilon}Q^{2n^{\prime}n+\frac{n^{\prime}}{2}+\varepsilon}T^{1+\frac{n^{\prime}n[F:\mathbb{Q}]}{4}+\varepsilon}$

follows from (3.6) (to bound the ramified Euler factors) and (3.11).
Temporarily writing $L=L(1+\frac{1}{\log Y}+iv)$ and
$M_{X}=M_{X}(1+\frac{1}{\log Y}+iv)$, we deduce from (3.11) that

$\displaystyle|1-LM_{X}|^{2}$ $\displaystyle=|L|^{2}|L^{-1}-M_{X}|^{2}$
$\displaystyle\ll_{\varepsilon}Q^{\varepsilon}C(\pi^{\prime})^{\varepsilon}T^{\varepsilon}|L^{-1}-M_{X}|^{2}$
$\displaystyle\ll_{\varepsilon}Q^{\varepsilon}C(\pi^{\prime})^{\varepsilon}T^{\varepsilon}(|L^{-1}-M_{X^{\log
Y}}|^{2}+|M_{X^{\log Y}}-M_{X}|^{2}).$

A computation similar to (5.3) shows that $|L^{-1}-M_{X^{\log Y}}|\ll 1$.
Recalling the definitions of $X$ and $M_{X}(s)$, we collect the above
estimates and find that $N_{\pi\times\pi^{\prime}}(\sigma,T)$ is

$\ll_{\varepsilon}C(\pi^{\prime})^{\varepsilon}Q^{\varepsilon}T^{\varepsilon}\Big{(}Y^{\frac{1}{2}-\sigma}\Big{(}C(\pi^{\prime})^{\frac{n}{2}}Q^{2n^{\prime}n+\frac{n^{\prime}}{2}}T^{1+\frac{n^{\prime}n[F:\mathbb{Q}]}{4}}+\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}\mathrm{N}\mathfrak{n}\leq
X\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{\frac{1}{2}+\frac{1}{\log
Y}+iv}}\Big{|}^{2}dv\Big{)}\\\
+Y^{2(1-\sigma)}\Big{(}1+\int_{-T}^{T}\Big{|}\sum_{\begin{subarray}{c}X<\mathrm{N}\mathfrak{n}\leq
X^{\log Y}\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\frac{\mu_{\pi\times\pi^{\prime}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{1+\frac{1}{\log
Y}+iv}}\Big{|}^{2}dv\Big{)}\Big{)}.$

We sum over $\pi\in\mathfrak{F}_{n}(Q)$ and apply Corollary 4.2 and (5.1) to
conclude that

(5.5)
$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{\varepsilon}C(\pi^{\prime})^{\varepsilon}Q^{\varepsilon}T^{\varepsilon}(Y^{\frac{1}{2}-\sigma}X+Y^{2(1-\sigma)})\ll_{\varepsilon}(C(\pi^{\prime})QT)^{\varepsilon}X^{\frac{4(1-\sigma)}{3-2\sigma}}.$

###### Proof of Theorem 1.1.

It follows from our choice for $X$ in (5.1) that (5.5) is

$\ll_{\varepsilon}(C(\pi^{\prime})QT)^{\varepsilon}((Q^{n^{2}+n}T^{[F:\mathbb{Q}]n^{2}+\varepsilon}+C(\pi^{\prime})^{n}Q^{2n^{\prime}n+n^{\prime}/2}T^{1+n^{\prime}n[F:\mathbb{Q}]/2})|\mathfrak{F}_{n}(Q)|)^{\frac{4(1-\sigma)}{3-2\sigma}}.$

Using (1.4), we find that

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{\varepsilon}C(\pi)^{\varepsilon}Q^{\varepsilon}T^{5\varepsilon}(C(\pi^{\prime})QT^{[F:\mathbb{Q}]})^{9(n+1)\max\\{n,n^{\prime}\\}(1-\sigma)}.$

The theorem follows once we rescale $\varepsilon$ to $\varepsilon/5$. ∎

##  6\. Zero-free regions and bounds for Rankin–Selberg $L$-functions

###### Proof of Theorem 2.1.

Fix $A\geq 1$, and define

$\delta\coloneqq\frac{\varepsilon}{9(n+1)\max\\{n,n^{\prime}\\}+1},\qquad
B\coloneqq(A+2)(9(n+1)\max\\{n,n^{\prime}\\}+1).$

Observe that if $T\leq Q^{A}$, then Theorem 1.1 implies that

(6.1)
$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(\sigma,T)\ll_{A,\varepsilon}(C(\pi^{\prime})Q^{A+1})^{\frac{\varepsilon}{B}}(C(\pi^{\prime})QT^{[F:\mathbb{Q}]})^{9(n+1)\max\\{n,n^{\prime}\\}(1-\sigma)}$

once we rescale $\varepsilon$. For each integer $j\in[2,A\log Q]$, we
iteratively apply (6.1) with

$T=T_{j}\coloneqq e^{j}-3,\qquad\sigma=\sigma_{j}\coloneqq 1-\frac{\delta\log
Q}{\log(C(\pi^{\prime})Q(T_{j}+3)^{[F:\mathbb{Q}]})},$

discarding
$C(\pi^{\prime})^{\frac{\varepsilon}{B}}Q^{9(n+1)\max\\{n,n^{\prime}\\}\delta+\frac{\varepsilon(A+1)}{B}}$
exceptions at most $A\log Q$ times. Thus, for all $\pi\in\mathfrak{F}_{n}(Q)$
with at most

$C(\pi^{\prime})^{\frac{\varepsilon}{B}}Q^{9(n+1)\max\\{n,n^{\prime}\\}\delta+\frac{\varepsilon(A+1)}{B}}A\log
Q\ll_{A,\varepsilon}(C(\pi^{\prime})Q)^{\varepsilon}$

exceptions, we dyadically build the zero-free region

$\textup{Re}(s)\geq 1-\frac{\varepsilon\log
Q}{19n\max\\{n,n^{\prime}\\}\log(C(\pi^{\prime})Q(|\textup{Im}(s)|+3)^{[F:\mathbb{Q}]})},\quad|\textup{Im}(s)|\leq
Q^{A}.$

for $L(s,\pi\times\pi^{\prime})$. Since $C(\pi)\leq Q$, we conclude that the
more restrictive zero-free region

$\textup{Re}(s)\geq 1-\frac{\varepsilon\log
C(\pi)}{19n\max\\{n,n^{\prime}\\}\log(C(\pi)C(\pi^{\prime})(|\textup{Im}(s)|+3)^{[F:\mathbb{Q}]})},\quad|\textup{Im}(s)|\leq
C(\pi)^{A}$

holds for all $\pi\in\mathfrak{F}_{n}(Q)$ with at most
$O_{B,\varepsilon}((C(\pi^{\prime})Q)^{\varepsilon})$ exceptions, as desired.
∎

###### Proof of Theorem 2.5.

Let $\varepsilon>0$ and $\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$. Rescale
$\varepsilon$ to $\varepsilon/(54n\max\\{n,n^{\prime}\\}+3)$ in Theorem 1.1 so
that

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(\sigma,|t|+6)\ll_{\varepsilon}(C(\pi^{\prime})(3+|t|)^{[F:\mathbb{Q}]}Q)^{9(n+1)\max\\{n,n^{\prime}\\}(1-\sigma)+\frac{\varepsilon}{54n\max\\{n,n^{\prime}\\}+3}}.$

Let

(6.2) $\alpha_{0}=\frac{\varepsilon}{54n\max\\{n,n^{\prime}\\}+3},$

so that

$\sum_{\pi\in\mathfrak{F}_{n}(Q)}N_{\pi\times\pi^{\prime}}(1-\alpha_{0},|t|+6)\ll_{\varepsilon}(C(\pi^{\prime})(3+|t|)^{[F:\mathbb{Q}]}Q)^{\frac{\varepsilon}{3}}.$

Consequently, we have

(6.3) $|\\{\pi\in\mathfrak{F}_{n}(Q)\colon
N_{\pi\times\pi^{\prime}}(1-\alpha_{0},|t|+6)>0\\}|\ll_{\varepsilon}\begin{cases}Q^{\varepsilon}&\parbox{86.72267pt}{if
$C(\pi^{\prime})\leq Q$ and $|t|\leq Q^{1/[F:\mathbb{Q}]}$,}\\\
(C(\pi^{\prime})(3+|t|)^{[F:\mathbb{Q}]}Q)^{\varepsilon}&\mbox{otherwise.}\end{cases}$

Let $0\leq\alpha\leq\alpha_{0}$ and $\pi\in\mathfrak{F}_{n}(Q)$. We mimic the
proof of [65, Theorem 1.1], while replacing $\pi^{\prime}$ with
$\pi^{\prime}\otimes\left|\det\right|^{it}$, which has the effect of adding
$it$ to all of the archimedean parameters
$\mu_{j,j^{\prime},\pi\times\pi^{\prime}}(v)$, and find that555There are some
small changes that are required in order to maintain uniformity in $F$.
Ultimately, these changes do not affect the proof in a meaningful way.

$\displaystyle\log|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|$
$\displaystyle\leq\Big{(}\frac{1}{4}-\frac{\alpha}{10^{9}}\Big{)}\log
C(\pi\times\pi^{\prime},t)+2\log|L(\tfrac{3}{2}+it,\pi\times\pi^{\prime})|$
$\displaystyle+\frac{\alpha}{10^{7}}|\\{\rho=\beta+i\gamma\colon\beta\geq
1-\alpha,~{}|\gamma-t|\leq 6\\}|+O(1)$
$\displaystyle\leq\Big{(}\frac{1}{4}-\frac{\alpha}{10^{9}}\Big{)}\log
C(\pi\times\pi^{\prime},t)+2\log|L(\tfrac{3}{2}+it,\pi\times\pi^{\prime})|$
$\displaystyle+\frac{\alpha}{10^{7}}N_{\pi\times\pi^{\prime}}(1-\alpha,|t|+6)+O(1).$

By (3.10) and (3.11), there is an effectively computable constant
$\Cl[abcon]{kappa_{c}onst}=\Cr{kappa_{c}onst}(n,n^{\prime},[F:\mathbb{Q}],\varepsilon)>0$
such that

(6.4)
$\log|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|\leq\Big{(}\frac{1}{4}-\frac{9\alpha}{10^{10}}\Big{)}\log(C(\pi\times\pi^{\prime},t))\\\
+\frac{\alpha}{10^{7}}N_{\pi\times\pi^{\prime}}(1-\alpha,|t|+6)+\Cr{kappa_{c}onst}.$

Theorem 2.5 now follows from (6.3) and (6.4). ∎

In our proofs of Theorems 2.6, 2.8, and 9.13, we will use the following
result, which is more convenient and slightly weaker than Theorem 2.5.

######  Corollary 6.1.

Let $n,n^{\prime}\geq 1$ be integers and
$\pi^{\prime}\in\mathfrak{F}_{n^{\prime}}$. If $\varepsilon>0$,
$C(\pi^{\prime})\leq Q$, and $|t|\leq Q^{1/[F:\mathbb{Q}]}$, then

$|\\{\pi\in\mathfrak{F}_{n}(Q)\colon|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|\geq(C(\pi)^{n^{\prime}}C(\pi^{\prime})^{n}(1+|t|)^{n^{\prime}n[F:\mathbb{Q}]})^{\frac{1}{4}-\frac{\varepsilon}{10^{11}n\max\\{n,n^{\prime}\\}}}\\}|\ll_{\varepsilon}Q^{\varepsilon}.$

######  Remark.

The same conclusion holds when $\mathfrak{F}_{n}(Q)$ is replaced by a
subfamily $S(Q)$ that grows polynomially in $Q$.

###### Proof.

The proof of Theorem 2.5, namely (6.3) and (6.4), shows that if
$C(\pi^{\prime})\leq Q$ and $|t|\leq Q^{1/[F:\mathbb{Q}]}$, then for all
$\varepsilon>0$, there exists an effectively computable constant
$\Cl[abcon]{6.1const}=\Cr{6.1const}(n,n^{\prime},[F:\mathbb{Q}],\varepsilon)>0$
such that for all except $\ll_{\varepsilon}Q^{\varepsilon}$ of the
$\pi\in\mathfrak{F}_{n}(Q)$, we have

$\log|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|\leq\Big{(}\frac{1}{4}-\frac{9\alpha}{10^{10}}\Big{)}\log
C(\pi\times\pi^{\prime},t)+\Cr{6.1const},$

where $0\leq\alpha\leq\alpha_{0}$ and $\alpha_{0}$ is defined by (6.2). By
(3.9), we have

$\log|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|\leq\Big{(}\frac{1}{4}-\frac{9\alpha}{10^{10}}\Big{)}\log\big{(}C(\pi)^{n^{\prime}}C(\pi^{\prime})^{n}(3+|t|)^{n^{\prime}n[F:\mathbb{Q}]}\big{)}+\Cr{6.1const}.$

once we enlarge $\Cr{6.1const}$ accordingly. If $C(\pi)$ is sufficiently large
with respect to $n$, $n^{\prime}$, $[F:\mathbb{Q}]$, and $\varepsilon$, then

$\log|L(\tfrac{1}{2}+it,\pi\times\pi^{\prime})|\leq\Big{(}\frac{1}{4}-\frac{\varepsilon}{10^{11}n\max\\{n,n^{\prime}\\}}\Big{)}\log\big{(}C(\pi)^{n^{\prime}}C(\pi^{\prime})^{n}(1+|t|)^{n^{\prime}n[F:\mathbb{Q}]}\big{)}.$

The number of $\pi\in\mathfrak{F}_{n}(Q)$ such that $C(\pi)$ is not
sufficiently large with respect to $n$, $n^{\prime}$, $[F:\mathbb{Q}]$, and
$\varepsilon$ is $O_{\varepsilon}(1)$ per (1.4), and the corollary follows. ∎

##  7\. Effective multiplicity one on average

In this section, we consider the family of cuspidal automorphic
representations $\mathfrak{F}_{n}(Q)$ over a number field $F$. Recall our
convention that implied constants are allowed to depend on $n$,
$[F:\mathbb{Q}]$, and $\varepsilon$ unless specifically mentioned otherwise.
For $\pi_{1},\pi_{2}\in\mathfrak{F}_{n}$ define the numbers
$\Lambda_{\pi_{1}\times\pi_{2}}(\mathfrak{n})$ by the Dirichlet series
identity (for $\textup{Re}(s)>1$)

(7.1)
$\sum_{\mathfrak{n}}\frac{\Lambda_{\pi_{1}\times\pi_{2}}(\mathfrak{n})}{\mathrm{N}\mathfrak{n}^{s}}=-\frac{L^{\prime}}{L}(s,\pi_{1}\times\pi_{2})=\sum_{\mathfrak{p}}\sum_{k=1}^{\infty}\frac{(\sum_{j,j^{\prime}=1}^{n}\alpha_{j,j^{\prime},\pi_{1}\times\pi_{2}}(\mathfrak{p})^{k})\log\mathrm{N}\mathfrak{p}}{\mathrm{N}\mathfrak{p}^{ks}}.$

######  Lemma 7.1.

Let $\pi^{\prime}\in\mathfrak{F}_{n}$. There exist absolute and effectively
computable constants $\Cl[abcon]{Hoh5}\in(0,1)$;
$\Cl[abcon]{Hoh6},\Cl[abcon]{Hoh1},\Cl[abcon]{Hoh2},\Cl[abcon]{Hoh3}\geq 1$;
and $\Cl[abcon]{Hoh4}\in(0,1)$ such that the following are true.

  1. (1)

The $L$-function $L(s,\pi^{\prime}\times\widetilde{\pi}^{\prime})$ has at most
one zero, say $\beta_{1}$, in the region

$\textup{Re}(s)\geq
1-\frac{\Cr{Hoh5}}{\log(C(\pi^{\prime})^{n}(|\textup{Im}(s)|+e)^{n^{2}[F:\mathbb{Q}]})}.$

If $\beta_{1}$ exists, then it must be real and simple and satisfy
$\beta_{1}\leq 1-C(\pi^{\prime})^{-\Cr{Hoh6}n}$.

  2. (2)

If $A\geq\Cr{Hoh1}$, $\log\log
C(\pi^{\prime})\geq\Cr{Hoh2}n^{4}[F:\mathbb{Q}]^{2}$, and $x\geq
C(\pi^{\prime})^{\Cr{Hoh3}A^{2}n^{3}[F:\mathbb{Q}]\log(en[F:\mathbb{Q}])}$,
then

$\sum_{\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq
x}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})=\begin{cases}\frac{x}{2}(1-\xi^{\beta_{1}-1})(1+O(e^{-\Cr{Hoh4}A}))&\mbox{if
$\beta_{1}$ exists,}\\\
\frac{x}{2}(1+O(e^{-\Cr{Hoh4}A}))&\mbox{otherwise.}\end{cases}$

The implied constants are absolute and effectively computable, and
$\xi\in[\frac{x}{2},x]$ satisfies
$x^{\beta_{1}}-(\frac{x}{2})^{\beta_{1}}=\beta_{1}\frac{x}{2}\xi^{\beta_{1}-1}$.

###### Proof.

This is [30, Theorem 2.1] with $\delta=0$ and $x$ replaced with $x/2$. ∎

######  Corollary 7.2.

Let $\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)$, and let $x=\frac{1}{2}(\log
Q)^{B}$, where $B\geq n^{2}+1$ is a suitably large constant depending on $n$
and $[F:\mathbb{Q}]$. If
$C(\pi^{\prime})\ll_{n,[F:\mathbb{Q}]}x^{1/(A^{2}\Cr{Hoh3}n^{4}[F:\mathbb{Q}]^{2})}$,
then

$\sum_{\begin{subarray}{c}\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq x\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\gg
x^{1-\frac{2\Cr{Hoh6}}{A^{2}\Cr{Hoh3}n^{3}[F:\mathbb{Q}]^{2}}}.$

###### Proof.

Let $\pi,\pi^{\prime}\in\mathfrak{F}_{n}(Q)$. In Lemma 7.1, the lower bound on
$C(\pi^{\prime})$ only serves to ensure that the implied constants are
absolute. This was pertinent in [30], but it is not pertinent here. Thus, we
may replace the two conditions $\log\log
C(\pi^{\prime})\geq\Cr{Hoh2}n^{4}[F:\mathbb{Q}]^{2}$ and $x\geq
C(\pi^{\prime})^{A^{2}\Cr{Hoh3}n^{3}[F:\mathbb{Q}]\log(en[F:\mathbb{Q}])}$
with the single condition $C(\pi^{\prime})\ll
x^{1/(A^{2}\Cr{Hoh3}n^{4}[F:\mathbb{Q}]^{2})}$.

We want to refine Lemma 7.1 so that one only sums over $\mathfrak{n}$ such
that
$\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}$.
Using (3.2) and (3.6), we find that the contribution to Lemma 7.1(2) arising
from $\mathfrak{n}$ such that
$\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})\neq(1)$ is

$\displaystyle\sum_{\begin{subarray}{c}\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq
x\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})\neq(1)\end{subarray}}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\ll\sum_{\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}\sum_{\frac{\log(x/2)}{\log\mathrm{N}\mathfrak{p}}<j\leq\frac{\log
x}{\log\mathrm{N}\mathfrak{p}}}\mathrm{N}\mathfrak{p}^{j(1-\frac{2}{n^{2}+1})}\ll\frac{\log
Q}{\log\log Q}x^{1-\frac{2}{n^{2}+1}},$

which is $\ll x^{1-\frac{1}{n^{2}+1}}$. Thus, in the worst case where
$\beta_{1}$ in Lemma 7.1(1), it follows from Lemma 7.1(2) and the above
discussion that if
$C(\pi^{\prime})\ll_{n,[F:\mathbb{Q}]}x^{1/(A^{2}\Cr{Hoh3}n^{3}[F:\mathbb{Q}]\log(en[F:\mathbb{Q}]))}$,
then

$\displaystyle\sum_{\begin{subarray}{c}x<\mathrm{N}\mathfrak{n}\leq 2x\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\gg
x(1-\xi^{\beta_{1}-1})\gg_{n,[F:\mathbb{Q}]}x\min\\{1,(1-\beta_{1})\log
C(\pi^{\prime})\\}\gg\frac{x\log
C(\pi^{\prime})}{C(\pi^{\prime})^{\Cr{Hoh6}n}}.$

This implies the desired result. ∎

######  Lemma 7.3.

Let $n\geq 3$, $A>0$, and $\varepsilon>0$. Let $B\geq n^{2}+1$ depend at most
on $n$ and $[F:\mathbb{Q}]$. Let $\pi\in\mathfrak{F}_{n}(Q)$,
$\pi^{\prime}\in\mathfrak{F}_{n}((\log Q)^{A})$, and $\pi\neq\pi^{\prime}$.
Let $x=\frac{1}{2}(\log Q)^{B}$. Let $\Phi$ be a fixed smooth function
supported on a compact subset of $[\frac{1}{4},2]$ such that $0\leq\Phi(t)\leq
1$ for all $t\in[\frac{1}{4},2]$ and $\Phi(t)=1$ for $t\in(\frac{1}{2},1]$. If
$L(s,\pi\times\pi^{\prime})$ does not vanish in the region $\textup{Re}(s)\geq
1-\frac{\varepsilon}{20n^{2}}$ and $|\textup{Im}(s)|\leq\log Q$, then

$\sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}}\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\Phi(\mathrm{N}\mathfrak{n}/x)\ll_{B}x^{1+\frac{2}{B}-\frac{\varepsilon}{20n^{2}}}.$

###### Proof.

Writing $\widehat{\Phi}(s)=\int_{0}^{\infty}\Phi(t)t^{s-1}dt$ for the Mellin
transform of $\Phi$ (which is an entire function of $s$), we observe via
Mellin inversion that

$\sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}}\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\Phi(\mathrm{N}\mathfrak{n}/x)=\frac{1}{2\pi
i}\int_{2-i\infty}^{2+i\infty}-\frac{(L^{S})^{\prime}}{(L^{S})}(s,\pi\times\widetilde{\pi}^{\prime})\widehat{\Phi}(s)x^{s}ds,$

where
$L^{S}(s,\pi\times\widetilde{\pi}^{\prime})=L(s,\pi\times\widetilde{\pi}^{\prime})\prod_{\mathfrak{p}\mid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}}L(s,\pi_{\mathfrak{p}}\times\widetilde{\pi}_{\mathfrak{p}}^{\prime})^{-1}$.
A standard contour integral calculation using the argument principle shows
that the above display equals

$-\sum_{L^{S}(\rho,\pi\times\widetilde{\pi}^{\prime})=0}\widehat{\Phi}(\rho)x^{\rho},$

where $\rho$ ranges over all zeros (trivial and nontrivial) of
$L^{S}(s,\pi\times\widetilde{\pi}^{\prime})$.

Since $\Phi$ is compactly supported and $\widehat{\Phi}$ is entire, it follows
that for any $R\geq 2$, we have
$|\widehat{\Phi}(s)|\ll_{R}\min\\{1,|s|^{-R}e^{\textup{Re}(s)}\\}$. Note that
the reciprocals of the Euler factors of
$L(s,\pi\times\widetilde{\pi}^{\prime})$ at prime ideals
$\mathfrak{p}|\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$ and all of the
trivial zeros of $L(s,\pi\times\widetilde{\pi}^{\prime})$ have no zeros with
real part greater than $1-2/(n^{2}+1)$ per (3.2) and (3.6). Since for any
$t\in\mathbb{R}$ there are $\ll\log Q+\log(|t|+2)$ zeros $\rho=\beta+i\gamma$
of $L^{S}(s,\pi\times\widetilde{\pi}^{\prime})$ that satisfy $0<\beta<1$ and
$|\gamma-t|\leq 1$, we find for any $T\geq 1$, $R\geq 2$, and $\sigma_{0}\geq
1-2/(n^{2}+1)$ that

$\displaystyle\sum_{L^{S}(\rho,\pi\times\widetilde{\pi}^{\prime})=0}\widehat{\Phi}(\rho)x^{\rho}=\sum_{\begin{subarray}{c}L^{S}(\rho,\pi\times\widetilde{\pi}^{\prime})=0\\\
\beta\geq\sigma_{0}\\\ |\gamma|\leq
T\end{subarray}}\widehat{\Phi}(\rho)x^{\rho}+O(T^{1-R}\log(QT)x+T\log(QT)x^{\sigma_{0}}).$

We choose $T=\log Q=(2x)^{1/B}$ and $\sigma_{0}=1-\varepsilon/(20n^{2})$, in
which case our hypotheses imply that the sum over zeros in the above display
is empty and

$\Big{|}\sum_{L^{S}(\rho,\pi\times\widetilde{\pi}^{\prime})=0}\widehat{\Phi}(\rho)x^{\rho}\Big{|}\ll_{B}x^{1+\frac{2-R}{B}}+x^{\frac{2}{B}+\sigma_{0}}.$

The desired result follows from choosing $R=\max\\{B(1-\sigma_{0}),3\\}$. ∎

###### Proof of Theorem 2.2.

Let $0<\varepsilon<1$, let $Q$ be sufficiently large with respect to
$\varepsilon$, and let

(7.2) $B=41n^{2}/\varepsilon,\quad x=(\log
Q)^{B}/2,\quad\pi\in\mathfrak{F}_{n}(Q),\quad\pi^{\prime}\in\mathfrak{F}_{n}((\log
Q)^{\frac{B}{A^{2}\Cr{Hoh3}n^{4}[F:\mathbb{Q}]^{2}}}).$

Suppose in order to obtain a contradiction that $\pi\neq\pi^{\prime}$ and
$\pi_{\mathfrak{p}}\cong\pi_{\mathfrak{p}}^{\prime}$ for all
$\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$ with
$\mathrm{N}\mathfrak{p}\leq 2x$. Then
$A_{\pi}(\mathfrak{p})=A_{\pi^{\prime}}(\mathfrak{p})$ for all
$\mathfrak{p}\nmid\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}}$ with
$\mathrm{N}\mathfrak{p}\leq x$. By (3.5) and (7.1), it follows that
$\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})=\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})$
for all $\mathfrak{n}$ such that
$\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}$
and $\mathrm{N}\mathfrak{n}\leq x$. Since
$\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\geq 0$ for
all such $\mathfrak{n}$, the same holds for
$\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})$. Therefore, we
have that

$\sum_{\begin{subarray}{c}\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq
x\end{subarray}}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})=\sum_{\begin{subarray}{c}\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq
x\end{subarray}}\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n}).$

Recall the parameter $A$ in Corollary 7.2. If

(7.3) $C(\pi^{\prime})\ll x^{1/(A^{2}\Cr{Hoh3}n^{4}[F:\mathbb{Q}]^{2})}$

and $L(s,\pi\times\widetilde{\pi}^{\prime})\neq 0$ in the region
$\\{s\in\mathbb{C}\colon\mathrm{Re}(s)\geq
1-\frac{\varepsilon}{20n^{2}},~{}|\textup{Im}(s)|\leq\log Q\\}$, then by
Corollary 7.2 and Lemma 7.3, we conclude that

$x^{1-\frac{2\Cr{Hoh6}}{A^{2}\Cr{Hoh3}n^{3}[F:\mathbb{Q}]^{2}}}\ll\sum_{\begin{subarray}{c}\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq
x\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\Lambda_{\pi^{\prime}\times\widetilde{\pi}^{\prime}}(\mathfrak{n})=\sum_{\begin{subarray}{c}\frac{x}{2}<\mathrm{N}\mathfrak{n}\leq
x\\\
\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}\end{subarray}}\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\\\
\leq\sum_{\gcd(\mathfrak{n},\mathfrak{q}_{\pi}\mathfrak{q}_{\pi^{\prime}})=\mathcal{O}_{F}}\Lambda_{\pi\times\widetilde{\pi}^{\prime}}(\mathfrak{n})\Phi(\mathrm{N}\mathfrak{n}/x)\ll_{B}x^{1+\frac{2}{B}-\frac{\varepsilon}{20n^{2}}}.$

In particular, we find that
$1-\frac{2\Cr{Hoh6}}{A^{2}\Cr{Hoh3}n^{2}}<1+\frac{2}{B}-\frac{\varepsilon}{20n^{2}}$.
Thus, we are led to a contradiction once

(7.4)
$B>\frac{40n^{2}}{\varepsilon}\Big{(}1+\frac{40\Cr{Hoh6}}{A^{2}\Cr{Hoh3}\varepsilon-40\Cr{Hoh6}}\Big{)}.$

It remains to choose $A$ so that we determine a suitable power of $\log Q$ in
our upper bound for $C(\pi^{\prime})$ while ensuring that
$B=41n^{2}/\varepsilon$ satisfies (7.4). Recall that $\beta_{1}\leq
1-C(\pi^{\prime})^{-\Cr{Hoh6}n}$ and $A\geq\Cr{Hoh1}$ in Lemma 7.1. Since the
bound on $\beta_{1}$ only worsens as $\Cr{Hoh6}$ grows, we may assume that
$\Cr{Hoh6}$ is large enough so that
$\sqrt{3240\Cr{Hoh6}}/\sqrt{\Cr{Hoh3}\varepsilon}\geq\Cr{Hoh1}$. We then
choose $A=\sqrt{3240\Cr{Hoh6}}/\sqrt{\Cr{Hoh3}\varepsilon}$ to ensure that the
desired contradiction is achieved for $B=41n^{2}/\varepsilon$. Per (7.2), our
choices for $A$ and $B$ ensure that $C(\pi^{\prime})\leq(\log
Q)^{41/(3240\Cr{Hoh6}n^{4}[F:\mathbb{Q}]^{2})}$. Our use of Lemma 7.3 assumes
the zero-free region for $L(s,\pi\times\widetilde{\pi}^{\prime})$ stated
therein. By Theorem 2.1, our choices of $A$ and $B$ ensure that
$L(s,\pi\times\widetilde{\pi}^{\prime})$ has the desired zero-free region for
all except $O_{\varepsilon}(Q^{\varepsilon})$ of the
$\pi\in\mathfrak{F}_{n}(Q)$, provided that $Q$ is sufficiently large with
respect to $\varepsilon$ and $|\textup{Im}(s)|\leq\log Q$. ∎

##  8\. Hybrid-aspect quantum ergodicity for $\mathrm{GL}_{2}$ Hecke–Maaß
newforms

Let $\varphi$ be a Hecke–Maaß newform on
$\Gamma_{0}(q_{\varphi})\backslash\mathbb{H}$ with trivial nebentypus and
Laplace eigenvalue $\lambda_{\varphi}=\frac{1}{4}+t_{\varphi}^{2}>0$. Then
$\varphi$ is an eigenfunction of the hyperbolic Laplacian $\Delta$, all of the
Hecke operators, and the involution $T_{-1}$ sending $\varphi(z)$ to
$\varphi(-\overline{z})$. The eigenvalue $W_{\varphi}$ of $\varphi$ for
$T_{-1}$ is either $1$ or $-1$, leading to the respective Fourier expansions

(8.1) $\displaystyle\varphi(x+iy)$
$\displaystyle=\rho(\varphi)\sqrt{y}\sum_{m=1}^{\infty}\lambda_{\varphi}(m)K_{it_{\varphi}}(2\pi
my)\cos(2\pi mx),$ $\displaystyle\varphi(x+iy)$
$\displaystyle=\rho(\varphi)\sqrt{y}\sum_{m=1}^{\infty}\lambda_{\varphi}(m)K_{it_{\varphi}}(2\pi
my)\sin(2\pi nx),$

where $\rho(\varphi)$ is a normalizing constant that we will soon specify. We
consider the family

$\mathpzc{F}(Q)\coloneqq\\{\varphi\colon\lambda_{\varphi}q_{\varphi}\leq
Q\\}.$

It follows from work of Brumley and Milićević [9] that $\mathpzc{F}(Q)\asymp
Q^{2}$.

Let $(\varphi_{j})_{j=1}^{\infty}$ be the sequence of Hecke–Maaß newforms on
$\Gamma\backslash\mathbb{H}$ normalized to have Petersson norm 1, and let
$E(\cdot,\frac{1}{2}+it)$ denote a real-analytic Eisenstein series. With
$z=x+iy$, let

$\langle
f,g\rangle_{q}\coloneqq\int_{\Gamma_{0}(q)\backslash\mathbb{H}}f(z)\overline{g(z)}d\mu(z)$

be the level $q$ Petersson inner product. For a bounded measurable function
$H$ on $\Gamma\backslash\mathbb{H}$ and a Hecke–Maaß newform $\varphi$ on
$\Gamma_{0}(q_{\varphi})\backslash\mathbb{H}$, we define

$\mu_{\varphi}(H)=\int_{\Gamma_{0}(q_{\varphi})\backslash\mathbb{H}}|\varphi(z)|^{2}H(z)d\mu(z).$

We always consider $\varphi$ to be normalized so that $\mu_{\varphi}$ is a
probability measure. Subject to this normalization, we take the constant
$\rho(\varphi)$ in (8.1) to be such that $\lambda_{\varphi}(1)=1$.

###  8.1. Preliminaries

Let $B(w,r)$ be an injective geodesic ball on $\Gamma\backslash\mathbb{H}$ of
radius $r$ centered at $w\in\Gamma\backslash\mathbb{H}$, and let
$\mathbf{1}_{B(w,r)}$ be its indicator function. We will study the discrepancy

$D(\mu_{\varphi})=\sup_{\begin{subarray}{c}B(w,r)\subseteq\Gamma\backslash\mathbb{H}\\\
r>0,~{}w\in\Gamma\backslash\mathbb{H}\end{subarray}}\Big{|}\mu_{\varphi}(\mathbf{1}_{B(w,r)})-\frac{3}{\pi}\mu(\mathbf{1}_{B(w,r)})\Big{|}.$

Choose $T\geq e$ and define

$\displaystyle
D^{T}(\mu_{\varphi})\coloneqq\sup_{\begin{subarray}{c}B(w,r)\subseteq\Gamma\backslash\mathbb{H}\\\
w\in\Gamma\backslash\mathbb{H},~{}\textup{Im}(w)\geq
T\end{subarray}}\Big{|}\mu_{\varphi}(\mathbf{1}_{B})-\frac{3}{\pi}\mu(\mathbf{1}_{B})\Big{|},\quad
D_{T}(\mu_{\varphi})\coloneqq\sup_{\begin{subarray}{c}B(w,r)\subseteq\Gamma\backslash\mathbb{H}\\\
w\in\Gamma\backslash\mathbb{H},~{}\textup{Im}(w)<T\end{subarray}}\Big{|}\mu_{\varphi}(\mathbf{1}_{B})-\frac{3}{\pi}\mu(\mathbf{1}_{B})\Big{|}.$

The inequality
$D(\mu_{\varphi})=\max\\{D_{T}(\mu_{\varphi}),D^{T}(\mu_{\varphi})\\}\leq
D_{T}(\mu_{\varphi})+D^{T}(\mu_{\varphi})$ implies that

$D(\mu_{\varphi})^{2}\ll D_{T}(\mu_{\varphi})^{2}+D^{T}(\mu_{\varphi})^{2}.$

We first bound $D^{T}(\mu_{\varphi})^{2}$ using the work of Soundararajan
[64].

######  Lemma 8.1.

Let $\varphi\in\mathpzc{F}(Q)$. If $T\geq e$, then
$D^{T}(\mu_{\varphi})^{2}\ll(\log T)^{2}/T$.

###### Proof.

Consider an injective geodesic ball $B(w,r)$ with $\textup{Im}(w)\geq T$.
Observe that

$\displaystyle\Big{|}\mu_{\varphi}(\mathbf{1}_{B(w,r)})-\frac{3}{\pi}\mu(\mathbf{1}_{B(w,r)})\Big{|}\leq\mu_{\varphi}(\mathbf{1}_{B(w,r)})+\frac{3}{\pi}\mu(\mathbf{1}_{B(w,r)})\leq\mu_{\varphi}(\mathbf{1}_{B(w,r)})+O(T^{-2})$

and

$\mu_{\varphi}(\mathbf{1}_{B(w,r)})\ll\int_{\begin{subarray}{c}|x|\leq\frac{1}{2}\\\
y\geq T\end{subarray}}|\varphi(x+iy)|^{2}\frac{dx~{}dy}{y^{2}}.$

We expand $\varphi$ according to (8.1) and apply Parseval’s identity to obtain

$\int_{\begin{subarray}{c}|x|\leq\frac{1}{2}\\\ y\geq
T\end{subarray}}|\varphi(x+iy)|^{2}\frac{dx~{}dy}{y^{2}}=\frac{\rho(\varphi)^{2}}{2}\int_{1}^{\infty}|K_{ir}(2\pi
t)|^{2}\sum_{m\leq t/T}|\lambda_{\varphi}(m)|^{2}\frac{dt}{t}.$

Since $\varphi$ is has trivial nebentypus, the Hecke relations [31, Section
8.5] state that

$\lambda_{\varphi}(m_{1})\lambda_{\varphi}(m_{2})=\sum_{\begin{subarray}{c}d|\gcd(m_{1},m_{2})\\\
\gcd(d,q_{\varphi})=1\end{subarray}}\lambda_{\varphi}\Big{(}\frac{m_{1}m_{2}}{d^{2}}\Big{)}.$

Consequently, for integers $m,m_{1},m_{2}\geq 1$ and a prime $p$, we have the
bounds

$\displaystyle|\lambda_{\varphi}(p)|^{2}\leq
1+|\lambda_{\varphi}(p^{2})|,\qquad|\lambda_{\varphi}(mp^{2})|\leq|\lambda_{\varphi}(p^{2})\lambda_{\varphi}(m)|+|\lambda_{\varphi}(m)|+|\lambda_{\varphi}(m/p^{2})|,$
$\displaystyle|\lambda_{\varphi}(m_{1})\lambda_{\varphi}(m_{2})|\leq\sum_{d|m}|\lambda_{\varphi}(m_{1}m_{2}/d^{2})|,\qquad|\lambda_{\varphi}(pm)|\leq|\lambda_{\varphi}(p)\lambda_{\varphi}(m)|+|\lambda_{\varphi}(m/p)|.$

With these inequalities along with the multiplicative structure of the Hecke
eigenvalues $\lambda_{\varphi}(m)$, we can mimic the proof of [64, Theorem 3]
and conclude that

$\sum_{m\leq
x/y}|\lambda_{\varphi}(m)|^{2}\ll\frac{\log(ey)}{\sqrt{y}}\sum_{m\leq
x}|\lambda_{\varphi}(m)|^{2},\qquad 1\leq y\leq x,$

hence

$\displaystyle\frac{\rho(\varphi)^{2}}{2}\int_{1}^{\infty}|K_{ir}(2\pi
t)|^{2}\sum_{m\leq t/T}|\lambda_{\varphi}(m)|^{2}\frac{dt}{t}$
$\displaystyle\ll\frac{\log
T}{\sqrt{T}}\frac{\rho(\varphi)^{2}}{2}\int_{1}^{\infty}|K_{ir}(2\pi
t)|^{2}\sum_{m\leq t}|\lambda_{\varphi}(m)|^{2}\frac{dt}{t}$
$\displaystyle=\frac{\log
T}{\sqrt{T}}\int_{\begin{subarray}{c}|x|\leq\frac{1}{2}\\\ y\geq
1\end{subarray}}|\varphi(x+iy)|^{2}\frac{dx~{}dy}{y^{2}}$
$\displaystyle=\frac{\log
T}{\sqrt{T}}\mu_{\varphi}(\\{z\in\mathbb{C}\colon|\textup{Re}(z)|\leq\tfrac{1}{2},~{}\textup{Im}(z)\geq
1\\}).$

Since $\varphi$ is normalized so that $\mu_{\varphi}$ is a probability measure
and there exists a fundamental domain of $\Gamma_{0}(q)\backslash\mathbb{H}$
containing the set
$\\{z\in\mathbb{C}\colon|\textup{Re}(z)|\leq\tfrac{1}{2},~{}\textup{Im}(z)\geq
1\\}$, it follows that the above display is $\ll(\log T)/\sqrt{T}$. Thus,
$D^{T}(\mu_{\varphi})\ll(\log T)/\sqrt{T}$, and the result follows. ∎

######  Lemma 8.2.

Let $\varphi\in\mathpzc{F}(Q)$. If $M,T\geq e$, then

$D_{T}(\mu_{\varphi})^{2}\ll M^{-2}+(1+M^{-3}T)\Big{(}\sum_{|t_{k}|\ll
M}|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}+\int_{|t|\leq
M}|\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}dt\Big{)}$

with an absolute implied constant.

###### Proof.

We follow the strategy in [48, Section 5], which we include for completeness.
Let $B(w,r)\subseteq\Gamma\backslash\mathbb{H}$ be an injective geodesic ball
centered at $w\in\Gamma\backslash\mathbb{H}$ satisfying $\textup{Im}(w)<T$.
Define

$k_{r}(z,w)=\begin{cases}1&\mbox{if $d(z,w)<r$,}\\\
0&\mbox{otherwise,}\end{cases}\qquad
K_{r}(z,w)=\sum_{\gamma\in\Gamma}k_{r}(\gamma z,w).$

Here

$d(z,w)\coloneqq\log\frac{|z-\overline{w}|+|z-w|}{|z-\overline{w}|-|z-w|}$

is the hyperbolic distance between two points $z$ and $w$ in $\mathbb{H}$. It
follows from these definitions that $K_{r}(z,w)=\mathbf{1}_{B(w,r)}(z)$. We
spectrally expand $K_{r}(z,w)$ using [32, Theorem 15.7]. If $h_{r}(t)$ is the
Selberg–Harish-Chandra transform of $k_{r}(z,w)$ (see [32, Lemma 15.6]), then

(8.2)
$K_{r}(z,w)=\frac{3}{\pi}h_{r}\Big{(}\frac{i}{2}\Big{)}+\sum_{k=1}^{\infty}h_{r}(t_{k})\varphi_{k}(z)\overline{\varphi_{k}(w)}+\frac{1}{4\pi}\int_{\mathbb{R}}h_{r}(t)E(z,\tfrac{1}{2}+it)\overline{E(w,\tfrac{1}{2}+it)}dt.$

We smooth the sum and the integral in (8.2) as follows. Let
$\psi_{\varepsilon}(z,w)$ be a nonnegative mollifier supported inside of a
ball of radius $\varepsilon$ with the property that
$\int_{\mathbb{H}}\psi_{\varepsilon}(z,w)d\mu(z)=1$. We can and will choose
$\psi_{\varepsilon}(z,w)$ so that $\psi_{\varepsilon}(z,w)\ll\varepsilon^{-2}$
and its Selberg–Harish-Chandra transform $h^{(\varepsilon)}$ satisfies
$|h^{(\varepsilon)}(t)|\ll 1$ for $|t|\leq\varepsilon^{-1}$ and is rapidly
decreasing for $|t|>\varepsilon^{-1}$.

Given $B(w,r)$ as above, we consider $B(\zeta,r-2\varepsilon)$ and
$B(\zeta,r+2\varepsilon)$, subject to the convention that if $r\leq
2\varepsilon$, then $\mathbf{1}_{B(\zeta,r-2\varepsilon)}$ is identically
zero. For a function $F(z)$ on $\Gamma\backslash\mathbb{H}$, we define

$\overline{\psi}_{\varepsilon}(z,w)=\sum_{\gamma\in\Gamma}\psi_{\varepsilon}(\gamma
z,w),\qquad(F*\overline{\psi}_{\varepsilon})(z)\coloneqq\int_{\Gamma\backslash\mathbb{H}}F(w)\overline{\psi}_{\varepsilon}(w,z)d\mu(w).$

It follows by construction that
$k_{r-2\varepsilon}*\overline{\psi}_{\varepsilon}(z)\leq\mathbf{1}_{B(w,r)}(z)\leq
k_{r+2\varepsilon}*\overline{\psi}_{\varepsilon}(z)$. These two convolutions
have the following expansions per [48, Equation 48]:

$\displaystyle k_{r\pm 2\varepsilon}*\overline{\psi}_{\varepsilon}(z)$
$\displaystyle=\frac{3}{\pi}h_{r\pm
2\varepsilon}\Big{(}\frac{i}{2}\Big{)}h^{(\varepsilon)}\Big{(}\frac{i}{2}\Big{)}+\sum_{k=1}^{\infty}h_{r\pm
2\varepsilon}(t_{k})h^{(\varepsilon)}(t_{k})\varphi_{k}(z)\overline{\varphi_{k}(w)}$
$\displaystyle+\frac{1}{4\pi}\int_{\mathbb{R}}h_{r\pm
2\varepsilon}(t)h^{(\varepsilon)}(t)E(z,\tfrac{1}{2}+it)\overline{E(w,\tfrac{1}{2}+it)}dt.$

For any $H\in L^{2}(\Gamma\backslash\mathbb{H})$, we by [32, Theorem 15.5] and
the definition of $\mu_{\varphi}$ that

$\displaystyle\mu_{\varphi}(H)$
$\displaystyle=\frac{3}{\pi}\mu(H)+\sum_{k=1}^{\infty}\langle
H,\varphi_{k}\rangle_{1}\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}$
$\displaystyle+\frac{1}{4\pi}\int_{-\infty}^{\infty}\langle
H,E(\cdot,\tfrac{1}{2}+it)\rangle_{1}\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}dt.$

Therefore, we have

$\mu_{\varphi}(k_{r\pm
2\varepsilon}*\overline{\psi}_{\varepsilon})=\frac{3}{\pi}h_{r\pm
2\varepsilon}\Big{(}\frac{i}{2}\Big{)}h^{(\varepsilon)}\Big{(}\frac{i}{2}\Big{)}+\sum_{k=1}^{\infty}h_{r\pm
2\varepsilon}(t_{k})h^{(\varepsilon)}(t_{k})\overline{\varphi_{k}(w)}\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}\\\
+\frac{1}{4\pi}\int_{-\infty}^{\infty}h_{r\pm
2\varepsilon}(t)h^{(\varepsilon)}(t)\overline{E(w,\tfrac{1}{2}+it)}\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}dt.$

The inversion formula for the Selberg–Harish-Chandra transform implies that
$h_{r\pm
2\varepsilon}(\frac{i}{2})=\int_{\mathbb{H}}\psi_{\varepsilon}(z,w)\,d\mu(z)=1$
while $h_{r\pm 2\varepsilon}(\frac{i}{2})=\mu(\mathbf{1}_{B(w,r\pm
2\varepsilon)})=\mu(\mathbf{1}_{B(w,r)})+O(\varepsilon)$, so

$\Big{|}\mu_{\varphi}(\mathbf{1}_{B(w,r)})-\frac{3}{\pi}\mu(\mathbf{1}_{B(w,r)})\Big{|}\ll\varepsilon+\sum_{\pm}\Big{|}\sum_{k=1}^{\infty}h_{r\pm
2\varepsilon}(t_{k})h^{(\varepsilon)}(t_{k})\overline{\varphi_{k}(w)}\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}\\\
+\int_{-\infty}^{\infty}h_{r\pm
2\varepsilon}(t)h^{(\varepsilon)}(t)\overline{E(w,\tfrac{1}{2}+it)}\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}dt\Big{|}.$

Consequently,

$D_{T}(\mu_{\varphi})^{2}\ll\varepsilon^{2}+\sum_{\pm}\Big{(}\Big{|}\sum_{k=1}^{\infty}h_{r\pm
2\varepsilon}(t_{k})h^{(\varepsilon)}(t_{k})\overline{\varphi_{k}(w)}\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}\Big{|}^{2}\\\
+\Big{|}\int_{-\infty}^{\infty}h_{r\pm
2\varepsilon}(t)h^{(\varepsilon)}(t)\overline{E(w,\tfrac{1}{2}+it)}\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}dt\Big{|}^{2}\Big{)}.$

We first handle the contribution from the cuspidal spectrum. Note that by the
Cauchy–Schwarz inequality and our aforementioned decay properties for
$h^{(\varepsilon)}$, we have

$\Big{|}\sum_{k=1}^{\infty}h_{r\pm
2\varepsilon}(t_{k})h^{(\varepsilon)}(t_{k})\overline{\varphi_{k}(w)}\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}\Big{|}^{2}\\\
\leq\Big{(}\sum_{k=1}^{\infty}|h_{r\pm
2\varepsilon}(t_{k})\varphi_{k}(w)|^{2}|h^{(\varepsilon)}(t_{k})|\Big{)}\sum_{k=1}^{\infty}|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}|h^{\varepsilon}(t_{k})|\\\
\ll\Big{(}\sum_{\begin{subarray}{c}|t_{k}|\ll 1/\varepsilon\\\ k\geq
1\end{subarray}}|h_{r\pm
2\varepsilon}(t_{k})\varphi_{k}(w)|^{2}\Big{)}\sum_{\begin{subarray}{c}|t_{k}|\ll
1/\varepsilon\\\ k\geq
1\end{subarray}}|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}+\varepsilon^{2}.$

It follows from (8.2) that if $\textup{Im}(w)\leq T$, then

$\sum_{\begin{subarray}{c}|t_{k}|\ll 1/\varepsilon\\\ k\geq
1\end{subarray}}|h_{r\pm
2\varepsilon}(t_{k})\varphi_{k}(w)|^{2}\ll\int_{\Gamma\backslash\mathbb{H}}|K_{r+2\varepsilon}(z,w)|^{2}d\mu(z)\ll
1+\varepsilon^{3}T,$

hence

$\Big{|}\sum_{k=1}^{\infty}h_{r\pm
2\varepsilon}(t_{k})h^{(\varepsilon)}(t_{k})\overline{\varphi_{k}(w)}\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}\Big{|}^{2}\ll(1+\varepsilon^{3}T)\sum_{\begin{subarray}{c}|t_{k}|\ll
1/\varepsilon\\\ k\geq
1\end{subarray}}|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}.$

A verbatim argument for the contribution from the continuous spectrum shows
that

$\Big{|}\int_{-\infty}^{\infty}h_{r\pm
2\varepsilon}(t)h^{(\varepsilon)}(t)\overline{E(w,\tfrac{1}{2}+it)}\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}dt\Big{|}^{2}\\\
\ll(1+\varepsilon^{3}T)\int_{|t|\leq 1/\varepsilon}|\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}dt+\varepsilon^{2}.$

The lemma now follows by replacing $\varepsilon$ with $1/M$. ∎

We now relate the inner products
$|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}$ and $|\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}$ in Lemma 8.2
to values of $L$-functions on the critical line. Let
$\pi_{\varphi},\pi_{k}\in\mathfrak{F}_{2}$ correspond with $\varphi$ and
$\varphi_{k}$, respectively. Abusing notation, we use $\varphi$ and
$\pi_{\varphi}$ interchangeably, and similarly for $\varphi_{k}$ and
$\pi_{\varphi_{k}}$. Let $q_{\mathrm{Ad}^{2}\varphi}$ be the arithmetic
conductor of $\mathrm{Ad}^{2}\varphi$. The positive integer
$q_{\mathrm{Ad}^{2}\varphi}$ is a perfect square satisfying
$\sqrt{q_{\mathrm{Ad}^{2}\varphi}}|q_{\varphi}$; moreover, we have
$\sqrt{q_{\mathrm{Ad}^{2}\varphi}}=q_{\varphi}$ if and only if $q_{\varphi}$
is squarefree [59, Proposition 2.5].

######  Lemma 8.3.

Let $\varepsilon^{\prime}>0$, and let $W_{k}=W(\varphi_{k})\in\\{-1,1\\}$ be
the root number of $\varphi_{k}$. We have

$|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}\ll_{\varepsilon^{\prime}}(q_{\varphi}(3+|t_{\varphi}|))^{\frac{\varepsilon^{\prime}}{2}}(3+|t_{k}|)^{\frac{\varepsilon^{\prime}}{2}}\frac{(1+W_{k})q_{\mathrm{Ad}^{2}\varphi}^{-\frac{1}{2}}\Big{(}\frac{\sqrt{q_{\mathrm{Ad}^{2}\varphi}}}{q_{\varphi}}\Big{)}^{1-2\theta}|L(\frac{1}{2},\mathrm{Ad}^{2}\varphi\times\varphi_{k})|}{(1+|t_{k}|)^{\frac{1}{2}}(1+|2t_{\varphi}-t_{k}|^{\frac{1}{2}})(1+|2t_{\varphi}+t_{k}|^{\frac{1}{2}})}.$

###### Proof.

Nelson, Pitale, and Saha [59, Cor. 2.8, Thm 3.1, and Prop. 3.3] proved that

$|\langle\varphi_{k},|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}\leq(1+W_{k})\frac{10^{5\omega(q_{\varphi}/\sqrt{q_{\mathrm{Ad}^{2}\varphi}})}}{8q_{\varphi}}\frac{\Lambda(\frac{1}{2},\mathrm{Ad}^{2}\varphi\times\varphi_{k})\Lambda(\frac{1}{2},\varphi_{k})}{\Lambda(1,\mathrm{Ad}^{2}\varphi)^{2}\Lambda(1,\mathrm{Ad}^{2}\varphi_{k})}\tau\Big{(}\frac{q_{\varphi}}{\sqrt{q_{\mathrm{Ad}^{2}\varphi}}}\Big{)}^{2}\Big{(}\frac{q_{\varphi}}{\sqrt{q_{\mathrm{Ad}^{2}\varphi}}}\Big{)}^{2\theta},$

where $\omega(n)$ is the number of prime divisors of $n$, $\tau(n)$ is the
number of divisors of $n$, and $\theta\in[0,7/64]$ is the best exponent
towards GRC for $\pi\in\mathfrak{F}_{2}$. While they state their results in
the case where $\varphi$ is in fact a holomorphic cuspidal newform of weight
$k$, level $q$, and trivial nebentypus, their calculations are purely local.
Therefore, their result carries over to Hecke–Maaß newforms having trivial
nebentypus without any changes. This uses [69, Theorem 2] to show that the
archimedean normalized local integral $I_{\infty}^{\ast}$ in [59, Theorem 3.1]
equals $1$ if $\varphi_{k}$ is even and $0$ if $\varphi_{k}$ is odd.

In [25], it is shown that for all $\varepsilon^{\prime}>0$, we have

$L(1,\mathrm{Ad}^{2}\varphi)^{-1}\ll_{\varepsilon^{\prime}}(q_{\varphi}(3+|t_{\varphi}|))^{\frac{\varepsilon^{\prime}}{2}},\qquad
L(1,\mathrm{Ad}^{2}\varphi_{k})^{-1}\ll_{\varepsilon^{\prime}}(3+|t_{k}|)^{\frac{\varepsilon^{\prime}}{2}}.$

If $\varphi$ is dihedral, then the upper bound on
$L(1,\mathrm{Ad}^{2}\varphi)^{-1}$ is ineffective because it relies on
Siegel’s ineffective upper bound on $L(1,\chi)^{-1}$, where $\chi$ is a
primitive quadratic Dirichlet character. The lemma now follows from the
definition of $q_{\mathrm{Ad}^{2}\varphi}$, Stirling’s formula (see [13] for a
similar computation), and the convexity bound for
$L(\frac{1}{2},\varphi_{k})$. ∎

######  Lemma 8.4.

If $t\in\mathbb{R}$ and $\varepsilon^{\prime}>0$, then

$|\langle
E(\cdot,\tfrac{1}{2}+it),|\varphi|^{2}\rangle_{q_{\varphi}}|^{2}\ll_{\varepsilon^{\prime}}(q_{\varphi}(3+|t_{\varphi}|))^{\varepsilon^{\prime}}(3+|t|)^{\frac{1}{6}}\log(3+|t|)\frac{|L(\tfrac{1}{2}+it,\mathrm{Ad}^{2}\varphi)|^{2}}{q_{\varphi}(1+|t_{\varphi}|)}.$

###### Proof.

This is proved via the unfolding method (as in [57]), standard bounds for the
Riemann zeta function (including Weyl subconvexity), and the lower bound for
$L(1,\mathrm{Ad}^{2}\varphi)$ in the proof of Lemma 8.3. ∎

In order to bound the desired averages of the inner products in Lemmata 8.3
and 8.4, we require an understanding of the central values of the pertinent
$L$-functions on average. We obtain such an understanding using Corollary 6.1.

######  Proposition 8.5.

Let $\mathpzc{F}(Q)\coloneqq\\{\varphi\colon\lambda_{\varphi}q_{\varphi}\leq
Q\\}$, let $\varepsilon>0$, and let $|t|\leq Q$. For all except
$O_{\varepsilon}(Q^{\frac{1}{2}+3\varepsilon})$ of the
$\varphi\in\mathpzc{F}(Q)$, we have the bound

$|L(\tfrac{1}{2}+it,\mathrm{Ad}^{2}\varphi)|<(q_{\mathrm{Ad}^{2}\varphi}(3+|t_{\varphi}|)^{2}(1+|t|)^{3})^{\frac{1}{4}-\frac{\varepsilon}{9\cdot
10^{11}}}.$

Similarly, let $\varphi_{k}$ be an $L^{2}$-normalized Hecke–Maaß newform on
$\Gamma\backslash\mathbb{H}$. If $C(\varphi_{k})\leq Q$, then for all except
$O_{\varepsilon}(Q^{\frac{1}{2}+3\varepsilon})$ of the
$\varphi\in\mathpzc{F}(Q)$, we have the bound

$|L(\tfrac{1}{2},\mathrm{Ad}^{2}\varphi\times\varphi_{k})|<(q_{\mathrm{Ad}^{2}\varphi}^{2}(3+|t_{\varphi}|)^{4}(3+|t_{k}|)^{6})^{\frac{1}{4}-\frac{\varepsilon}{9\cdot
10^{11}}}.$

###### Proof.

We give the details for the second part only. The details for the first part
are simpler. We will separately estimate the cardinalities

(8.3) $|\\{\varphi\in\mathpzc{F}(Q)\colon\textup{$\varphi$ nondihedral,
}|L(\tfrac{1}{2},\mathrm{Ad}^{2}\varphi\times\varphi_{k})|\geq(q_{\mathrm{Ad}^{2}\varphi}^{2}(3+|t_{\varphi}|)^{4}(3+|t_{k}|)^{6})^{\frac{1}{4}-\frac{\varepsilon}{9\cdot
10^{11}}}\\}|$

and

(8.4) $|\\{\varphi\in\mathpzc{F}(Q)\colon\textup{$\varphi$ dihedral,
}|L(\tfrac{1}{2},\mathrm{Ad}^{2}\varphi\times\varphi_{k})|\geq(q_{\mathrm{Ad}^{2}\varphi}^{2}(3+|t_{\varphi}|)^{4}(3+|t_{k}|)^{6})^{\frac{1}{4}-\frac{\varepsilon}{9\cdot
10^{11}}}\\}|.$

If $\varphi_{1},\varphi_{2}\in\mathpzc{F}(Q)$ are such that
$\mathrm{Ad}^{2}\varphi_{1}=\mathrm{Ad}^{2}\varphi_{2}$, then by [17, Theorem
2], there exists a primitive quadratic Dirichlet character $\chi_{D}$ modulo
$D$ with $D\leq Q^{2}$ such that $\varphi_{1}=\varphi_{2}\otimes\chi_{D}$.
Consequently, there are $O_{\varepsilon}(Q^{1/2+\varepsilon})$ elements of
$\mathfrak{F}(Q)$ having the same given adjoint square lift.

If $\varphi\in\mathpzc{F}(Q)$ is nondihedral, then it follows from work of
Gelbart and Jacquet [20] that
$\mathrm{Ad}^{2}\pi_{\varphi}\in\mathfrak{F}_{3}$. Therefore, since
$C(\varphi_{k})=C(\pi_{k})\leq Q$ and $C(\mathrm{Ad}^{2}\varphi)\ll
C(\varphi)^{2}$, the above discussion and Corollary 6.1 imply that (8.3) is

$\ll_{\varepsilon}Q^{\frac{1}{2}+\varepsilon}|\\{\pi\in\mathfrak{F}_{3}(Q^{2})\colon|L(\tfrac{1}{2},\pi\times\varphi_{k})|\geq(C(\pi)^{2}C(\varphi_{k})^{3})^{\frac{1}{4}-\frac{\varepsilon}{9\cdot
10^{11}}}\\}|\ll_{\varepsilon}Q^{\frac{1}{2}+3\varepsilon}.$

If $\varphi\in\mathpzc{F}(Q)$ is dihedral, then there exists a real quadratic
extension $E/\mathbb{Q}$ of discriminant $D>1$ and a Hecke character $\chi$ of
$E$ such that $\varphi$ is the automorphic induction of $\chi$; in particular,
$\varphi$ has arithmetic conductor $D^{2}\leq Q$ [28, Lemma 4.2]. The adjoint
square lift of $\varphi$ has the isobaric decomposition
$\mathrm{Ad}^{2}\varphi=\chi_{D}\boxplus\varphi^{\prime}\otimes\chi_{D}$,
where $\varphi^{\prime}$ is the automorphic induction of $\chi^{2}$ and