abconsymbol=c

# Effective Sato–Tate distributions for Surfaces Arising from Products of
Elliptic Curves

Quanlin Chen  Department of Mathematics, Stanford University
<EMAIL_ADDRESS>and  Eric Shen  Department of Mathematics, Harvard
University<EMAIL_ADDRESS>
###### Abstract.

We prove, with an unconditional effective error bound, the Sato–Tate
distributions for two families of surfaces arising from products of elliptic
curves, namely a one-parameter family of K3 surfaces and double quadric
surfaces. To prove these effective Sato-Tate distributions, we prove an
effective form of the joint Sato–Tate distribution for two twist-inequivalent
elliptic curves, along with an effective form of the Sato–Tate distribution
for an elliptic curve for primes in arithmetic progressions. The former
completes the previous work [28] of Thorner by including the cases in which
one of the elliptic curves has CM.

##  1\. Introduction and Statement of Result

Let $E/\mathbb{Q}$ be an elliptic curve over $\mathbb{Q}$ of conductor $N_{E}$
without complex multiplication (CM). Hasse proved that for each prime $p$, the
group $E(\mathbb{F}_{p})$ of $\mathbb{F}_{p}$-rational points on the reduction
of $E$ modulo $p$ satisfies the bound

$|p+1-\\#E(\mathbb{F}_{p})|<2\sqrt{p}.$

For $p\nmid N_{E}$, we define $a_{E}(p)\mathrel{\mathop{\mathchar
58\relax}}=p+1-\\#E(\mathbb{F}_{p})$. Hasse’s bound implies that there exists
$\theta_{E}(p)\in[0,\pi]$ such that

$a_{E}^{*}(p)\mathrel{\mathop{\mathchar
58\relax}}=\frac{a_{E}(p)}{\sqrt{p}}=2\cos\theta_{E}(p).$

In 2011, Barnet-Lamb, Geraghty, Harris, and Taylor [4] proved the celebrated
Sato–Tate conjecture, which states that for a fixed subinterval
$[a,b]\subseteq[-2,2]$, we have that

$\lim_{x\to\infty}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{E}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}=\frac{1}{\pi}\int_{a}^{b}\sqrt{1-\Big{(}\frac{t}{2}\Big{)}^{2}}dt.$

Thorner [28] quantified the rate of convergence with effective dependence on
$E$. In particular, for every subinterval $[a,b]\subset[-2,2]$, we have
that111In this paper, all implied constants are positive, absolute, and
effectively computable. The sequence $c_{1},c_{2},c_{3},\ldots$ denotes a
sequence of certain positive, absolute, and effectively computable constants.

$\bigg{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{E}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\frac{1}{\pi}\int_{a}^{b}\sqrt{1-\Big{(}\frac{t}{2}\Big{)}^{2}}dt\bigg{|}\ll\frac{\log(N_{E}\log
x)}{\sqrt{\log x}},\qquad x\geq 3.$

The proof crucially relies on the work of Newton and Thorne [18, 19], proving
that for all integers $m\geq 1$, the $m$-th symmetric power $L$-function
$L(s,\mathrm{Sym}^{m}E)$ is the $L$-function of a unitary cuspidal automorphic
representation of $\mathrm{GL}_{m+1}$. If for each $m\geq 1$ the generalized
Riemann hypothesis is known for $L(s,\mathrm{Sym}^{m}E)$, then one can prove a
more rapid rate of convergence.

For a closed subinterval $I\subseteq[0,\pi]$, a change of variables yields the
equivalent statement

$\bigg{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}\theta_{E}(p)\in I\right\\}}{\\#\\{p\leq
x\\}}-\frac{2}{\pi}\int_{I}(\sin\theta)^{2}d\theta\bigg{|}\leq\Cl[abcon]{satotatestatement}\frac{\log(N_{E}\log
x)}{\sqrt{\log x}},\qquad x\geq 3.$

Note that $\frac{2}{\pi}(\sin\theta)^{2}d\theta$ is the pushforward of the
Haar measure on $\text{SU}_{2}(\mathbb{C})$ under the trace map. This
observation leads to a natural generalization of the Sato–Tate conjecture to
other abelian varieties $A/\mathbb{Q}$. One might hope to find a suitable
topological group $\mathrm{ST}(A)$ (the Sato–Tate group of $A$) such that the
sequence $(x_{p})$ of conjugacy classes of normalized images of Frobenius
elements in $\mathrm{ST}(A)$ at primes $p$ of good reduction are
equidistributed with respect to the pushforward of the Haar measure on
$\mathrm{ST}(A)$ to its space of conjuacy classes. See Sutherland [27, Section
3] for further discussion, including a recipe for how one might construct a
good candidate for $\mathrm{ST}(A)$.

In this paper, we provide the first unconditional effective bound of the error
on the Sato–Tate distribution for two families of surfaces that arise from
products of elliptic curves. The first family of surfaces we consider is the
one-parameter family of K3 surfaces first described in [1], defined by

$\displaystyle X_{\lambda}\mathrel{\mathop{\mathchar
58\relax}}s^{2}=xy(x+1)(y+1)(x+\lambda y),$

where $\lambda\in\mathbb{Q}-\\{0,-1\\}$. It is not immediately evident that
$X_{\lambda}$ is naturally related to a product of elliptic curves, but as
detailed in [1], the zeta function of $X_{\lambda}$ is related to the
symmetric square of the zeta function of the Clausen elliptic curve
$E_{\lambda}^{\text{Cl}}$, which is given by

(1.1) $E_{\lambda}^{\text{Cl}}\mathrel{\mathop{\mathchar
58\relax}}y^{2}=(x-1)(x^{2}+\lambda).$

To more explicitly state the relation between the Clausen elliptic curve
$E_{\lambda}^{\text{Cl}}$ and $X_{\lambda}$, we will introduce some notation.
For a prime $p$, let $\phi_{p}$ be the unique quadratic character modulo $p$.
For any two Dirichlet characters $\chi,\chi^{\prime}$ over
$(\mathbb{Z}/p\mathbb{Z})^{\times}$, define the normalized Jacobi sum by

$\binom{\chi}{\chi^{\prime}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{\chi^{\prime}(-1)}{p}\sum_{x\in\mathbb{F}_{p}}\chi(x)\overline{\chi^{\prime}}(1-x).$

Define

${}_{3}F_{2}(\lambda,p)\mathrel{\mathop{\mathchar
58\relax}}=\frac{p}{p-1}\sum_{\chi(p)}\binom{\phi_{p}\chi}{\chi}^{3}\chi(\lambda),$

where the summation is taken over all Dirichlet characters modulo $p$. Note
that this is a specific evaluation of the more general Gaussian hypergeometric
function over finite fields, first defined by Greene in [8], [9]. Now, the
Clausen elliptic curves $E_{\lambda}^{\textup{Cl}}$ and K3 surfaces
$X_{\lambda}$ are related by

$\displaystyle|X_{\lambda}(\mathbb{F}_{p})|=1+p^{2}+19p+p^{2}\leftidx{{}_{3}}{F}_{2}(-\lambda,p$

and

(1.2) $\displaystyle
p+p^{2}\phi_{p}(\lambda+1)_{3}F_{2}\left(\frac{\lambda}{\lambda+1},p\right)=a_{\lambda}^{\text{Cl}}(p)^{2},$

where $a_{\lambda}^{\text{Cl}}(p)$ is the trace of Frobenius of
$E_{\lambda}^{\text{Cl}}$ [20, Theorem 11.18], [1, Proposition 4.1]. Now, the
normalized trace of Frobenius of $X_{\lambda}$ is

$a_{X_{\lambda}}^{*}(p)=p\leftidx{{}_{3}}{F}_{2}(-\lambda,p).$

The limiting distribution of $a_{X_{\lambda}}^{*}(p)$ varied over all
$\lambda\in\mathbb{F}_{p}$ as $p\to\infty$ was studied by Ono, Saad, and
Saikia in [21] and [24]. Let $B(x)$ be the “Batman” distribution defined over
$[-3,3]$ by

$\displaystyle B(x)$
$\displaystyle=\begin{cases}\frac{3+x}{\sqrt{3-2x-x^{2}}}+\frac{3-x}{\sqrt{3+2x-x^{2}}}&|x|<1,\\\
\frac{3-|x|}{\sqrt{3+2|x|-x^{2}}}&1\leq|x|\leq 3,\\\
0&\text{otherwise.}\end{cases}$

In [24, Corollary 1.4], Saad proved that for any prime $p\geq 5$ and sub-
interval $[a,b]\subset[-3,3]$,

$\bigg{|}\frac{\\#\\{\lambda\in\mathbb{F}_{p}\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\\}}{p}-\frac{1}{4\pi}\int_{a}^{b}B(t)dt\bigg{|}\leq\frac{110.84}{p^{1/4}}.$

In this paper, we prove the Sato–Tate distribution of $a_{X_{\lambda}}^{*}(p)$
for a fixed K3 surface $X_{\lambda}$ when $p$ varies, with an effective error
bound. In the generic case that $\lambda+1$ is not a rational square and
$\lambda\not\in\\{1/8,1,-1/4,-1/64,-4,-64\\}$, we recover the “Batman”
distribution in [21].

######  Theorem 1.1.

Let $\lambda\in\mathbb{Q}$ satisfy

$\displaystyle\lambda\notin\\{r\in\mathbb{Q}\colon\sqrt{r+1}\in\mathbb{Q}\\}\cup\\{-64,-4,-\tfrac{1}{4},-\tfrac{1}{64},\tfrac{1}{8},1\\}.$

Let $\lambda_{1},\lambda_{2}\in\mathbb{Z}$ satisfy
$\gcd(\lambda_{1},\lambda_{2})=1$ and
$\lambda+1=\frac{\lambda_{1}}{\lambda_{2}}$. Let $q_{\lambda}$ be the
squarefree part of $\lambda_{1}\lambda_{2}$. Let $N_{\lambda}$ be the
conductor of the Clausen elliptic curve
$E_{-\lambda/(\lambda+1)}^{\mathrm{Cl}}$ defined by (1.1). Then there exists
an absolute constant $\Cl[abcon]{11}$ such that if $x\geq 3$ and
$[a,b]\subseteq[-3,3]$, then

$\Big{|}\frac{\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}B(t)dt\Big{|}\ll
x^{-\Cr{11}/\sqrt{q_{\lambda}}}+\frac{\log(N_{\lambda}q_{\lambda}\log
x)}{\sqrt{\log x}}.$

######  Theorem 1.2.

Let
$\lambda\in\\{r\in\mathbb{Q}\colon\sqrt{r+1}\in\mathbb{Q}\\}-\\{0,-1,8\\}$,
and let $N_{\lambda}$ be the conductor of the Clausen elliptic curve
$E_{-\lambda/(\lambda+1)}^{\mathrm{Cl}}$ defined by (1.1). If $x\geq 3$ and
$[a,b]\subseteq[-3,3]$, then

$\Big{|}\frac{\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}\frac{1}{2\pi}\sqrt{\frac{3-t}{1+t}}dt\Big{|}\ll\frac{\log(N_{\lambda}\log
x)}{\sqrt{\log x}}.$

######  Remark 1.3.

The “Batman” distribution and the distribution given by
$\frac{1}{2\pi}\sqrt{\frac{3-t}{1+t}}$ are the pushforwards of the Haar
measures on the corresponding Sato-Tate groups $\textup{O}_{3}(\mathbb{R})$
and $\textup{SO}_{3}(\mathbb{R})$ under the trace map.

######  Remark 1.4.

The corresponding results for all other $\lambda\in\mathbb{Q}-\\{0,-1\\}$ are
given in Theorem 2.1.

Now, consider any two twist-inequivalent elliptic curves

$E\mathrel{\mathop{\mathchar
58\relax}}y^{2}z=x^{3}+\Lambda_{1}xz^{2}+\Lambda_{2}z^{3}$

and

$E^{\prime}\mathrel{\mathop{\mathchar 58\relax}}y^{2}z^{\prime}=x^{\prime
3}+\Lambda_{3}x^{\prime}z^{\prime 2}+\Lambda_{4}z^{\prime 3}$

over $\mathbb{Q}$. The second family of surfaces that we consider are the
_double quadric surfaces_

$\mathcal{Z}=\mathcal{Z}(E,E^{\prime})\mathrel{\mathop{\mathchar
58\relax}}y^{2}=zz^{\prime}(x^{3}+\Lambda_{1}xz^{2}+\Lambda_{2}z^{3})(x^{\prime
3}+\Lambda_{3}x^{\prime}z^{\prime 2}+\Lambda_{4}z^{\prime 3}).$

The normalized trace of Frobenius for $\mathcal{Z}$ is

$a_{\mathcal{Z}}^{*}(p)=\frac{1}{p}\sum_{x,x^{\prime}}\left(\frac{x^{3}+\Lambda_{2}x+\Lambda_{3}}{p}\right)\left(\frac{x^{\prime
3}+\Lambda_{4}x^{\prime}+\Lambda_{5}}{p}\right)=a_{E}^{*}(p)a_{E^{\prime}}^{*}(p).$

Our effective form of the Sato–Tate distribution for $\mathcal{Z}$ may be
stated as follows.

######  Theorem 1.5.

Let $E$ and $E^{\prime}$ be two twist-inequivalent non-CM elliptic curves with
conductors $N_{E}$ and $N_{E^{\prime}}$, respectively. Let $C_{1}(t)$ be given
by

$C_{1}(t)=\frac{2}{\pi^{2}}\int_{|t|/2}^{2}\frac{1}{u}\sqrt{\bigg{(}1-\Big{(}\frac{u}{2}\Big{)}^{2}\bigg{)}\left(1-\Big{(}\frac{|t|}{2u}\Big{)}^{2}\right)}du.$

If $x\geq 16$ and $[a,b]\subset[-4,4]$, then

$\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{\mathcal{Z}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}C_{1}(t)dt\Big{|}\ll{\frac{\sqrt{\log(N_{E}N_{E^{\prime}}\log\log
x)}}{\sqrt[4]{\log\log x}}}.$

######  Remark 1.6.

The corresponding results in the cases where $E,E^{\prime}$ are possibly CM
are given in Theorem 2.2.

In order to prove Theorems 1.1, 1.2, and 1.5, we make effective the joint
Sato–Tate distribution for any pair of elliptic curves $E$ and $E^{\prime}$.
While generically $E$ will not be a twist of $E^{\prime}$, the effective joint
Sato–Tate distribution for $E$ and $E^{\prime}$ when they are twist-equivalent
can be recovered by understanding the effective Sato–Tate distribution for $E$
with the primes restricted to certain arithmetic progressions. With this
reduction in mind, we state our main result as being a classification of the
effective joint Sato–Tate distributions for arbitrary pairs of elliptic
curves. We will deduce Theorems 1.1, 1.2, and 1.5 as corollaries.

To state our main result, we first introduce some notation. Given coprime
integers $a,q\geq 1$, define

$\displaystyle\pi(x)$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\\#\\{p\leq x\\}$ $\displaystyle\pi(x;q,a)$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}=\\#\\{p\leq
x\mathrel{\mathop{\mathchar 58\relax}}p\equiv a\bmod q\\}.$

Now, let $E$ and $E^{\prime}$ be twist-inequivalent elliptic curves of
conductors $N_{E}$ and $N_{E^{\prime}}$, respectively. Given any interval $I$
and real number $r$, let $\mathbf{1}_{I}$ be the indicator function for $I$
and let $\mathbf{1}_{r\in I}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{1}_{I}(r)$. Assuming $I,I^{\prime}\subseteq[0,\pi]$, define

$\pi_{E,I}(x)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{p\leq
x}\mathbf{1}_{I}(\theta_{E}(p)),$ $\pi_{E,I}(x;q,a)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv
a\bmod{q}\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p)),$
$\pi_{E,E^{\prime},I,I^{\prime}}(x)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{p\leq
x}\mathbf{1}_{I}(\theta_{E}(p))\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p)).$

Define

$\mu_{\mathrm{CM}}(I)\mathrel{\mathop{\mathchar
58\relax}}=\frac{|I|}{2\pi}+\frac{1}{2}\mathbf{1}_{\frac{\pi}{2}\in
I},\qquad\mu_{\mathrm{ST}}(I)=\frac{2}{\pi}\int_{I}(\sin\theta)^{2}d\theta.$

Finally, if $E$ has complex multiplication (CM) over an imaginary quadratc
field $K$, then let $D_{K}$ be the discriminant of $K$ and define

$\delta(K,q)=\begin{cases}1&\mbox{if $D_{K}|\thinspace q$,}\\\
0&\mbox{otherwise.}\end{cases}$

We may now state our main result as follows.

######  Theorem 1.7.

Let $E,E^{\prime}$ be two twist-inequivalent elliptic curves over
$\mathbb{Q}$. Let $a,q$ be coprime positive integers, let
$I=[\alpha,\beta]\subseteq[0,\pi]$, and let
$I^{\prime}=[\alpha^{\prime},\beta^{\prime}]\subseteq[0,\pi]$. Finally, if
$E,E^{\prime}$ both have complex multiplication over a (possibly distinct)
imaginary quadratic field, denote those two fields as $K$ and $K^{\prime}$,
respectively.

  1. (1)

The following are true for all $x\geq 16$.

    1. (a)

If $E$ and $E^{\prime}$ are both non-CM,

$\displaystyle\Big{|}\pi_{E,E^{\prime},I,I^{\prime}}(x)-\mu_{\textup{ST}}(I)\mu_{\textup{ST}}(I^{\prime})\pi(x)\Big{|}\ll\pi(x){\frac{\log(N_{E}N_{E^{\prime}}\log\log
x)}{\sqrt{\log\log x}}}.$

    2. (b)

If $E$ is CM but $E^{\prime}$ is non-CM,

$\displaystyle\Big{|}\pi_{E,E^{\prime},I,I^{\prime}}(x)-\mu_{\mathrm{CM}}(I)\mu_{\textup{ST}}(I^{\prime})\pi(x)\Big{|}\ll\pi(x)\frac{\log(N_{E})^{4}\log(N_{E^{\prime}}\log
x)}{\sqrt{\log x}}.$

    3. (c)

If $E,E^{\prime}$ are both CM:

      1. (i)

When the discriminants of $K$ and $K^{\prime}$ are coprime, there exists an
absolute constant $\Cl[abcon]{161c1}$ such that

$\displaystyle\Big{|}\pi_{E,E^{\prime},I,I^{\prime}}(x)-\mu_{\mathrm{CM}}(I)\mu_{\mathrm{CM}}(I^{\prime})\pi(x)\Big{|}\ll\pi(x)\exp\Big{(}\frac{-\Cr{161c1}\log
x}{\sqrt{\log x}+3\log N_{E}N_{E^{\prime}}}\Big{)}.$

      2. (ii)

When the discriminants of $K$ and $K^{\prime}$ are not coprime, there exists
an absolute constant $\Cl[abcon]{161c2}$ such that

$\displaystyle\Big{|}\pi_{E,E^{\prime},I,I^{\prime}}(x)-\frac{1}{2}\Big{(}\frac{|I||I^{\prime}|}{\pi^{2}}+\mathbf{1}_{\frac{\pi}{2}\in
I}\mathbf{1}_{\frac{\pi}{2}\in
I^{\prime}}\Big{)}\pi(x)\Big{|}\ll\pi(x)\exp\Big{(}\frac{-\Cr{161c2}\log
x}{\sqrt{\log x}+3\log N_{E}N_{E^{\prime}}}\Big{)}.$

  2. (2)

The following are true for all $x\geq 16$.

    1. (a)

If $E$ is non-CM, then there exists an absolute constant $\Cl[abcon]{162b1}$
such that

$\Big{|}\pi_{E,I}(x;q,a)-\mu_{\textup{ST}}(I)\frac{\pi(x)}{\varphi(q)}\bigg{|}\ll\frac{\pi(x)}{\varphi(q)}\Big{(}x^{-\Cr{162b1}/\sqrt{q}}+\frac{\log(N_{E}q\log
x)}{\sqrt{\log x}}\bigg{)}.$

    2. (b)

If $E$ is CM, then there exists an absolute constant $\Cl[abcon]{162b2}$ such
that

$\displaystyle\Big{|}\pi_{E,I}(x;q,a)$
$\displaystyle-\mu_{\textup{CM}}(I)\frac{\pi(x)}{\varphi(q)}-\chi_{K}(a)\delta(K,q)\frac{\pi(x)}{2\varphi(q)}\Big{(}\frac{|I|}{\pi}-\mathbf{1}_{\pi/2\in
I}\Big{)}\Big{|}$
$\displaystyle\ll\frac{\pi(x)}{\varphi(q)}\Big{(}x^{-\Cr{162b1}/\sqrt{q}}+(\log
x)^{9/2}\exp\Big{(}\frac{-\Cr{162b2}\log x}{\sqrt{\log
x}+\log(N_{E}q)}\Big{)}\Big{)}.$

This paper is organized as follows. In Section 2 we state and prove the
general results for the effective Sato-Tate distributions for K3 surfaces and
double quadric surfaces, including Theorem 1.1, Theorem 1.2 and Theorem 1.5.
The proof applies Theorem 1.7 which will be proven later. In Section 3, we
present background information on Beurling-Selberg polynomials, elliptic
curves, and symmetric power $L$-functions that will be required to prove
Theorem 1.7. In Sections 4 and 5, we prove all the zero-free regions and prime
number theorems (respectively), used in the proof of Theorem 1.7. In Sections
6, 7, 8, 9, we prove parts 2.a, 2.b, 1.b, and 1.c of Theorem 1.7,
respectively. In Appendix A, we plot the Sato–Tate distributions for all
surfaces covered in Theorem 2.1 and Theorem 2.2 against numerical examples.

## Acknowledgements

The authors would like to thank Jesse Thorner for advising this project and
for many helpful comments and discussions, and Ken Ono and Hasan Saad for many
helpful discussions and valuable comments. The authors were participants in
the 2022 UVA REU in Number Theory. They are grateful for the support of grants
from the National Science Foundation (DMS-2002265, DMS-2055118, DMS-2147273),
the National Security Agency (H98230-22-1-0020), and the Templeton World
Charity Foundation. The authors used Wolfram Mathematica for computations.

##  2\. Proofs of Theorem 1.1, Theorem 1.2, and Theorem 1.5

###  2.1. Statement of the general results

We will start by stating the most general results for the effective Sato-Tate
distributions of K3 surfaces and double quadric surfaces. Throughout this
section, let $\lambda\in\mathbb{Q}-\\{0,1\\}$ denote a rational number. Let
$\lambda_{1},\lambda_{2}\in\mathbb{Z}$ satisfy
$\gcd(\lambda_{1},\lambda_{2})=1$ and
$\lambda+1=\frac{\lambda_{1}}{\lambda_{2}}$, and let $q_{\lambda}$ be the
squarefree part of $\lambda_{1}\lambda_{2}$. Let $N_{\lambda}$ be the
conductor of the Clausen elliptic curve
$E_{-\lambda/(\lambda+1)}^{\mathrm{Cl}}$ defined in (1.1). For a real
$r\in\mathbb{R}$ and closed interval $I=[a,b]\subset\mathbb{R}$, define
$\mathbf{1}_{r\in I}$ to be $1$ if $r\in I$ and 0 otherwise. Let $E$ and
$E^{\prime}$ be two elliptic curves of conductors $N_{E}$ and
$N_{E^{\prime}}$, respectively. Finally, define the “flying Batman”
distribution $B_{1}(x)$ by

$\displaystyle B_{1}(x)$
$\displaystyle=\begin{cases}\frac{1}{4\pi\sqrt{3-2x-x^{2}}}+\frac{1}{4\pi\sqrt{3+2x-x^{2}}}&|x|<1,\\\
\frac{1}{4\pi\sqrt{3+2|x|-x^{2}}}&1\leq|x|\leq 3,\\\
0&\text{otherwise.}\end{cases}$

######  Theorem 2.1.

Fix the notation in Section 1. Then there exists absolute constants
$\Cr{162b1},\Cr{162b2}$ such that the following are true for all $x\geq 16$.

  1. (1)

If $\lambda\not\in(\mathbb{Q}^{2}-1)\cup\\{1/8,1,-1/4,-1/64,-4,-64\\}$, then
for every subinterval $[a,b]\subset[-3,3]$,

$\displaystyle\frac{\\#\left\\{p\leq X\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
X\\}}-\int_{a}^{b}B(t)dt$ $\displaystyle\ll
x^{-\frac{\Cr{162b1}}{\sqrt{q}}}+\frac{\log(N_{E}q\log x)}{\sqrt{\log x}}.$

  2. (2)

If $\lambda\in(\mathbb{Q}^{2}-1)-\\{0,-1,8\\}$, then for every subinterval
$[a,b]\subset[-1,3]$,

$\displaystyle\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}\frac{1}{2\pi}\sqrt{\frac{3-t}{1+t}}dt\Big{|}\ll\frac{\log(N_{E}\log
x)}{\sqrt{\log x}}.$

  3. (3)

If $\lambda\in\\{1/8,1,-1/4,-1/64\\}$, then for every subinterval
$[a,b]\subset[-3,3]$,

$\displaystyle\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\frac{\mathbf{1}_{1\in[a,b]}+\mathbf{1}_{-1\in[a,b]}}{4}-\int_{a}^{b}B_{1}(t)dt\Big{|}$
$\displaystyle\ll x^{-\Cr{162b1}/\sqrt{q}}+(\log
x)^{9/2}\exp\Big{(}\frac{-\Cr{162b2}\log x}{\sqrt{\log
x}+\log(N_{E}q)}\Big{)}.$

  4. (4)

If $\lambda\in\\{-4,-64\\}$, then for every subinterval $[a,b]\subset[-1,3]$,

$\displaystyle\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\frac{\mathbf{1}_{1\in[a,b]}}{2}-\int_{a}^{b}\frac{dt}{2\pi\sqrt{3+2t-t^{2}}}\Big{|}$
$\displaystyle\ll x^{-\Cr{162b1}/\sqrt{q}}+(\log
x)^{9/2}\exp\Big{(}\frac{-\Cr{162b2}\log x}{\sqrt{\log
x}+\log(N_{E}q)}\Big{)}.$

  5. (5)

If $\lambda=8$, then for every subinterval $[a,b]\subset[-1,3]$,

$\displaystyle\Big{|}\frac{\\#\left\\{p\leq X\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
X\\}}-\frac{\mathbf{1}_{-1\in[a,b]}}{2}-\int_{a}^{b}\frac{dt}{2\pi\sqrt{3+2t-t^{2}}}\Big{|}$
$\displaystyle\ll x^{-\Cr{162b1}/\sqrt{q}}+(\log
x)^{9/2}\exp\Big{(}\frac{-\Cr{162b2}\log x}{\sqrt{\log
x}+\log(N_{E}q)}\Big{)}.$

To state the Sato-Tate distributions for double quadric surfaces, we define
the following functions

$\displaystyle C_{1}(t)$
$\displaystyle=\frac{2}{\pi^{2}}\int_{|t|/2}^{2}\frac{1}{u}\sqrt{\bigg{(}1-\Big{(}\frac{u}{2}\Big{)}^{2}\bigg{)}\left(1-\Big{(}\frac{|t|}{2u}\Big{)}^{2}\right)}du,$
$\displaystyle C_{2}(t)$
$\displaystyle=\frac{1}{2\pi^{2}}\int_{|t|/2}^{2}\frac{1}{u}\sqrt{\frac{1-(u/2)^{2}}{1-(|t|/2u)^{2}}}du,$
$\displaystyle C_{3}(t)$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{|t|/2}^{2}\frac{1}{u\sqrt{(1-(u/2)^{2})(1-(|t|/2u)^{2})}}du.$

######  Theorem 2.2.

Let $E$ and $E^{\prime}$ be two twist-inequivalent elliptic curves of
conductors $N_{E}$ and $N_{E^{\prime}}$. The following are true for all $x\geq
16$.

  1. (1)

If $E$ and $E^{\prime}$ are both non-CM, then

$\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{\mathcal{Z}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}C_{1}(t)dt\Big{|}\ll{\frac{\sqrt{\log(N_{E}N_{E^{\prime}}\log\log
x)}}{\sqrt[4]{\log\log x}}}.$

  2. (2)

If $E$ is non-CM and $E^{\prime}$ is CM, then

$\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{\mathcal{Z}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}C_{2}(t)dt-\frac{1}{2}\mathbf{1}_{0\in
I}\Big{|}\ll\frac{\sqrt[3]{\log(N_{E})^{4}\log(N_{E^{\prime}}\log
x)}}{\sqrt[6]{\log x}}.$

  3. (3)

If $E,E^{\prime}$ are both CM, and the discriminants of their CM fields are
coprime, then there exists an absolute constant $\Cl[abcon]{2quad1}$ such that

$\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{\mathcal{Z}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-\int_{a}^{b}C_{3}(t)dt-\frac{3}{4}\mathbf{1}_{0\in
I}\Big{|}\ll\exp\left(\frac{-\Cr{2quad1}\log x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}}\right).$

  4. (4)

If $E,E^{\prime}$ are both CM elliptic curves, and the discriminants of their
CM fields are not coprime, then there exists an absolute constant
$\Cl[abcon]{2quad2}$ such that

$\Big{|}\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{\mathcal{Z}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
x\\}}-2\int_{a}^{b}C_{3}(t)dt-\frac{1}{2}\mathbf{1}_{0\in
I}\Big{|}\ll\exp\left(\frac{-\Cr{2quad2}\log x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}}\right).$

It is clear that Theorem 1.1 and 1.2 are special cases of Theorem 2.1, and
Theorem 1.5 is a special case of Theorem 2.2. In the rest of this section, we
will present proofs for Theorem 2.1 and Theorem 2.2 using Theorem 1.7, which
will be proven in Sections 6 to 9.

###  2.2. Proof of Theorem 2.1

For simplicity, we only present the proof of Theorem 2.1 for the intervals
$[a,b]\subset[0,3]$; the other cases are proved in the same way. In this
proof, we fix $\lambda$ and denote

$E\mathrel{\mathop{\mathchar
58\relax}}=E_{-\lambda/(\lambda+1)}^{\text{Cl}},\qquad
a(p)\mathrel{\mathop{\mathchar
58\relax}}=a_{E_{-\lambda/(\lambda+1)}^{\text{Cl}}}^{*}(p),\qquad
q\mathrel{\mathop{\mathchar 58\relax}}=q_{\lambda}.$

Let $p$ be a prime that does not divide $q$ and let $\phi_{p}$ be the unique
quadratic character modulo $p$.

###### Proof of Theorem 2.1 (1).

In this case, we have

(2.1)
$\lambda\notin\\{r\in\mathbb{Q}\colon\sqrt{r+1}\in\mathbb{Q}\\}\cup\\{-64,-4,-\tfrac{1}{4},-\tfrac{1}{64},\tfrac{1}{8},1\\}.$

From [20, Page 191] we find that condition (2.1) implies $E$ is a non-CM
elliptic curve. Note that
$\lambda\notin\\{r\in\mathbb{Q}\colon\sqrt{r+1}\in\mathbb{Q}\\}$ implies
$q>1$. Applying the relations between $X_{\lambda}$ and its Clausen elliptic
curve $E$ ((1) and (1.2)), we have

$a_{X_{\lambda}}^{*}(p)=\phi_{p}(\lambda+1)(a(p)^{2}-1).$

Note that when $\phi_{p}(\lambda+1)=1$, $a_{X_{\lambda}}^{*}(p)\in[a,b]$ is
equivalent to

$\displaystyle a(p)\in\Big{[}\sqrt{1+a},\sqrt{1+b}\Big{]}$

and when $\phi_{p}(\lambda+1)=-1$, $a_{X_{\lambda}}^{*}(p)\in[a,b]$ is
equivalent to

$\displaystyle a(p)\in\Big{[}\sqrt{1-b},\sqrt{1-a}\Big{]},$

where each endpoint is set to be $0$ if it is not real. By quadratic
reciprocity, $\phi_{p}(\lambda+1)$ as a function of $p$ is periodic with
period dividing $4q$ when $p>4q$. Since $q\neq 1$ and square-free, when $j$
spans $(\mathbb{Z}/T\mathbb{Z})^{\times}$, exactly half of the $\phi_{j}(q)$
will be equal to $1$ and the other half will equal $-1$. Apply Theorem 1.7
(2.a) to modulus $4q$; with a simple calculation, we obtain

$\displaystyle\frac{\\#\left\\{p\leq X\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq X\\}}$
$\displaystyle=\int_{a}^{b}B(t)dt+O\Big{(}x^{-\frac{\Cr{162b1}}{\sqrt{q}}}+\frac{\log(N_{E}q\log
x)}{\sqrt{\log x}}\Big{)}.$

∎

###### Proof of Theorem 2.1 (2).

This proof differs from the proof of Theorem 2.1 (1) only in that $\lambda+1$
is a rational square, and thus $\phi_{p}(q)=1$ for all $p$ not dividing $q$.
Hence we may apply [28, Theorem 1.1] to derive that for all $x\geq 3$,

$\frac{\\#\left\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}a_{X_{\lambda}}^{*}(p)\in[a,b]\right\\}}{\\#\\{p\leq
X\\}}=\frac{1}{2\pi}\int_{a}^{b}\sqrt{\frac{3-t}{1+t}}dt+O\Big{(}\frac{\log(N_{E}\log
x)}{\sqrt{\log x}}\Big{)}.$

∎

###### Proof of Theorem 2.1 (3, 4, 5).

By [20, Page 191], when $\lambda\in\\{8,1/8,1,-4,-1/4,-64,-1/64\\}$, $E$ is a
CM elliptic curve. Let $K$ be the CM field of $E$ and $D_{K}$ the discriminant
of $K$. Also, let $T$ be the period of $\phi_{p}(\lambda+1)$. All the possible
values of $\lambda$ and the corresponding $T$ and $D_{K}$ are presented in
Table 1.

Table 1. Values of $T$ and $D_{K}$ for $\lambda$ satisfying that $E_{-\lambda/(1+\lambda)}^{\text{Cl}}$ has CM.  $\lambda$ |  $8$ |  $1/8$ |  $1$ |  $-4$ |  $-1/4$ |  $-64$ |  $-1/64$
---|---|---|---|---|---|---|---
$T$ |  $1$ |  $8$ |  $8$ |  $6$ |  $12$ |  $14$ |  $28$
$D_{K}$ |  $-4$ |  $-4$ |  $-8$ |  $-3$ |  $-3$ |  $-7$ |  $-7$

The proof now proceeds in the same way as that of Theorem 2.1 (1). The only
difference is that we apply Theorem 1.7 (2.b) to modulus $T$, instead of (2.a)
as in the above two cases. ∎

###  2.3. Proof of Theorem 2.2

Let $E$ and $E^{\prime}$ be two twist-inequivalent elliptic curves. For all
subintervals $I\subset[-2,2]$, define the semicircular measure $S(I)$ as

$S(I)=\int_{I}\frac{1}{\pi}\sqrt{1-(t/2)^{2}}dt,$

and the reciprocal $T(I)$ of the semicircular distribution as

$T(I)=\int_{I}\frac{1}{2\pi\sqrt{1-(t/2)^{2}}}dt.$

Consider

$R=\\{(x,y)\in[-2,2]\times[-2,2]\mathrel{\mathop{\mathchar 58\relax}}a\leq
xy\leq b\\}.$

For an odd integer $L>0$ that will be specified differently in each case,
consider the $2L-2$ lines $x=-2+4k/L$ and $y=-2+4k/L$ for $1\leq k\leq L-1$.
They divide the cell $[-2,2]\times[-2,2]$ into the $L^{2}$ smaller cells

$\left[-2+\frac{4i-4}{L},-2+\frac{4i}{L}\right]\times\left[-2+\frac{4i^{\prime}-4}{L},-2+\frac{4i^{\prime}}{L}\right],\qquad
1\leq i,j\leq L.$

Since $a_{E}^{*}(p)$ and $a_{E^{\prime}}^{*}(p)$ are both irrational whenever
they are nonzero, no point $(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))$ lies on the
boundary of any of the above $L^{2}$ cells.

###### Proof of Theorem 2.2 (1).

Consider any $1\leq i,i^{\prime}\leq L$, and define $I=[-2+(4i-4)/L,-2+4i/L]$
and $I^{\prime}=[-2+(4j-4)/L,-2+4j/L]$. Applying Theorem 1.7 (1.a) to $I$ and
$I^{\prime}$, we have

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in I\times I^{\prime}\\}$
$\displaystyle=S(I)S(I^{\prime})\pi(x)+O\bigg{(}\pi(x){\frac{\log(N_{E}N_{E^{\prime}}\log\log
x)}{\sqrt{\log\log x}}}\bigg{)}$ (2.5)
$\displaystyle\ll\pi(x)\bigg{(}\frac{1}{L^{2}}+{\frac{\log(N_{E}N_{E^{\prime}}\log\log
x)}{\sqrt{\log\log x}}}\bigg{)}.$

Let $\mathcal{S}$ be the union of all the smaller cells whose interior is
contained in the interior of $R$. Let $\mathcal{T}$ be the union of all the
smaller cells whose interior has non-empty intersection with
$\\{xy=a\\}\cup\\{xy=b\\}$. It is easy to check that $\mathcal{T}$ contains
$O(L)$ of the smaller cells. Now, we have

$\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in R\\}=\\#\\{p\leq
x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in\mathcal{S}\\}+\\#\\{p\leq
x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in\mathcal{T}\cap R\\}.$

The $L-1$ vertical lines divide $\mathcal{S}$ into at most $L$ vertical strips
each with width $1/L$. Applying Theorem 1.7 (1.a) to those strips of
$\mathcal{S}$, we obtain

(2.6) $\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in\mathcal{S}\\}$
$\displaystyle=\pi(x)\int_{\mathcal{S}}\frac{1}{\pi^{2}}\sqrt{\bigg{(}1-\bigg{(}\frac{x}{2}\bigg{)}^{2}\bigg{)}\bigg{(}1-\bigg{(}\frac{y}{2}\bigg{)}^{2}\bigg{)}}dxdy$
$\displaystyle+O\bigg{(}\pi(x)\bigg{(}L{\frac{\log(N_{E}N_{E^{\prime}}\log\log
x)}{\sqrt{\log\log x}}}\bigg{)}\bigg{)}.$

Note that

(2.7)
$\displaystyle\int_{R-\mathcal{S}}\frac{1}{\pi^{2}}\sqrt{\bigg{(}1-\bigg{(}\frac{x}{2}\bigg{)}^{2}\bigg{)}\bigg{(}1-\bigg{(}\frac{y}{2}\bigg{)}^{2}\bigg{)}}dxdy$
$\displaystyle\ll\frac{1}{L}$

Applying (2.5) to each cell of $\mathcal{T}$, we have

(2.8) $\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in\mathcal{T}\cap
R\\}\ll\pi(x)\bigg{(}L{\frac{\log(N_{E}N_{E^{\prime}}\log\log
x)}{\sqrt{\log\log x}}}+\frac{1}{L}\bigg{)}.$

Collating (2.6), (2.7), and (2.8), we obtain

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in R\\}$
$\displaystyle=\pi(x)\int_{a}^{b}C_{1}(t)+O\bigg{(}\pi(x)\bigg{(}L{\frac{\log(N_{E}N_{E^{\prime}}\log\log
x)}{\sqrt{\log\log x}}}+\frac{1}{L}\bigg{)}\bigg{)}.$

Taking $L=2\left\lfloor\frac{\sqrt[4]{\log\log
x}}{\sqrt{\log(N_{E}N_{E^{\prime}}\log\log x)}}\right\rfloor+1$ concludes the
proof. ∎

###### Proof of Theorem 2.2 (2).

The proof proceeds similarly to that of Theorem 2.2 (1). For simplicity, we
only present the proof in the case that $0\notin[a,b]$; the full theorem may
be proven in the same way. When $0\notin I^{\prime}$, we have that by applying
Theorem 1.7 (1.b) and elementary calculus,

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in I\times I^{\prime}\\}$
$\displaystyle=S(I)T(I^{\prime})\pi(x)+O\bigg{(}\pi(x)\frac{\log(N_{E})^{4}\log(N_{E^{\prime}}\log
x)}{\sqrt{\log x}}\bigg{)}$
$\displaystyle\ll\pi(x)\bigg{(}\frac{1}{L^{3/2}}+\pi(x)\frac{\log(N_{E})^{4}\log(N_{E^{\prime}}\log
x)}{\sqrt{\log x}}\bigg{)}.$

Define $R$ similarly as in case (1). Then by a similar calculation as in case
(1),

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in
R\\}=\pi(x)\int_{a}^{b}C_{2}(t)+O\bigg{(}\pi(x)\bigg{(}L\frac{\log(N_{E})^{4}\log(N_{E^{\prime}}\log
x)}{\sqrt{\log x}}+\frac{1}{\sqrt{L}}\bigg{)}\bigg{)}.$

Taking $L=2\left\lfloor\frac{\sqrt[3]{\log
x}}{\sqrt[3]{\log(N_{E})^{8}\log(N_{E^{\prime}}\log x)^{2}}}\right\rfloor+1$
now concludes the proof. ∎

###### Proof of Theorem 2.2 (3), (4).

The proofs of cases (3) and (4) proceed in exactly the same way, so we only
present the proof for case (3). Define $\mathcal{T}$ and $R$ similarly as in
the previous cases. Similar to case (2), we only consider the case when
$0\notin[a,b]$. Now when $0\notin I$ and $0\notin I^{\prime}$, we have

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in I\times I^{\prime}\\}$
$\displaystyle=T(I)T(I^{\prime})\pi(x)+O\bigg{(}\pi(x)\exp\bigg{(}\frac{-\Cr{161c1}\log
x}{\sqrt{\log x}+3\log N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}$
$\displaystyle\ll\pi(x)\bigg{(}\frac{\min\\{T(I),T(I^{\prime})\\}}{\sqrt{L}}+\exp\bigg{(}\frac{-\Cr{161c1}\log
x}{\sqrt{\log x}+3\log N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}.$

Now, consider any $c\in(-4,4)$. Let $\mathcal{T}_{c}$ be the set of cells that
have a nonempty intersection with the curve $\\{xy=c\\}$; evidently
$|\mathcal{T}_{c}|\ll L(4-|c|)$. By elementary calculus, for any $I\times
I^{\prime}\in S_{c}$,

$\displaystyle\min\\{T(I),T(I^{\prime})\\}$
$\displaystyle\ll\frac{1}{L\sqrt{4-\Big{(}\sqrt{c}+\frac{1}{L}\Big{)}^{2}}}.$

Thus

$\sum_{I\times
I^{\prime}\in\mathcal{T}_{c}}\frac{\min\\{T(I),T(I^{\prime})\\}}{\sqrt{L}}\ll\frac{1}{\sqrt{L}}.$

Hence, we have that

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in\mathcal{T}\cap
R\\}\ll\pi(x)\bigg{(}L\exp\bigg{(}\frac{-\Cr{161c1}\log x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}+\frac{1}{\sqrt{L}}\bigg{)}.$

Thus,

$\displaystyle\\#\\{p\leq x\mathrel{\mathop{\mathchar
58\relax}}(a_{E}^{*}(p),a_{E^{\prime}}^{*}(p))\in R\\}$
$\displaystyle=\pi(x)\bigg{(}\int_{a}^{b}C_{3}(t)+\frac{3}{4}\mathbf{1}_{0\in[a,b]}\bigg{)}$
$\displaystyle+O\bigg{(}\pi(x)\bigg{(}L\exp\bigg{(}\frac{-\Cr{161c1}\log
x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}+\frac{1}{\sqrt{L}}\bigg{)}\bigg{)}.$

Taking $L=2\left\lfloor\exp\bigg{(}\frac{2\Cr{161c1}\log x}{3\sqrt{\log
x}+9\log N_{E}N_{E^{\prime}}}\bigg{)}\right\rfloor+1$ concludes the proof. ∎

##  3\. Preliminaries for the Proof of Theorem 1.7

In this section we introduce preliminary information that is needed for the
proof of Theorem 1.7. Throughout this section, when $E$ (resp. $E^{\prime}$)
is an elliptic curve with complex multiplication over an imaginary quadratic
field $K$ (resp. $K^{\prime}$), let $\chi_{K}$ (resp. $\chi_{K^{\prime}}$) and
$D_{K}$ (resp. $D_{K^{\prime}}$) respectively denote the Kronecker character
and the absolute discriminant of $K$ (resp. $K^{\prime}$). Finally, for an
interval $I$, let $\chi_{I}$ denote the characteristic function of $I$.

###  3.1. Beurling-Selberg polynomials

As important technical tools, we will need both the original one-dimensional
Beurling-Selberg polynomials, along with a two-dimensional analogue. The
following two Lemmas follow from [15, Theorem 1] and [6, Theorem 2].

######  Lemma 3.1.

Let $I\subseteq[0,\pi]$ be a closed subinterval and $M$ a positive integer.
Then there exists an absolute constant $\Cl[abcon]{bspoly0}>0$, and two
polynomials

$F^{\pm}_{I,M}(\theta)=\sum_{0\leq m\leq
M}\hat{F}_{I,M}^{\pm}(m)\cos(m\theta)$

such that

  1. (1)

for all $\theta\in[0,\pi]$

$F^{-}_{I,M}(\theta)\leq\mathbf{1}_{I}(\theta)\leq F^{+}_{I,M}(\theta);$

  2. (2)

we have that

$\Big{|}\hat{F}_{I,M}^{\pm}(0)-\frac{|I|}{\pi}\Big{|}\leq\frac{\Cr{bspoly0}}{M};$

  3. (3)

if $m\neq 0$, then

$\Big{|}\hat{F}_{I,M}^{\pm}(m)\Big{|}\leq\frac{\Cr{bspoly0}}{m}.$

######  Lemma 3.2.

Let $I,I^{\prime}\subseteq[0,\pi]$ be two closed subintervals and $M$ a
positive integer. Then there exists an absolute constant
$\Cl[abcon]{bspoly00}>0$, and two polynomials

$F^{\pm}_{I,I^{\prime},M}(\theta,\theta^{\prime})=\sum_{0\leq m,m^{\prime}\leq
M}\hat{F}_{I,I^{\prime},M}^{\pm}(m,m^{\prime})\cos(m\theta)\cos(m^{\prime}\theta^{\prime})$

such that

  1. (1)

for all $\theta,\theta^{\prime}\in[0,\pi]$,

$F^{-}_{I,I^{\prime},M}(\theta,\theta^{\prime})\leq\chi_{I}(\theta)\chi_{I^{\prime}}(\theta^{\prime})\leq
F^{+}_{I,I^{\prime},M}(\theta,\theta^{\prime});$

  2. (2)

we have that

$\Big{|}\hat{F}_{I,I^{\prime},M}^{\pm}(0,0)-\frac{|I|}{\pi}\Big{|}\leq\frac{\Cr{bspoly00}}{M};$

  3. (3)

if $m\neq 0$, then

$\Big{|}\hat{F}_{I,I^{\prime},M}^{\pm}(m,0)\Big{|},\thickspace\Big{|}\hat{F}_{I,I^{\prime},M}^{\pm}(0,m)\Big{|}\leq\frac{\Cr{bspoly00}}{m};$

  4. (4)

if $mm^{\prime}\neq 0$, then

$\Big{|}\hat{F}_{I,I^{\prime},M}^{\pm}(m,m^{\prime})\Big{|}\leq\frac{\Cr{bspoly00}}{mm^{\prime}}.$

The above two lemmas can be viewed as Fourier expansions of the characteristic
functions with respect to the basis $\cos(m\theta)$. When we deal with
elliptic curves without complex multiplication, it will be useful to change
the basis of trigonometric polynomials in Lemma 3.1 to the basis of the $m$-th
Chebyshev polynomials of the second type $U_{m}(\cos(\theta))$, which form an
orthonormal basis for $L^{2}([0,\pi],\mu_{ST})$ with respect to the usual
inner product $\langle f,g\rangle=\int_{0}^{\pi}f(\theta)g(\theta)d\mu_{ST}$.
For a demonstration of said base change, see [22, Section 3].

###  3.2. Newforms

Here we briefly recall the notion of newforms. For a complete treatment, see
Section 2.5 in [20]. For a positive integer $N$, recall that the level $N$
congruence subgroup $\Gamma_{0}(N)$ is defined by

$\Gamma_{0}(N)\mathrel{\mathop{\mathchar
58\relax}}=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\textup{SL}_{2}(\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}c\equiv 0\bmod N\right\\}.$

Throughout the rest of the paper, let $\mathcal{M}_{k}(\Gamma_{0}(N),\chi)$
denote the space of modular forms of weight $k$, level $N$, and nebentypus
$\chi$, and let
$S_{k}(\Gamma_{0}(N),\chi)\subset\mathcal{M}_{k}(\Gamma_{0}(N),\chi)$ denote
the subspace of cusp forms, both with respect to $\Gamma_{0}(N)$. When $\chi$
is trivial, write
$\mathcal{M}_{k}(\Gamma_{0}(N),\chi)=\mathcal{M}_{k}(\Gamma_{0}(N))$,
$\mathcal{S}_{k}(\Gamma_{0}(N),\chi)=\mathcal{S}_{k}(\Gamma_{0}(N))$. Now,
given a positive integer $d$, define the $V$-operator $V(d)$ as

$\Bigg{(}\sum_{n\geq n_{0}}c(n)q^{n}\Bigg{)}|V(d)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{n\geq n_{0}}c(n)q^{dn}.$

If $f(z)\in S_{k}(\Gamma_{0}(N))$ and $d>1$, then both $f(z)$ and
$f(dz)=f(z)|V(d)$ are in $S_{k}(\Gamma_{0}(dN))$ (Proposition 2.22, [20]).
Thus there are at least two natural ways for a function in
$S_{k}(\Gamma_{0}(dN))$ to come from lower levels. Motivated by this, we
define the space of oldforms $S_{k}^{\textup{old}}(\Gamma_{0}(N))\subset
S_{k}(\Gamma_{0}(N))$ as

$S_{k}^{\textup{old}}(\Gamma_{0}(N))\mathrel{\mathop{\mathchar
58\relax}}=\bigoplus_{\begin{subarray}{c}dM|N\\\ M\neq
N\end{subarray}}S_{k}(\Gamma_{0}(M))|V(d),$

where we sum over pairs of positive integers $(d,M)$ satisfying $dM|N$ and
$M\neq N$. Now, let $\mathfrak{F}_{N}$ denote the fundamental domain for the
action of $\Gamma_{0}(N)$ on the upper half of the complex plane, and recall
the Petersson inner product between cusp forms $f(z),g(z)\in
S_{k}(\Gamma_{0}(N))$, defined as

$\displaystyle\langle f,g\rangle\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{[\textup{SL}_{2}(\mathbb{Z})\mathrel{\mathop{\mathchar
58\relax}}\Gamma_{0}(N)]}\int_{\mathfrak{F}_{N}}f(z)\overline{g(z)}y^{k-2}dxdy$

where $z=x+iy$.

######  Definition 3.3.

Define $S_{k}^{\textup{new}}(\Gamma_{0}(N))$, the _space of newforms_ , to be
the orthogonal complement of $S_{k}^{\textup{old}}(\Gamma_{0}(N))$ in
$S_{k}(\Gamma_{0}(N))$ with respect to the Petersson inner product.

######  Definition 3.4.

A _newform_ in $S_{k}^{\textup{new}}(\Gamma_{0}(N))$ is a normalized cusp form
that is an eigenform for all the Hecke operators on
$S_{k}^{\textup{new}}(\Gamma_{0}(N))$, the Atkin-Lehner involution
$|_{k}W(N)$, and all of the Atkin-Lehner involutions $|_{k}W(Q_{p})$ for each
prime $p|N$ (for more details, see [20, Section 2.5]).

###  3.3. CM elliptic curves, Grössencharacters, and automorphic induction

Let $K$ be an imaginary quadratic field with absolute discriminant $D_{K}$,
ring of integers $\mathcal{O}_{K}$, and absolute norm
$\mathrm{N}=\mathrm{N}_{K/\mathbb{Q}}$ defined as
$\mathrm{N}\mathfrak{a}=|\mathcal{O}_{K}/\mathfrak{a}|$ for all nonzero ideals
$\mathfrak{a}\subset\mathcal{O}_{K}$. Let $\chi_{K}$ be the Kronecker
character associated to $K$; note that for all primes $p\nmid D_{K}$,

$\displaystyle\chi_{K}(p)$ $\displaystyle=\begin{cases}1&p\text{ splits in
}K\\\ -1&p\text{ inert in }K\end{cases}$

######  Definition 3.5 ([13], Section 12.2).

Given a $u_{\xi}\in\mathbb{Z}$, define $\xi_{\infty}\mathrel{\mathop{\mathchar
58\relax}}K^{\times}\rightarrow S^{1}$ to be the group homomorphism satisfying

$\xi_{\infty}(a)=\bigg{(}\frac{a}{|a|}\bigg{)}^{u_{\xi}}.$

Let $\mathfrak{m}$ be an ideal of $\mathcal{O}_{K}$. Define a _Hecke
Grössencharacter_ $\xi$ with modulus $\mathfrak{m}$ to be a group homomorphism
from

$I_{\mathfrak{m}}=\\{\mathfrak{a}\text{ fractional ideal in
}\mathcal{O}_{K}\mathrel{\mathop{\mathchar
58\relax}}(\mathfrak{a},\mathfrak{m})=1\\}$

to

$\\{z\in\mathbb{C}\mathrel{\mathop{\mathchar 58\relax}}|z|=1\\}$

that agrees with $\xi_{\infty}$ on

$P_{\mathfrak{m}}=\\{a\mathcal{O}_{K}\mathrel{\mathop{\mathchar 58\relax}}a\in
K^{\times},a\equiv 1\bmod\mathfrak{m}\\}.$

Note that there may exist another Grössencharacter $\xi^{\ast}$ of $K$ with
modulus $\mathfrak{m}^{\ast}\supset\mathfrak{m}$ which satisfies that
$\xi^{\ast}(\mathfrak{a})=\xi(\mathfrak{a})$ for all $\mathfrak{a}\in
I_{\mathfrak{m}}$. The largest ideal $\mathfrak{n}$ which is a modulus of such
a Grössencharacter is defined as the conductor of $\xi$. If
$\mathfrak{n}=\mathfrak{m}$, $\xi$ is called primitive. The following theorem
characterizes the $L$-function $L(s,E)$ of an elliptic curve $E/\mathbb{Q}$
with CM over an imaginary quadratic field $K$ as the $L$-function of a
Grössencharacter defined over $K$.

######  Theorem 3.6 ([25, Theorem II.10.5]).

Fix the notation above. Let $E/\mathbb{Q}$ be an elliptic curve with complex
multiplication over an imaginary quadratic field $K$ with absolute
discriminant $D_{K}$. Then there exists a primitive Grössencharacter $\xi$
with conductor $\mathfrak{m}\subset\mathcal{O}_{K}$ and $u_{\xi}=1$ such that
$\textup{N}\mathfrak{m}=N_{E}/D_{K}$ and $L(s,E)=L(s,\xi).$

It will often be convenient to interpret the $L$-functions of Hecke
Grössencharacters as $L$-functions associated with modular forms via
automorphic induction.

######  Theorem 3.7 ([13, Theorem 12.5]).

Let $K$ be an imaginary quadratic field with absolute discriminant $D_{K}$ and
let $\xi$ be a primitive Grössencharacter with conductor $\mathfrak{m}$ and
$u_{\xi}$ non-negative. Let $\chi$ be the Dirichlet character given by
$\chi(n)=\chi_{D}(n)\xi((n))$ for all $n\in\mathbb{Z}^{+}$. Then the modular
form

$f(z)=\sum_{\mathfrak{a}}\xi(\mathfrak{a})(\textup{N}\mathfrak{a})^{u_{\xi}/2}e(z\textup{N}\mathfrak{a})\in\mathcal{M}_{u+1}(\Gamma_{0}(D_{K}\textup{N}\mathfrak{m}),\chi)$

is a newform and satisfies that

$\displaystyle L(s,f)$ $\displaystyle=L(s,\xi).$

When $u_{\xi}>0$, $f$ is a cusp form.

Now, given a CM elliptic curve $E$, consider $\xi$ as defined in Theorem 3.6.
Define $\xi_{m}(\bmod\text{~{}}\mathfrak{m}_{m})$ to be the primitive
Grössencharacter which induces $\xi^{m}$, and let $\chi_{m}$ be the Dirichlet
character given by $\chi_{m}(n)=\chi_{D}(n)\xi_{m}((n))$ for all
$n\in\mathbb{Z}^{+}$. Note that $D_{K}\textup{N}\mathfrak{m}_{m}|N_{E}$. Let
$f_{m}$ be the holomorphic cusp form associated with $\xi_{m}$ as in Theorem
3.7. Then $f_{m}\in
S_{m+1}(\Gamma_{0}(|D_{K}|\textup{N}\mathfrak{m}_{m}),\chi_{m})$.

###  3.4. Automorphic _L_ -functions

Let $\mathbb{A}_{\mathbb{Q}}$ be the adele ring of $\mathbb{Q}$. Let
$\mathfrak{F}_{m}$ denote the set of all cuspidal representations of
$\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character that
are trivial on the diagonally embedded copy of the positive real numbers. For
$\pi\in\mathfrak{F}_{m}$, let $\widetilde{\pi}$ denote the representation
contragredient to $\pi$ and let $q_{\pi}$ denote the conductor of $\pi$. Now,
there exists a standard $L$-function $L(s,\pi)$ attached to $\pi$; let the
conductor of this $L$-function be $q_{\pi}$. The local parameters of this
$L$-function are known as the Satake parameters, and for each prime $p$ the
Satake parameters $\alpha_{1,\pi}(p),\dots,\alpha_{m,\pi}(p)\in\mathbb{C}$
satisfy that when $p\nmid q_{\pi}$, $\alpha_{j,\pi}(p)\neq 0$ for all $j$. For
every $n\in\mathbb{Z}^{+}$, denote the $n$th Dirichlet coefficient of
$L(s,\pi)$ as $a_{\pi}(n)$. The Dirichlet series of $L(s,\pi)$ is given by the
formula

$L(s,\pi)=\prod_{p}\prod_{j=1}^{m}\Big{(}1-\frac{\alpha_{j,\pi}(p)}{p^{s}}\Big{)}^{-1}=\sum_{n=1}^{\infty}\frac{a_{\pi}(n)}{n^{s}},$

which converges absolutely when $\mathrm{Re}(s)>1$. Define

$\Gamma_{\mathbb{R}}(s)=\pi^{-s/2}\Gamma(s/2),$

where $\Gamma$ is the usual gamma function. The local parameters at infinity,
$\mu_{\pi}(1),\dots,\mu_{\pi}(n)$, of $L(s,\pi)$, are known as the Langlands
parameters. The gamma factor $\gamma(s,\pi)$ of $L(s,\pi)$ is by definition

$\gamma(s,\pi)=\prod_{j=1}^{m}\Gamma_{\mathbb{R}}(s+\mu_{\pi}(j)).$

Note that all of the relevant automorphic representations within this paper
will satisfy the Generalized Ramanujan conjecture, so we have the following
bounds on the Satake and Langlands parameters,

(3.1) $|\alpha_{j,\pi}(p)|\leq 1,\qquad\mathrm{Re}(\mu_{\pi}(j))\geq 0.$

Note that as particular examples of automorphic $L$-functions, any Dirichlet
$L$-function $L(s,\chi)$ corresponds to a one-dimensional cuspidal
representation $\chi$ of $\text{GL}_{1}(\mathbb{A}_{\mathbb{Q}})$ and thus is
an automorphic $L$-function.

Recall that $L(s,\pi)$ is always meromorphic; if $\pi$ is the trivial
representation $\mathbbm{1}$ of $\text{GL}_{1}(\mathbb{A}_{\mathbb{Q}})$ then
$L(s,\pi)$ has a simple pole at $s=1$, otherwise $L(s,\pi)$ is entire. Denote
the order of the pole of $L(s,\pi)$ at $s=1$ as $r_{\pi}$. Then $r_{\pi}=1$
when $\pi=\mathbbm{1}$ and $r_{\pi}=0$ otherwise. Now, the complete
$L$-function

$\Lambda(s,\pi)=(s(s-1))^{r_{\pi}}q_{\pi}^{s/2}L(s,\pi)\gamma(s,\pi)$

is entire of order 1. Moreover, there exists a $W(\pi)\in\mathbb{C}$
satisfying $|W(\pi)|=1$, for which the functional equation

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

holds true. Note that $q_{\widetilde{\pi}}=q_{\pi}$, and the following sets
are equal:

$\\{\alpha_{j,\widetilde{\pi}}(p)\\}=\\{\overline{\alpha_{j,\pi}(p)}\\},\qquad\\{\mu_{\widetilde{\pi}}(j)\\}=\\{\overline{\mu_{\pi}(j)}\\}.$

The analytic conductor of $L(s,\pi)$ is defined as

$C(\pi)=q_{\pi}\prod_{j=1}^{m}(3+|\mu_{\pi}(j)|).$

Note that throughout this paper, we will use $C(\cdot)$ to generally refer to
the analytic conductor of an $L$-function (see [12, Page 95] for the
definition). Moreover, we will define $\Lambda_{\pi}(n)$ to be the $n$th
coefficient of the logarithmic derivative of $-L(s,\pi)$. Specifically, we
have the formula

$\sum_{n=1}^{\infty}\frac{\Lambda_{\pi}(n)}{n^{s}}=-\frac{L^{\prime}}{L}(s,\pi)=\sum_{p}\sum_{\ell=1}^{\infty}\frac{\sum_{j=1}^{m}\alpha_{j,\pi}(p)^{\ell}\log
p}{p^{\ell s}}.$

###  3.5. Rankin-Selberg _L_ -functions

Given any two $\pi\in\mathfrak{F}_{m}$ and
$\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$, it is often useful to consider the
Rankin-Selberg convolution of their $L$-functions. This convolution is an
$L$-function itself, with Satake parameters denoted as
$\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)$. A complete description of
these parameters is given in [26, Appendix]. Note that if $m^{\prime}=m$ and
$\pi^{\prime}=\widetilde{\pi}$, then we call the resulting Rankin-Selberg
convolution the Rankin-Selberg square of $\pi$. Note that for any prime
$p\nmid q_{\pi}q_{\pi^{\prime}}$, we have that

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

The Dirichlet series of the Rankin-Selberg convolution of $L(s,\pi)$ and
$L(s,\pi^{\prime})$ is given by

$L(s,\pi\times\pi^{\prime})=\prod_{p}\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\Big{(}1-\frac{\alpha_{j,j^{\prime},\pi\times\pi^{\prime}}(p)}{p^{s}}\Big{)}^{-1}=\sum_{n=1}^{\infty}\frac{a_{\pi\times\pi^{\prime}}(n)}{n^{s}},$

is associated to the tensor product $\pi\times\pi^{\prime}$, and converges
absolutely for $\mathrm{Re}(s)>1$. Let $q_{\pi\times\pi^{\prime}}$ be the
conductor of $L(s,\pi\times\pi^{\prime})$. Note that
$q_{\pi\times\pi^{\prime}}|q_{\pi}^{m^{\prime}}q_{\pi^{\prime}}^{m}$ (see
[3]). Now, denote the Langlands parameters of $L(s,\pi\times\pi^{\prime})$ as
$\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})\in\mathbb{C}$; a complete
description of these Langlands parameters can be found in [26, Proof of Lemma
2.1]. By definition, the gamma factor of $L(s,\pi\times\pi)$ is given by

$\gamma(s,\pi\times\pi^{\prime})=\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}\Gamma_{\mathbb{R}}(s+\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})).$

It is known that $L(s,\pi\times\pi^{\prime})$ is entire if
$\pi^{\prime}\neq\widetilde{\pi}$. If $\pi^{\prime}=\widetilde{\pi}$,
$L(s,\pi\times\pi^{\prime})$ has a simple pole at $s=1$ and is holomorphic
elsewhere. Let $r_{\pi\times\pi^{\prime}}$ denote the order of the pole of
$L(s,\pi\times\pi^{\prime})$ at $s=1$; then $r_{\pi\times\pi^{\prime}}=1$ if
$\pi$ and $\pi^{\prime}$ are dual to each other and
$r_{\pi\times\pi^{\prime}}=0$ otherwise. The completed $L$-function of
$L(s,\pi\times\pi^{\prime})$ is given by

$\Lambda(s,\pi\times\pi^{\prime})=(s(s-1))^{r_{\pi\times\pi^{\prime}}}q_{\pi\times\pi^{\prime}}^{s/2}L(s,\pi\times\pi^{\prime})\gamma(s,\pi\times\pi^{\prime}).$

Note that $\Lambda(s,\pi\times\pi^{\prime})$ is entire and of order $1$.
Moreover, there exists a $W(\pi\times\pi^{\prime})\in\mathbb{C}$ satisfying
$|W(\pi\times\pi^{\prime})|=1$ such that the functional equation

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

holds. Define the analytic conductor of $L(s,\pi\times\pi^{\prime})$ as

$C(\pi\times\pi^{\prime})=q_{\pi\times\pi^{\prime}}\prod_{j=1}^{m}\prod_{j^{\prime}=1}^{m^{\prime}}(3+|\mu_{\pi\times\pi^{\prime}}(j,j^{\prime})|).$

By the work of Bushnell and Henniart [3] and Brumley [10, Appendix], we have
the following bound for the analytic conductor

(3.2) $\displaystyle\log C(\pi\times\pi^{\prime})\ll m^{\prime}\log
C(\pi)+m\log C(\pi^{\prime})$

Finally, we denote the $n$th coefficient of the negative log derivative of
$L(s,\pi\times\pi^{\prime})$ as $\Lambda_{\pi\times\pi^{\prime}}(n)$. By
definition, we have

$\sum_{n=1}^{\infty}\frac{\Lambda_{\pi\times\pi^{\prime}}(n)}{n^{s}}=-\frac{L^{\prime}}{L}(s,\pi\times\pi^{\prime})=\sum_{p}\sum_{\ell=1}^{\infty}\frac{\sum_{j=1}^{m}\alpha_{j,\pi\times\pi^{\prime}}(p)^{\ell}\log
p}{p^{\ell s}}.$

###  3.6. Isobaric sums

Here we recall some basic facts about the isobaric sum operation $\boxplus$,
first introduced by Langlands in [14]. Let $k\geq 1$ be an integer, let
$m_{1},\ldots m_{k}\geq 1$ be integers, let $\pi_{i}\in\mathfrak{F}_{m_{i}}$,
let $r=\sum_{i=1}^{k}m_{i}$, and let $t_{1},t_{2},\ldots t_{k}\in\mathbb{R}$.
Consider the isobaric automorphic representation $\Pi$ of
$\mathrm{GL}_{r}(\mathbb{A})$, defined by

$\displaystyle\Pi$
$\displaystyle=\pi_{1}\otimes|\det|^{it_{1}}\boxplus\ldots\boxplus\pi_{k}\otimes|\det|^{it_{k}}$

The $L$-function associated to $\Pi$ is

$\displaystyle L(s,\Pi)$ $\displaystyle=\prod_{j=1}^{k}L(s+it_{j},\pi_{j}).$

The analytic conductor of this $L$-function is defined as

$\displaystyle C(\Pi,t)$ $\displaystyle=\prod_{j=1}^{k}C(\pi_{j},t+t_{j})$
$\displaystyle C(\Pi)$ $\displaystyle=C(\Pi,0).$

Now, let $k^{\prime}\geq 1$ be an integer, let $m^{\prime}_{1},\ldots
m^{\prime}_{k^{\prime}}\geq 1$ be integers, let
$\pi^{\prime}_{i}\in\mathfrak{F}_{m^{\prime}_{i}}$, let
$r^{\prime}=\sum_{i=1}^{k^{\prime}}m_{i}^{\prime}$, and let
$t^{\prime}_{1},\ldots t^{\prime}_{k^{\prime}}\in\mathbb{R}$. Consider the
isobaric automorphic representation
$\Pi^{\prime}\in\mathrm{GL}_{r^{\prime}}(\mathbb{A})$, defined by

$\displaystyle\Pi$
$\displaystyle=\pi^{\prime}_{1}\otimes|\det|^{it^{\prime}_{1}}\boxplus\ldots\boxplus\pi_{k^{\prime}}\otimes|\det|^{it^{\prime}_{k^{\prime}}}.$

Then, the Rankin-Selberg convolution of $L(s,\pi)$ and $L(s,\pi^{\prime})$ is
given by

$\displaystyle L(s,\Pi\times\Pi^{\prime})$
$\displaystyle=\prod_{j=1}^{k}\prod_{j^{\prime}=1}^{k^{\prime}}L(s+it_{j}+it^{\prime}_{j^{\prime}},\pi_{j}\times\pi^{\prime}_{j^{\prime}})$

and has analytic conductor

$\displaystyle C(\Pi\times\Pi^{\prime},t)$
$\displaystyle=\prod_{j=1}^{d}\prod_{j^{\prime}=1}^{d^{\prime}}C(\pi_{j}\times\pi^{\prime}_{j^{\prime}},t+t_{j}+t^{\prime}_{j^{\prime}})$
$\displaystyle C(\Pi\times\Pi^{\prime})$
$\displaystyle=C(\Pi\times\Pi^{\prime},0).$

In our establishment of zero-free regions below we will implicitly need the
following lemma, which is evident from [26, Section A]:

######  Lemma 3.8.

Given any unitary isobaric automorphic representation $\Pi$ with $L$-function
$L(s,\pi)$, the Dirichlet coefficients of
$-\frac{L^{\prime}}{L}(s,\Pi\times\widetilde{\Pi})$ are all nonnegative.

###  3.7. Symmetric power _L_ -functions

A particular type of $L$-function of interest to us is the symmetric power
$L$-function. Consider an elliptic curve $E/\mathbb{Q}$. Recall that by the
modularity theorem [2], there exists a non-CM cusp form $f(z)\in
S_{2}^{\text{new}}(\Gamma_{0}(N_{E}))$ of weight $2$ and level $N_{E}$
corresponding to $E$. Now, let the Fourier expansion of $f$ be
$f(z)=\sum_{n=1}^{\infty}a_{f}(n)e^{2\pi inz}$. We have that $a_{f}(p)$ agrees
with the trace of Frobenius of $E$ modulo $p$, so thus we may write
$a_{f}(p)=2\sqrt{p}\cos\theta_{E}(p)$ (where $\theta_{E}(p)$ is as defined in
Section 1).

For each non-negative integer $m$ and prime $p$ consider the Satake parameters
$\alpha_{0,\operatorname{\textup{Sym}}^{m}E}(p),\dots$ αm,Symmf(p)∈C. When
$p\nmid N_{E}$, the Satake parameters satisfy the useful identity

$\\{\alpha_{0,\operatorname{\textup{Sym}}^{m}E}(p),\ldots,\alpha_{m,\operatorname{\textup{Sym}}^{m}E}(p)\\}=\\{e^{i(m-2j)\theta_{E}(p)}\colon
0\leq j\leq m\\}.$

A complete description of the values of
$\alpha_{j,\operatorname{\textup{Sym}}^{m}E}(p)$ can be derived from [26,
Appendix]. Now, the $m$th symmetric power $L$-function associated to $f$
(denoted
$L(s,\operatorname{\textup{Sym}}^{m}f)=L(s,\operatorname{\textup{Sym}}^{m}E)$)
is the $L$-function with local parameters

$\\{\alpha_{0,\operatorname{\textup{Sym}}^{m}f}(p),\dots,\alpha_{m,\operatorname{\textup{Sym}}^{m}f}(p)\\}$

If we denote the $n$th Dirichlet coefficient of the $m$th symmetric power
$L$-function as $a_{\operatorname{\textup{Sym}}^{m}E}(n)$, then by definition
we have the Euler product and Dirichlet series expansions

$L(s,\operatorname{\textup{Sym}}^{m}E)=\prod_{p}\prod_{j=0}^{m}\Big{(}1-\frac{\alpha_{j,\operatorname{\textup{Sym}}^{m}E}(p)}{p^{s}}\Big{)}^{-1}=\sum_{n=1}^{\infty}\frac{a_{\operatorname{\textup{Sym}}^{m}E}(n)}{n^{s}},$

which converge for $\mathrm{Re}(s)>1$. When $p\nmid N_{E}$, we may readily
compute $a_{\operatorname{\textup{Sym}}^{m}E}(p)=U_{m}(\cos\theta_{p})$, where
$U_{m}$ is the $m$-th Chebyshev polynomial of the second kind. Note that
$L(s,\operatorname{\textup{Sym}}^{m}E)$ is a self-dual $L$-function.

Define $\Gamma_{\mathbb{C}}(s)\mathrel{\mathop{\mathchar
58\relax}}=\Gamma_{\mathbb{R}}(s)\Gamma_{\mathbb{R}}(s+1)=2(2\pi)^{-s}\Gamma(s)$,
let $q_{\operatorname{\textup{Sym}}^{m}E}$ be the conductor of
$L(s,\operatorname{\textup{Sym}}^{m}E)$, and for even $m$, let $r\in\\{0,1\\}$
such that $r\equiv m/2\bmod 2$. The gamma factor of
$L(s,\operatorname{\textup{Sym}}^{m}E)$ is given by

$\gamma(s,\operatorname{\textup{Sym}}^{m}E)=\begin{cases}\prod_{j=1}^{(m+1)/2}\Gamma_{\mathbb{C}}(s+(j-\frac{1}{2}))&\text{if
}m\text{ is odd }\\\
\Gamma_{\mathbb{R}}(s+r)\prod_{j=1}^{m/2}\Gamma_{\mathbb{C}}(s+j)&\text{if
}m\text{ is even. }\end{cases}$

One may define the analytic conductor $C(\operatorname{\textup{Sym}}^{m}E)$ of
$L(s,\operatorname{\textup{Sym}}^{m}E)$ similarly to the previous subsections.
Note that the complete $L$-function of $L(s,\operatorname{\textup{Sym}}^{m}E)$
is entire of order $1$, and is given by

$\Lambda(s,\operatorname{\textup{Sym}}^{m}E)=q_{\operatorname{\textup{Sym}}^{m}E}^{s/2}\gamma(s,\operatorname{\textup{Sym}}^{m}E)L(s,\operatorname{\textup{Sym}}^{m}E).$

Also, there exists a $W(\operatorname{\textup{Sym}}^{m}E)\in\mathbb{C}$
satisfying $|W(\operatorname{\textup{Sym}}^{m}E)|=1$ for which the functional
equation

$\Lambda(s,\operatorname{\textup{Sym}}^{m}E)=W(\operatorname{\textup{Sym}}^{m}E)\Lambda(1-s,\operatorname{\textup{Sym}}^{m}E)$

holds.

Let $\pi_{f}\in\mathfrak{F}_{2}$ be the cuspidal form corresponding to $f$.
Then as detailed in [28, Theorem 6.1], due to the work by Newton and Thorne
([19, Theorem B] and [18, Theorem A]) and the work in [16] and [5], we know
that $L(s,\operatorname{\textup{Sym}}^{m}E)$ is the standard $L$-function
associated to the representation
$\operatorname{\textup{Sym}}^{m}\pi_{f}\in\mathfrak{F}_{m+1}$ with the same
gamma factor, complete $L$-function, and functional equation (thus henceforth
we will often write
$\operatorname{\textup{Sym}}^{m}E=\operatorname{\textup{Sym}}^{m}\pi_{f}$).
Now, by [7, Section A.2], we have

$\log q_{\operatorname{\textup{Sym}}^{m}E}\ll m\log N_{E}.$

Through a straightforward calculation using Stirling’s formula and the above
properties, we may now obtain

(3.3) $\displaystyle\log C(\operatorname{\textup{Sym}}^{m}E)\ll
m\log(N_{E}m).$

##  4\. Zero-free regions

Throughout this section we will fix the notation from Sections 1 and 3. The
following proposition, taken from [28], will be useful in establishing many of
the zero-free regions used in this paper.

######  Proposition 4.1 ([28, Proposition 4.1]).

Let $\Pi$ be an isobaric automorphic representation of
$\mathrm{GL}_{r}(\mathbb{A})$. If $L(s,\Pi\times\widetilde{\Pi})$ has a pole
of order $r_{\Pi\times\widetilde{\Pi}}\geq 1$ at $s=1$, then
$L(1,\Pi\times\widetilde{\Pi})\neq 0$, and there exists a constant
$\Cl[abcon]{prop41}$ such that $L(s,\Pi\times\widetilde{\Pi})$ has at most
$r_{\Pi\times\widetilde{\Pi}}$ real zeroes in the interval

$\displaystyle s$ $\displaystyle\geq
1-\frac{\Cr{prop41}}{(r_{\Pi\times\widetilde{\Pi}}+1)\log
C(\Pi\times\widetilde{\Pi})}.$

We will also use the following zero-free region throughout the proof of
Theorem 1.7.

######  Theorem 4.2 ([28, Corollary 4.2]).

Let $\pi\in\mathfrak{F}_{m}$ and $\pi^{\prime}\in\mathfrak{F}_{m^{\prime}}$.
Suppose that both $\pi$ and $\pi^{\prime}$ are self-dual. There exists a
constant $\Cl[abcon]{thorner}>0$ for which the following results hold.

  1. (1)

$L(s,\pi)\neq 0$ in the region

$\displaystyle\mathrm{Re}(s)$ $\displaystyle\geq
1-\frac{\Cr{thorner}}{m\log(C(\pi)(3+|\mathrm{Im}(s)|)}$

apart from at most one zero. If the exceptional zero exists, then it is real
and simple.

  2. (2)

$L(s,\pi\times\pi^{\prime})\neq 0$ in the region

$\displaystyle\mathrm{Re}(s)$ $\displaystyle\geq
1-\frac{\Cr{thorner}}{(m+m^{\prime})\log(C(\pi)C(\pi^{\prime})(3+|\mathrm{Im}(s)|)^{\min(m,m^{\prime})})}$

apart from at most one zero. If the exceptional zero exists, then it is real
and simple.

To remove the possibility of an exceptional zero, we will frequently use the
following proposition.

######  Proposition 4.3.

Let $f$ be a holomorphic newform with complex multiplication, and let $\chi$
be the Dirichlet character such that $f=f\otimes\chi$. Let
$\pi_{f}\in\mathfrak{F}_{2}$ be the representation corresponding with $f$. Let
$\pi$ correspond with a cuspidal automorphic representation of
$\mathfrak{F}_{m}$, and suppose that $\pi\otimes\chi\neq\pi$. Then there
exists an absolute constant $\Cl[abcon]{propB}$ such that
$L(s,\pi_{f}\times\pi)$ has no zeroes in the interval

$\displaystyle\mathrm{Re}(s)$ $\displaystyle\geq
1-\frac{\Cr{propB}}{m\log(C(\pi_{f})C(\pi)(|\mathrm{Im}(s)|+3))}.$

###### Proof.

Consider the isobaric sum

$\Pi=\pi\boxplus\pi\otimes\chi\boxplus\widetilde{\pi}_{f}\otimes|\cdot|^{-i\gamma}.$

By hypothesis, $\pi_{f}\otimes\chi=\pi_{f}$ and $\pi\otimes\chi\neq\pi$. This
implies that $\widetilde{\pi}_{f}\otimes\bar{\chi}=\widetilde{\pi}_{f}$ and
$\pi\otimes\bar{\chi}\neq\pi$. It follows that $L(s,\Pi\times\widetilde{\Pi})$
factors as

$\displaystyle
L(s,\pi_{f}\times\widetilde{\pi}_{f})L(s,\pi\times\widetilde{\pi})^{2}L(s,\pi\times(\widetilde{\pi}\otimes\chi))L(s,\pi\times(\widetilde{\pi}\otimes\overline{\chi}))L(s+i\gamma,\pi\times\pi_{f})L(s-i\gamma,\widetilde{\pi}\times\widetilde{\pi}_{f})$
$\displaystyle\times$ $\displaystyle
L(s+i\gamma,\pi\times(\pi_{f}\otimes\chi))L(s-i\gamma,\widetilde{\pi}\times(\widetilde{\pi}_{f}\otimes\overline{\chi}))$
$\displaystyle=$ $\displaystyle
L(s,\pi_{f}\times\widetilde{\pi}_{f})L(s,\pi\times\widetilde{\pi})^{2}L(s,\pi\times(\widetilde{\pi}\otimes\chi))L(s,\pi\times(\widetilde{\pi}\otimes\overline{\chi}))L(s+i\gamma,\pi\times\pi_{f})^{2}L(s-i\gamma,\widetilde{\pi}\times\widetilde{\pi}_{f})^{2}.$

This $L$-function has a pole of order $3$ at $s=1$ (due to the
$L(s,\pi_{f}\times\widetilde{\pi}_{f})$ and $L(s,\pi\times\widetilde{\pi})$
terms). Moreover, if $\beta$ is a real zero of
$L(s+i\gamma,\pi\times\pi_{f})$, then $\beta$ is also a real zero of
$L(s-i\gamma,\widetilde{\pi}\times\widetilde{\pi}_{f})$. Hence a zero
$\beta+i\gamma$ of $L(s,\pi\times\pi_{f})$ would imply a real zero of order
$4$ of $L(s,\Pi\times\widetilde{\Pi})$. If $\beta+i\gamma$ is in the interval
specified above, then this would now contradict Proposition 4.1 by a simple
conductor calculation using (3.2). Note that the contribution from $C(\chi)$
may be neglected as the conductor of $\chi$ divides the square of the
conductor of $\pi_{f}$, by [23, Theorem A]. ∎

Note that in both applications of this Proposition within this paper, the
newform $f$ is taken to correspond to an elliptic curve with CM. Hence there
are finitely many choices for $\chi$ (as it must be the Kronecker character of
the field over which the elliptic curve has CM), and so the contribution of
$C(\chi)$ above can be absorbed into the constant $\Cr{propB}$.

###  4.1. Non-CM newforms twisted by Dirichlet characters

######  Lemma 4.4.

Let $\pi\in\mathfrak{F}_{m+1}$ ($m\geq 1$), and $\chi$ be a primitive
Dirichlet character. Suppose that $\pi$ is self dual. Then there exists an
absolute constant $\Cl[abcon]{41}$ such that $L(s,\pi\otimes\chi)\neq 0$ for
all $s$ in the region

$\displaystyle\textnormal{Re}(s)\geq
1-\frac{\Cr{41}}{m\log(C(\pi)C(\chi)(3+|\textnormal{Im}(s)|))},$

apart from at most one Siegel zero. If such a Siegel zero exists, it is real
and simple, and $\chi$ is self-dual.

###### Proof.

When $\chi$ is real, this is a special case of Theorem 4.2. Hence assume
$\chi$ is not real. Suppose for the sake of contradiction that
$\rho=\sigma+it$ is a zero in that region. Let
$\Pi=\chi|\cdot|^{it}\boxplus\overline{\chi}|\cdot|^{-it}\boxplus\pi$. If
$\psi$ denotes the primitive Dirichlet character that induces $\chi^{2}$, then

$L(s,\Pi\times\overline{\Pi})=\zeta(s)^{2}L(s,\pi\times\overline{\pi})L(s+it,\pi\otimes\chi)^{2}L(s-it,\pi\otimes\overline{\chi})^{2}L(s+2it,\psi)L(s-2it,\overline{\psi}),$

which has exactly 3 poles at $s=1$. However, since $\pi$ is self-dual, both
$L(s+it,\pi\otimes\chi)$ and $L(s-it,\pi\otimes\overline{\chi})$ have a zero
at $\sigma$, so thus $L(s,\Pi\times\widetilde{\Pi})$ has at least four zeroes
at $\sigma$. Combined with the bound on the conductor in (3.2), this now
contradicts Proposition 4.1. Note that this proof works even when $t=0$; hence
a Siegel zero cannot exist in this case. ∎

######  Lemma 4.5.

Consider a non-CM elliptic curve $E$, and let $f$ be the cusp form
corresponding to $E$. Let $\pi_{f}$ be the representation corresponding to
$f$, and let $\pi=\textup{Sym}^{m}\pi_{f}\in\mathfrak{F}_{m+1}$. Then we may
replace $\Cr{41}$ in Lemma 4.4 with an absolute constant $\Cl[abcon]{42}$ such
that Lemma 4.4 holds true for $\pi$ without the possibility of a Siegel zero.

###### Proof.

We only need to consider the case when $\chi$ is self-dual. Consider the
isobaric representation
$\Pi_{m}=\mathbf{1}\boxplus\text{Sym}^{2}\pi_{f}\boxplus(\text{Sym}^{m}\pi_{f}\otimes\chi)$.
Recall the identities

$\displaystyle
L(s,\operatorname{\textup{Sym}}^{m}\pi_{f}\times\operatorname{\textup{Sym}}^{m}\pi_{f})$
$\displaystyle=\zeta(s)\prod_{j=1}^{m}L(s,\operatorname{\textup{Sym}}^{2j}\pi_{f}),$
$\displaystyle\operatorname{\textup{Sym}}^{m}\pi_{f}\times\operatorname{\textup{Sym}}^{2}\pi_{f}$
$\displaystyle=\boxplus_{j=0}^{2}\operatorname{\textup{Sym}}^{m+2-2j}\pi_{f},$

These identities imply that

$\displaystyle
L(s,\Pi_{m}\times\overline{\Pi}_{m})=\zeta(s)^{3}L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)^{4}$
$\displaystyle
L(s,\text{Sym}^{2}\pi_{f})^{3}L(s,\text{Sym}^{4}\pi_{f})L(s,\text{Sym}^{m+2}\pi_{f}\otimes\chi)^{2}$
$\displaystyle\times
L(s,\text{Sym}^{m-2}\pi_{f}\otimes\chi)^{2}\prod_{j=1}^{m}L(s,\text{Sym}^{2j}\pi_{f}).$

Now, this function has a pole of order 3 at $s=1$. If
$L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)$ has a Siegel zero at $\rho$,
$L(s,\Pi_{m}\times\overline{\Pi}_{m})$ will have a zero of order at least 4 at
$\rho$. Combined with the bound on the conductor in (3.2), this would again
contradict Proposition 4.1. ∎

###  4.2. CM newforms twisted by Dirichlet characters

Consider an elliptic curve $E$ which has complex multiplication over an
imaginary quadratic field $K$. Let $m\geq 1$ be an integer, $\xi$ denote the
primitive Grössencharacter which satisfies $L(s,E)=L(s,\xi)$, $f_{m}$ denote
the cusp form which induces $\xi_{m}$ (where $\xi_{m}$ is as defined in
Section 3.3), and $\pi_{f_{m}}\in\mathfrak{F}_{2}$ denote the representation
corresponding to $f_{m}$. Finally, let $\chi$ be a primitive Dirichlet
character that induces a character $\bmod\thinspace q$.

######  Lemma 4.6.

There exists an absolute constant $\Cl[abcon]{cmapzerofree}>0$ such that
$L(s,\pi_{f_{m}}\otimes\chi)$ has no zeroes in the region

$\displaystyle\mathrm{Re}(s)$ $\displaystyle\geq
1-\frac{\Cr{cmapzerofree}}{\log(N_{E}qm(|3+\mathrm{Im}(s)|))}.$

###### Proof.

By [13, Theorem 7.5], there exists an integer $M\leq N_{E}q^{2}$ such that
$f_{m}\otimes\chi$ is a cusp newform in
$S_{m+1}(\Gamma_{0}(M),\chi_{m}\chi^{2})$ (where $\chi_{m}$ is as defined in
Section 3.3). Hence we have that $C(\pi_{f_{m}}\otimes\chi)\ll
N_{E}q^{2}m^{2}$. Moreover, by [12, Section 5.11], we have that the Rankin-
Selberg convolutions $L(s,\pi_{f_{m}}\otimes\chi\times\pi_{f_{m}}\otimes\chi)$
and $L(\pi_{f_{m}}\otimes\chi\times\overline{\pi_{f_{m}}\otimes\chi})$ both
exist, with the former being entire and the latter having a simple pole at
$s=1$. Now, using the information above and from Section 3.3, we may apply
[12, Theorem 5.10], which proves the above except for the possibility of an
exceptional zero. We may now remove this possibility using [11, Theorem C]. ∎

###  4.3. Rankin-Selberg convolutions of a CM and Non-CM newform

Consider two elliptic curves $E,E^{\prime}$, the former having complex
multiplication over an imaginary quadratic field $K$, and the latter without
complex multiplication over any imaginary quadratic field. Let $\xi$ denote
the primitive Grössencharacter which satisfies $L(s,E)=L(s,\xi)$, $g_{m}$
(instead of $f_{m}$) denote the cusp form which induces $\xi_{m}$, and
$\pi_{g_{m}}\in\mathfrak{F}_{2}$ denote the representation corresponding to
$g_{m}$. Also let $f^{\prime}$ denote the cusp form which corresponds to
$E^{\prime}$, and let $\pi_{f^{\prime}}\in\mathfrak{F}_{2}$ be the
representation which corresponds to $f^{\prime}$.

######  Lemma 4.7.

There exists an absolute constant $\Cl[abcon]{461}>0$ such that for all
integers $1\leq m,m^{\prime}\leq M$,

$L(s,\pi_{g_{m}}\times\operatorname{\textup{Sym}}^{m^{\prime}}\pi_{f^{\prime}})\neq
0$ in the region

$\displaystyle\textup{Re}(s)$ $\displaystyle\geq
1-\frac{\Cr{461}}{M^{2}\log(N_{E}N_{E^{\prime}}(3+|\textup{Im}(s)|))}.$

###### Proof.

When $\mathrm{Im}(s)\neq 0$, we may apply Theorem 4.2, while in the case where
$\mathrm{Im}(s)=0$, we may apply Proposition 4.3. In both cases, the result
follows from a conductor calculation using (3.3) and the properties discussed
in Section 3.3. ∎

###  4.4. Rankin-Selberg convolutions of two CM newforms

Consider two twist-inequivalent elliptic curves $E,E^{\prime}$ having complex
multiplication over two imaginary quadratic fields $K,K^{\prime}$
(respectively). Let $m,m^{\prime}\geq 1$ be integers. Consider the
Grössencharacters $\xi,\xi^{\prime}$ corresponding to $E,E^{\prime}$, let
$f=f_{m},f^{\prime}=f^{\prime}_{m^{\prime}}$ be the cusp forms corresponding
to $\xi_{m},\xi^{\prime}_{m^{\prime}}$ (as defined in 3.3), and let
$\pi_{f},\pi_{f^{\prime}}\in\mathfrak{F}_{2}$ denote the representations
corresponding to $f,f^{\prime}$ respectively. Finally, let
$\mathfrak{q}=N_{E}N_{E^{\prime}}mm^{\prime}$.

######  Lemma 4.8.

There exists an absolute constant $\Cl[abcon]{CMCMzerofree}>0$ such that
$L(s,\pi_{f}\times\pi_{f^{\prime}})\neq 0$ in the region

$\displaystyle\textup{Re}(s)$ $\displaystyle\geq
1-\frac{\Cr{CMCMzerofree}}{\log(\mathfrak{q}(3+|\textup{Im}(s)|))}.$

###### Proof.

Note that as a special case of the work due to Bushnell and Henniart [3], we
know that $q(\pi_{f}\times\pi_{f^{\prime}})$ divides
$q(\pi_{f})^{2}q(\pi_{f^{\prime}})^{2}$ (where $q(\cdot)$ denotes the
conductor), and hence $C(\pi_{f}\times\pi_{f^{\prime}})\ll\mathfrak{q}^{4}$.
Now, since $\pi_{f},\pi_{f^{\prime}}$ are both self-dual, Lemma 4.8 follows
from Theorem 4.2 with the exception of a possible Siegel zero. We may remove
this possibility using Proposition 4.3. ∎

##  5\. Prime Number Theorems

Throughout this section, fix the notation from Sections 1 and 3.

###  5.1. Representations of $\mathrm{GL}_{m}(\mathbb{A})$ twisted by
Dirichlet characters

######  Proposition 5.1.

Let $\pi\in\mathfrak{F}_{m}$ be self-dual, and let $\chi$ be a primitive
Dirichlet character. Let $\beta_{1}$ denote the possible Siegel zero from
Lemma 4.4. For each prime $p$, let $a_{\pi}(p)$ be the coefficient of $p^{-s}$
in the Dirichlet series expansion of $L(s,\pi)$. Suppose that $\pi$ satisfies
the generalized Ramanujan conjecture at all primes $p\nmid C(\pi)$ and all its
Langlands parameters satisfy either $\mu_{\pi}(j)=0$ or
$\textnormal{Re}(\mu_{\pi}(j))\geq\frac{1}{2}$ for each $j$. Then there exists
an absolute constant $\Cl[abcon]{43}>2$ such that if
$2\leq(C(\pi)C(\chi))^{m}\leq x^{1/\Cr{43}}$ and
$\Cr{43}mx^{-\frac{1}{\Cr{43}m}}<\frac{1}{4}$, we have

$\displaystyle\Bigg{|}\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
q_{\pi}q_{\chi}\end{subarray}}a_{\pi}(p)\chi(p)\log
p\bigg{)}-r_{\pi\otimes\chi}x+\frac{x^{\beta_{1}}}{\beta_{1}}\Bigg{|}$
$\displaystyle\ll
m^{2}x^{1-\frac{1}{\Cr{43}m}}+m^{2}x\Big{(}\exp\Big{(}-\frac{\Cr{41}\log
x}{2m\log C(\pi)C(\chi)}\Big{)}+\exp\Big{(}-\frac{\sqrt{\Cr{41}\log
x}}{2\sqrt{m}}\Big{)}\Big{)}.$

where the $\frac{x^{\beta_{1}}}{\beta_{1}}$ term is omitted if $\beta_{1}$
does not exist.

###### Proof.

The proof is identical to the proof of [28, Proposition 5.1], except for the
zero-free region [28, Corollary 4.2] is replaced by Lemma 4.4. Note that by
definition, the coefficient $a_{\pi\otimes\chi}(p)$ of $p^{-s}$ in the
Dirichlet series expansion of $L(s,\pi)$ is precisely $a_{\pi}(p)\chi(p)$ for
$p\nmid q_{\pi}q_{\chi}$. ∎

######  Lemma 5.2.

Let $E$ be a non-CM elliptic curve, and let $\chi$ be a Dirichlet character
which induces a character of modulus $q$. Then, there exists a sufficiently
small absolute constant $\Cl[abcon]{2aM}$ such that if
$M=\frac{\Cr{2aM}\sqrt{\log x}}{\log(N_{E}q\log x))}$, then for all $1\leq
m\leq M,$

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\chi(p)U_{m}(\cos\theta_{E}(p))\log p\ll
m^{2}x\Big{(}x^{-\frac{1}{\Cr{43}m}}+\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}qm)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)}\Big{)}.$

###### Proof.

Consider any integer $m\geq 1$. Let $f$ be the cusp form corresponding to $E$,
and let $\pi_{f}\in\mathfrak{F}_{2}$ be the representation corresponding to
$f$. We first establish the following properties about
$L(s,\operatorname{\textup{Sym}}^{m}\pi_{f}\otimes\chi)$.

  1. (1)

The conductor of $\text{Sym}^{m}\pi_{f}\otimes\chi$ satisfies $\log
C(\text{Sym}^{m}\pi_{f}\otimes\chi)\ll m\log(N_{E}qm)$.

  2. (2)

All of the Langlands parameters at infinity of
$L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)$ are nonnegative, and either integers
or half integers.

  3. (3)

$L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)$ is the $L$-function of a cuspidal
automorphic representation in $\mathfrak{F}_{m+1}$.

  4. (4)

$L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)$ is entire for $m\geq 1$.

  5. (5)

$L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)$ has no zero in the region

$\textnormal{Re}(s)\geq
1-\frac{\Cr{41}}{m^{2}\log(N_{E}qm(3+|\textnormal{Im}(s)|))}.$

  6. (6)

$L(s,\text{Sym}^{m}\pi_{f}\otimes\chi)$ satisfies the generalized Ramanujan
conjecture (GRC).

(1) follows from (3.2) and (3.3). (2), (3), and (4) all follow from the work
due to Newton and Thorne in [18] and [19]. (5) follows from Lemma 4.4 and
Lemma 4.5. (6) follows from the definition of the symmetric power
$L$-function, given in Section 3.7.

Now, given (1)-(6), we have that evidently there exists a sufficiently small
absolute constant $\Cr{2aM}$ such that if $M=\frac{\Cr{2aM}\sqrt{\log
x}}{\log(N_{E}q\log x))}$, then for all $1\leq m\leq M$,
$\text{Sym}^{m}\pi_{f}\otimes\chi\in\mathfrak{F}_{m+1}$ satisfies the
conditions in Proposition 5.1 for all $x\geq 3$. Applying the proposition, we
thus obtain

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\chi(p)U_{m}(\cos\theta_{E}(p))\log p\ll
m^{2}x^{1-\frac{1}{\Cr{43}m}}+m^{2}x\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}qm)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)},$

as desired. ∎

###  5.2. CM Newforms twisted by Dirichlet characters

Consider an elliptic curve $E$ which has complex multiplication over an
imaginary quadratic field $K$. Let $m\geq 1$ be an integer, $\xi$ denote the
primitive Grössencharacter which satisfies $L(s,E)=L(s,\xi)$, $f_{m}$ denote
the cusp form which induces $\xi_{m}$ (where $\xi_{m}$ is as defined in
Section 3.3), and $\pi_{f_{m}}\in\mathfrak{F}_{2}$ denote the representation
corresponding to $f_{m}$.

######  Lemma 5.3.

Given any positive integer $m$, there exists an absolute constant
$\Cl[abcon]{44}>0$ such that for all $x\geq N_{E}$,

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\textup{ splits in }K\end{subarray}}\cos(m\theta_{E}(p))\log p$
$\displaystyle\ll x\Big{(}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}qm)}\Big{)}\log(xN_{E}qm)^{4}\Big{)}.$

###### Proof.

Consider a primitive Dirichlet character $\chi$ which induces a Dirichlet
character mod $q$. We first establish a prime number theorem for
$L(s,f_{m}\otimes\chi)$. Let $\Lambda(n)$ denote the Von Mangoldt function.
Note that by [13, Section 12.3], we have that if we write the logarithmic
derivative of $L(s,f_{m}\otimes\chi)$ as

$\displaystyle-\frac{L^{\prime}(s,f_{m}\otimes\chi)}{L(s,f_{m}\otimes\chi)}$
$\displaystyle=\sum_{n}a_{m,\chi}(n)\Lambda(n)n^{-s},$

then

$\displaystyle a_{m,\chi}(p)$
$\displaystyle=\begin{cases}2\chi(p)\cos(m\theta_{E}(p))&p\text{ splits in
}K\\\ 0&\text{otherwise}\end{cases}$

and $|a_{m,\chi}(p^{j})|\leq 2$ for all prime powers $p^{j},j\geq 2$. Note
that when $\chi$ is trivial, we will write $a_{m,\chi}=a_{m}$. Now, this
evidently implies that $\sum_{n\leq x}|a_{m,\chi}(n)\Lambda(n)|^{2}\ll
x\log^{2}(x)$. Hence, given the above and Lemma 4.6, we may apply [12, Theorem
5.13] and obtain that there exists an absolute constant $\Cr{44}>0$ for which

(5.1) $\displaystyle\sum_{n\leq x}a_{m,\chi}(n)\Lambda(n)$ $\displaystyle\ll
x\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}qm)}\Big{)}\log(xN_{E}qm)^{4}.$

Now, note that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in }K\end{subarray}}\cos(m\theta_{E}(p))\log
p-\frac{1}{2}\sum_{\begin{subarray}{c}n\leq x\\\ n\equiv
a(q)\end{subarray}}a_{m}(n)\Lambda(n)$
$\displaystyle=-\frac{1}{2}\sum_{\begin{subarray}{c}p\leq x\\\ r\geq 2\\\
p^{r}\equiv a(q)\end{subarray}}a_{m}(p^{r})\log
p-\frac{1}{2}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p|N_{E}\end{subarray}}a_{m}(p)\log p$ $\displaystyle\ll\sqrt{N_{E}}\log
N_{E}+\sqrt{x}\log x.$

Here we used the fact that by (4), $a_{m}(p)=2\cos(m\theta_{E}(p))$ for all
$p\nmid N_{E}$ and $|a_{m}(p)|\leq 2\sqrt{p}$ for all $p$. Hence we have that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in }K\end{subarray}}\cos(m\theta_{E}(p))\log p$
$\displaystyle\ll\sqrt{N_{E}}\log N_{E}+\sqrt{x}\log
x+\sum_{\begin{subarray}{c}n\leq x\\\ n\equiv
a(q)\end{subarray}}a_{m}(n)\Lambda(n)$ $\displaystyle\ll\sqrt{N_{E}}\log
N_{E}+\sqrt{x}\log
x+\frac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi(a)}\sum_{n\leq
x}a_{m,\chi}(n)\Lambda(n)$ $\displaystyle\ll x\exp\Big{(}\frac{-\Cr{44}\log
x}{\sqrt{\log x}+3\log(N_{E}qm)}\Big{)}\log(xN_{E}qm)^{4}.$

when $x\geq N_{E}$, as desired. Note that in the summation $\chi$ ranges over
all the primitive characters which induce the characters $\bmod\thinspace q$,
while to get the last inequality we replace these characters with all the
(possibly imprimitive) characters $\bmod\thinspace q$. This generates a
negligible error which is absorbed into other error terms. ∎

For the rest of the paper, the symbol $\chi(q)$ in the subscript for a
summation means that the sum ranges through all Dirichlet characters $\chi$
with modulus $q$. Again, we will often replace these characters by the
primitive characters that induce them; in every case the error will be
negligble.

For the proof of Theorem 1.7 (2.b), it will also be necessary to derive an
estimate for the number of primes in an arithmetic progression which split or
remain inert in $K$. Throughout the following proof, let $D_{K}$ be the
discriminant of $K$.

######  Lemma 5.4.

There exists absolute constants $\Cl[abcon]{451},\Cl[abcon]{452}$ such that

$\displaystyle(1)\thickspace\thickspace\thickspace$ $\displaystyle\\#\\{p\leq
x\mathrel{\mathop{\mathchar 58\relax}}p\equiv a\bmod q,p\textup{ splits in
}K\\}$
$\displaystyle=\frac{\pi(x)}{2\varphi(q)}\left(1+\chi_{K}(a)\delta(K,q)\right)$
$\displaystyle+O\Big{(}\pi(x)\Big{(}\frac{x^{-\Cr{451}/\sqrt{q|D_{K}|}}}{\varphi(q)}+\exp\Big{(}\frac{-\Cr{452}\log
x}{\sqrt{\log x}+3\log(q|D_{K}|)}\Big{)}(\log(xq|D_{K}|))^{4}\Big{)}\Big{)}.$
$\displaystyle(2)\thickspace\thickspace\thickspace$ $\displaystyle\\#\\{p\leq
x\mathrel{\mathop{\mathchar 58\relax}}p\equiv a\bmod q,p\textup{ inert in
}K\\}$
$\displaystyle=\frac{\pi(x)}{2\varphi(q)}\left(1-\chi_{K}(a)\delta(K,q)\right)$
$\displaystyle+O\Big{(}\pi(x)\Big{(}\frac{x^{-\Cr{451}/\sqrt{q|D_{K}|}}}{\varphi(q)}+\exp\Big{(}\frac{-\Cr{452}\log
x}{\sqrt{\log x}+3\log(q|D_{K}|)}\Big{)}(\log(xq|D_{K}|))^{4}\Big{)}\Big{)}.$

###### Proof.

The two estimates are proved analogously, so we only prove (1). First, note
that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in }K\end{subarray}}1$ $\displaystyle=\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv a\bmod q\end{subarray}}\frac{\chi_{K}(p)+1}{2}$
$\displaystyle=\frac{1}{2}\pi(x;q,a)+\frac{1}{2}\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv a\bmod q\end{subarray}}\chi_{K}(p).$

Now, given the possibility of a Siegel zero, we have that by [17, Section
11.3] and [12, Theorem 5.28], there exists absolute constants $\Cr{451}$ and
$\Cl[abcon]{453}$ such that $\pi(x;q,a)$ satisfies the following:

(5.2)
$\displaystyle\pi(x;q,a)=\frac{\operatorname{\textup{Li}}(x)}{\varphi(q)}+O\Big{(}\frac{x}{\log
x}\Big{(}\frac{x^{-\Cr{451}/\sqrt{q}}}{\varphi(q)}+(\log
x)e^{-\Cr{453}\sqrt{\log x}}\Big{)}\Big{)}.$

where $\operatorname{\textup{Li}}(x)\mathrel{\mathop{\mathchar
58\relax}}=\int_{2}^{x}\frac{dt}{\ln t}$. Note that by [17, Theorem 6.14], we
may replace the $\operatorname{\textup{Li}}(x)$ in (5.2) with $\pi(x)$ and
generate an error that is absorbed into the other error terms. Now, consider
any primitive character $\chi$ that induces $\chi^{\prime}\chi_{K}$ for some
$\chi^{\prime}$ mod $q$. Define $\delta_{\chi}$ as $1$ if $\chi$ is trivial
and $0$ otherwise. Then, by [12, Theorem 5.13, Theorem 5.28], there exists an
absolute constant $\Cl[abcon]{454}$ such that

$\displaystyle\sum_{n\leq x}\chi(n)\Lambda(n)$
$\displaystyle=\delta_{\chi}x+O\Big{(}x^{1-\Cr{451}/\sqrt{q}}+x\exp\Big{(}\frac{-\Cr{454}\log
x}{\sqrt{\log x}+3\log q}\Big{)}(\log xq)^{4}\Big{)}.$

Here we adjust $\Cr{451}$ if necessary to absorb the contribution from
$\sqrt{D_{K}}$, which can only take finitely many values. Also note that by
[12, Theorem 5.28], the Siegel zero term $x^{1-\Cr{451}/\sqrt{q}}$ can only
exist for at most one of the primitive $\chi$ we consider. Now, note that

$\displaystyle\sum_{p\leq x}\chi(p)\log p=\sum_{n\leq
x}\chi(n)\Lambda(n)+O(\sqrt{x}\log x).$

We thus obtain

(5.3) $\displaystyle\sum_{p\leq x}\chi(p)\log p$
$\displaystyle=\delta_{\chi}x+O\Big{(}x^{1-\Cr{451}/\sqrt{q}}+x\exp\Big{(}\frac{-\Cr{454}\log
x}{\sqrt{\log x}+3\log q}\Big{)}(\log xq)^{4}\Big{)}.$

We now bound $\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod
q\end{subarray}}\chi_{K}(p)\log p$. We split into two cases. Note that by [17,
Theorem 9.13], we have that $\chi_{K}$ is primitive with conductor $|D_{K}|$.
Hence, when $\delta(K,q)=0$, we have that $\chi^{\prime}\chi_{K}$ is
nontrivial for all $\chi^{\prime}\bmod q$ with conductor dividing $q|D_{K}|$,
so thus by (5.3),

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod
q\end{subarray}}\chi_{K}(p)\log p$
$\displaystyle=\frac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi(a)}\sum_{p\leq
x}\chi(p)\chi_{K}(p)\log p$
$\displaystyle\ll\frac{x^{1-\Cr{451}/\sqrt{q|D_{K}|}}}{\varphi(q)}+x\exp\Big{(}\frac{-\Cr{454}\log
x}{\sqrt{\log x}+3\log(q|D_{K}|)}\Big{)}(\log(xq|D_{K}|))^{4}.$

When $\delta(K,q)=1$, $\chi^{\prime}\chi_{K}$ is trivial for exactly one
character $\chi^{\prime}\bmod q$. For this $\chi^{\prime}$, we have
$\overline{\chi^{\prime}(a)}=1/\overline{\chi_{K}(a)}=\chi_{K}(a)$. By (5.3),
we thus have

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod
q\end{subarray}}\chi_{K}(p)\log p$
$\displaystyle=\frac{1}{\varphi(q)}\sum_{\chi(q)}\overline{\chi(a)}\sum_{p\leq
x}\chi(p)\chi_{K}(p)\log p$
$\displaystyle=\frac{\chi_{K}(a)}{\varphi(q)}x+O\Big{(}\frac{x^{1-\Cr{451}/\sqrt{q}}}{\varphi(q)}+x\exp\Big{(}\frac{-\Cr{454}\log
x}{\sqrt{\log x}+3\log q}\Big{)}(\log xq)^{4}\Big{)}.$

We now collate the bounds above. By partial summation, we have

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod
q\end{subarray}}\chi_{K}(p)$
$\displaystyle=\frac{\pi(x)}{\varphi(q)}\chi_{K}(a)\delta(K,q)$
$\displaystyle+O\bigg{(}\frac{1}{\log
x}\Big{(}\frac{x^{1-\Cr{451}/\sqrt{q|D_{K}|}}}{\varphi(q)}+x\exp\Big{(}\frac{-\Cr{454}\log
x}{\sqrt{\log x}+3\log(q|D_{K}|)}\Big{)}(\log(xq|D_{K}|))^{4}\Big{)}\bigg{)}.$

Hence, there exists an absolute constant $\Cr{452}$ such that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in }K\end{subarray}}1$
$\displaystyle=\frac{1}{2}\pi(x;q,a)+\frac{1}{2}\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv a\bmod q\end{subarray}}\chi_{K}(p)$
$\displaystyle=\frac{\pi(x)}{2\varphi(q)}\left(1+\chi_{K}(a)\delta(K,q)\right)$
$\displaystyle+O\Big{(}\pi(x)\Big{(}\frac{x^{-\Cr{451}/\sqrt{q|D_{K}|}}}{\varphi(q)}+\exp\Big{(}\frac{-\Cr{452}\log
x}{\sqrt{\log x}+3\log(q|D_{K}|)}\Big{)}(\log(xq|D_{K}|))^{4}\Big{)}\Big{)}.$

∎

###  5.3. Rankin-Selberg convolutions of a CM and Non-CM newform

Consider two elliptic curves $E,E^{\prime}$, the former having complex
multiplication over an imaginary quadratic field $K$, and the latter without
complex multiplication over any imaginary quadratic field. Let $\xi$ denote
the primitive Grössencharacter which satisfies $L(s,E)=L(s,\xi)$, $g_{m}$
(instead of $f_{m}$) denote the cusp form which induces $\xi_{m}$, and
$\pi_{g_{m}}\in\mathfrak{F}_{2}$ denote the representation corresponding to
$g_{m}$. Also let $f^{\prime}$ denote the cusp form which corresponds to
$E^{\prime}$, and let $\pi_{f^{\prime}}\in\mathfrak{F}_{2}$ be the
representation which corresponds to $f^{\prime}$.

######  Lemma 5.5.

There exists an absolute constant $\Cl[abcon]{161}>0$ such that if
$M=\frac{\Cr{161}\sqrt{\log x}}{\log(N_{E}N_{E^{\prime}}\log x)}$, then for
all $1\leq m,m^{\prime}\leq M$,

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}N_{E^{\prime}}\end{subarray}}\cos(m\theta_{E}(p))U_{m}(\cos\theta_{E^{\prime}}(p))\log
p$ $\displaystyle\ll m^{\prime
2}x\Big{(}x^{-\frac{1}{32\Cr{43}M^{2}}}+\exp\Big{(}-\frac{\Cr{461}\log
x}{4M^{2}\log(N_{E}{N_{E^{\prime}}}M)}\Big{)}+\exp\Big{(}-\frac{\sqrt{\Cr{461}\log
x}}{2M}\Big{)}\Big{)}.$

###### Proof.

By the information in Sections 3.3 and 3.7, along with [12, Equation 5.86] and
[28, Equation 3.3], we have that the Langlands parameters
$\mu_{\pi_{g_{m}}\times\operatorname{\textup{Sym}}^{m^{\prime}}\pi_{f^{\prime}}}$
of
$L(s,\pi_{g_{m}}\times\operatorname{\textup{Sym}}^{m^{\prime}}\pi_{f^{\prime}})$
satisfy that
$\mu_{\pi_{g_{m}}\times\operatorname{\textup{Sym}}^{m^{\prime}}\pi_{f^{\prime}}}=0$
or
$\textup{Re}(\mu_{\pi_{g_{m}}\times\operatorname{\textup{Sym}}^{m^{\prime}}\pi_{f^{\prime}}})\geq\frac{1}{2}$.
Combining this with (3.1), we find that there exists an absolute constant
$\Cr{161}>0$ such that if $M$ is defined as above, then
$\pi_{g_{m}}\times\text{Sym}^{m^{\prime}}\pi_{f^{\prime}}$ satisfies the
conditions of [28, Proposition 5.1] for all $1\leq m,m^{\prime}\leq M$.
Applying the Proposition, we find that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}N_{E^{\prime}}\end{subarray}}\cos(m\theta_{E}(p))U_{m}(\cos\theta_{E^{\prime}}(p))\log
p$ $\displaystyle\ll m^{\prime 2}x^{1-\frac{1}{32\Cr{43}M^{2}}}+m^{\prime
2}x\Big{(}\exp\Big{(}-\frac{\Cr{461}\log
x}{4M^{2}\log(N_{E}{N_{E^{\prime}}}M)}\Big{)}+\exp\Big{(}-\frac{\sqrt{\Cr{461}\log
x}}{2M}\Big{)}\Big{)},$

as desired. ∎

###  5.4. Rankin-Selberg convolutions of Two CM newforms

Consider two twist-inequivalent elliptic curves $E,E^{\prime}$ having complex
mulitplication over two imaginary quadratic fields $K,K^{\prime}$
(respectively). Let $m,m^{\prime}\geq 1$ be integers. Consider the
Grössencharacters $\xi,\xi^{\prime}$ corresponding to $E,E^{\prime}$, let
$f=f_{m},f^{\prime}=f^{\prime}_{m^{\prime}}$ be the cusp forms corresponding
to $\xi_{m},\xi^{\prime}_{m^{\prime}}$ (as defined in 3.3), and let
$\pi_{f},\pi_{f^{\prime}}\in\mathfrak{F}_{2}$ denote the representations
corresponding to $f,f^{\prime}$ respectively. Finally, let
$\mathfrak{q}=N_{E}N_{E^{\prime}}mm^{\prime}$.

######  Lemma 5.6.

There exists an absolute constant $\Cl[abcon]{471}>0$ such that

$\sum_{\begin{subarray}{c}p\leq x\\\ p\textup{ splits in
}K\end{subarray}}\cos(m\theta_{E}(p))\cos(m^{\prime}\theta_{E^{\prime}}(p))\log
p\ll x\exp\bigg{(}\frac{-\Cr{471}\log x}{\sqrt{\log
x}+3\log\mathfrak{q}}\bigg{)}(\log x\mathfrak{q})^{4}.$

###### Proof.

This is trivial for $x<\mathfrak{q}$; thus assume $x\geq\mathfrak{q}$. Let
$\Lambda(n)$ denote the Von Mangoldt function, and let the Dirichlet series
expansion of $L(s,\pi_{f}\times\pi_{f^{\prime}})$ be $\sum_{n\geq
1}a_{\pi_{f}\times\pi_{f^{\prime}}}(n)n^{-s}$. Define

$\displaystyle\psi(x,\pi_{f}\times\pi_{f^{\prime}})$
$\displaystyle=\sum_{n\leq x}a_{\pi_{f}\times\pi_{f^{\prime}}}(n)\Lambda(n).$

Note that by definition
$a_{\pi_{f}\times\pi_{f^{\prime}}}(p)=\cos(m\theta_{E}(p))\cos(m^{\prime}\theta_{E^{\prime}}(p))$
when $p\nmid N_{E}N_{E^{\prime}}$, and moreover this quantity equals $0$ when
$p$ is inert in either $K$ or $K^{\prime}$. Hence

$\displaystyle\psi(x,\pi_{f}\times\pi_{f^{\prime}})$
$\displaystyle=\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}N_{E^{\prime}}\\\ p\text{ splits in
}K\textup{~{}and~{}}K^{\prime}\end{subarray}}\cos(m\theta_{E}(p))\cos(m\theta_{E^{\prime}}(p))\log
p+O\bigg{(}\sum_{p^{j}\leq x,j\geq 2}\log
p+\sum_{p|N_{E}N_{E^{\prime}}}\sqrt{p}\log p\bigg{)}$
$\displaystyle=\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}N_{E^{\prime}}\\\ p\textup{ splits in
}K\textup{~{}and~{}}K^{\prime}\end{subarray}}\cos(m\theta_{E}(p))\cos(m\theta_{E^{\prime}}(p))\log
p+O\left(\sqrt{x}\log
x+\sqrt{N_{E}N_{E^{\prime}}}\log(N_{E}N_{E^{\prime}})\right).$

Now, note that [12, Equation 5.48] holds trivially for
$L(s,\pi_{f}\times\pi_{f^{\prime}})$. Hence, given the information above along
with Lemma 4.8, we may apply [12, Theorem 5.13] (substituting [12, Equation
5.39] with (4.8)) and get that there exists an absolute constant $\Cr{471}>0$
such that

$\psi(x,\pi_{f}\times\pi_{f^{\prime}})\ll x\exp\bigg{(}-\frac{\Cr{471}\log
x}{\sqrt{\log x}+3\log\mathfrak{q}}\bigg{)}(\log x\mathfrak{q})^{4},$

Substituting in $\psi$, we now get our desired result. ∎

##  6\. Proof of Theorem 1.7 (2.a)

Fix the notation from Sections 1, 3, and 4.1. In particular, let $a,q$ be
coprime positive integers, and consider an elliptic curve $E$ without complex
multiplication over any imaginary quadratic field.

###### Proof of Theorem 1.7 (2.a).

We prove the Theorem for $x\geq N_{E}$; for $x<N_{E}$ the result is trivial.
We start with some preliminary manipulations. Let $U_{m}$ denote the $m$th
Chebyshev polynomial of the second kind. By Lemma 3.1, we have

$\displaystyle\pi_{I}(x;q,a)$ $\displaystyle=\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv a(q)\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))$
$\displaystyle\leq\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid N_{E}\\\ p\equiv
a(q)\end{subarray}}\Bigg{(}\mu_{\text{ST}}(I)+\Cr{bspoly0}\bigg{(}\frac{1}{M}+\sum_{m=1}^{M}\bigg{(}\frac{1}{m}U_{m}(\cos\theta_{E}(p))\bigg{)}\bigg{)}\Bigg{)}$
$\displaystyle=\mu_{\text{ST}}(I)\pi(x;q,a)+O\Bigg{(}\frac{\pi(x;q,a)}{M}+\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv
a(q)\end{subarray}}\sum_{m=1}^{M}\bigg{(}\frac{1}{m}U_{m}(\cos\theta_{E}(p))\bigg{)}\Bigg{)},$

We may similarly bound $\pi_{I}(x;q,a)$ from below to obtain that

(6.1) $\displaystyle\pi_{I}(x;q,a)$
$\displaystyle=\mu_{\text{ST}}(I)\pi(x;q,a)+O\Bigg{(}\frac{\pi(x;q,a)}{M}+\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv
a(q)\end{subarray}}\sum_{m=1}^{M}\bigg{(}\frac{1}{m}U_{m}(\cos\theta_{E}(p))\bigg{)}\Bigg{)}.$

Moreover, note that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv
a(q)\end{subarray}}\sum_{m=1}^{M}\frac{1}{m}U_{m}(\cos\theta_{E}(p))$
$\displaystyle=\frac{1}{\varphi(q)}\sum_{\chi\bmod
q}\overline{\chi(a)}\Bigg{(}\sum_{m=1}^{M}\frac{1}{m}\bigg{(}\sum_{p\leq
x}\chi(p)U_{m}(\cos\theta_{E}(p))\bigg{)}\Bigg{)}.$

Now, by Lemma 5.2, we have that there exists a sufficiently small absolute
constant $\Cr{2aM}$ such that if $M=\frac{\Cr{2aM}\sqrt{\log
x}}{\log(N_{E}q\log x))}$, then for all $1\leq m\leq M,$

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}\end{subarray}}\chi(p)U_{m}(\cos\theta_{E}(p))\log p\ll
m^{2}x\Big{(}x^{-\frac{1}{\Cr{43}m}}+\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}qm)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)}\Big{)}.$

Applying partial summation and inserting the contributions from primes
dividing $N_{E}$ (which is negligble for $x>N_{E}$), we obtain

$\displaystyle\sum_{p\leq
x}\chi(p)U_{m}(\cos\theta_{E}(p))\ll\frac{m^{2}x}{\log
x}\Big{(}x^{-\frac{1}{\Cr{43}m}}+\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}qm)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)}\Big{)}.$

Substituting this into (6.1), we obtain that

$\displaystyle\pi_{I}(x;q,a)-\mu_{\text{ST}}(I)\pi(x;q,a)-\frac{\pi(x)}{\varphi(q)M}$
$\displaystyle\ll\frac{1}{\varphi(q)}\sum_{\chi\bmod
q}\overline{\chi(a)}\sum_{m=1}^{M}\bigg{(}\frac{mx}{\log
x}\Big{(}x^{-\frac{1}{\Cr{43}m}}+\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}qm)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)}\Big{)}\bigg{)}$
$\displaystyle\ll\frac{M^{2}x}{\log
x}\Big{(}x^{-\frac{1}{\Cr{43}M}}+\exp\Big{(}-\frac{\Cr{42}\log
x}{2M^{2}\log(N_{E}qM)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{M}}\Big{)}\Big{)}.$

Adjusting $\Cr{2aM}$ to be suitably small, we find that for all $x\geq 2$,

$\displaystyle\pi(x)\bigg{(}\frac{1}{\varphi(q)M}+M^{2}\Big{(}x^{-\frac{1}{\Cr{43}M}}+\exp\Big{(}-\frac{\Cr{42}\log
x}{2M^{2}\log(N_{E}qM)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{M}}\Big{)}\Big{)}\bigg{)}$
$\displaystyle\ll\pi(x;q,a)\frac{\log(N_{E}q\log x)}{\sqrt{\log x}},$

Finally, recall that by (5.2),

$\displaystyle\pi(x;q,a)=\frac{\pi(x)}{\varphi(q)}+O\bigg{(}\frac{x}{\log
x}\Big{(}\frac{x^{-\Cr{451}/\sqrt{q}}}{\varphi(q)}+(\log
x)e^{-\Cr{453}\sqrt{\log x}}\Big{)}\bigg{)}.$

Collating the above bounds, we get our desired estimate. ∎

##  7\. Proof of Theorem 1.7 (2.b)

Throughout this section, we will fix the notation from Sections 1, 3, and 4.2.
In particular, we will consider an elliptic curve $E$ which has complex
multiplication over an imaginary quadratic field $K$, with its theta values
$\theta_{E}(p)$ corresponding to its traces of Frobenius modulo $p$.

###### Proof of Theorem 1.7 (2.b).

The bound is trivially true in the case $x<N_{E}q$, so we only prove the case
$x\geq N_{E}q$. Note that

(7.1) $\displaystyle\pi_{I}(x;q,a)$
$\displaystyle=\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\textup{ splits in
}K\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))+\mathbf{1}_{\pi/2\in
I}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\textup{
inert}\end{subarray}}1.$

We first estimate the first term on the right side of (7.1). By Lemma 3.1, we
have that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\textup{ splits in }K\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))$
$\displaystyle\leq\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\textup{ splits in
}K\end{subarray}}\bigg{(}\frac{|I|}{\pi}+\Cr{bspoly0}\bigg{(}\frac{1}{M}+\sum_{1\leq
m\leq M}\frac{\cos(m\theta_{E}(p))}{m}\bigg{)}\bigg{)}$
$\displaystyle=\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\text{ splits in
}K\end{subarray}}\frac{|I|}{\pi}+O\Big{(}\frac{1}{M}\Big{)}\bigg{)}+O\bigg{(}\sum_{1\leq
m\leq M}\frac{1}{m}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\text{ splits in }K\end{subarray}}\cos(m\theta_{E}(p))\bigg{)}.$

We may similarly bound the left hand side from below, to obtain that

(7.2) $\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\textup{ splits in }K\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))$
$\displaystyle=\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\text{ splits in
}K\end{subarray}}\frac{|I|}{\pi}+O\Big{(}\frac{1}{M}\Big{)}\bigg{)}+O\bigg{(}\sum_{1\leq
m\leq M}\frac{1}{m}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\text{ splits in }K\end{subarray}}\cos(m\theta_{E}(p))\bigg{)}.$

Now, using Lemma 5.3 and partial summation, we find that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in }K\end{subarray}}\cos(m\theta_{E}(p))$
$\displaystyle\ll\frac{1}{\log x}\Big{(}x\exp\Big{(}\frac{-\Cr{44}\log
x}{\sqrt{\log x}+3\log(N_{E}qm)}\Big{)}(\log(xN_{E}qm)^{4}\Big{)}$
$\displaystyle+\Big{(}\int_{1}^{x}(\log t)^{2}\exp\Big{(}\frac{-\Cr{44}\log
t}{\sqrt{\log t}+3\log(N_{E}qm)}\Big{)}\log(N_{E}qm)^{4}dt\Big{)}$ (7.3)
$\displaystyle\ll\frac{1}{\log x}\Big{(}x\exp\Big{(}\frac{-\Cr{44}\log
x}{\sqrt{\log x}+3\log(N_{E}qm)}\Big{)}(\log(xN_{E}qm)^{4}\Big{)}.$

Here we use that for $x\geq N_{E}q$,

$\displaystyle\int_{1}^{x}(\log t)^{2}\exp\Big{(}\frac{-\Cr{44}\log
t}{\sqrt{\log t}+3\log(N_{E}qm)}\Big{)}\log(N_{E}qm)^{4}dt$
$\displaystyle=\int_{0}^{\log
x}\exp\Big{(}t-\frac{-\Cr{44}t}{\sqrt{t}+3\log(N_{E}qm)}\Big{)}t^{2}\log(N_{E}qm)^{4}dt$
$\displaystyle\ll x(\log x)^{3}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}qm)}\Big{)}\log(N_{E}qm)^{4}.$

We now use (7.3) and Lemma 5.4 to estimate the error in (7.2). We first
investigate the contribution from the $p$ which split in $K$. Recall that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in
}K\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))-\sum_{\begin{subarray}{c}p\leq
x\\\ p\equiv a\bmod q\\\ p\text{ splits in }K\end{subarray}}\frac{|I|}{\pi}$
$\displaystyle\ll\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\
p\text{ splits in }K\end{subarray}}\frac{1}{M}+\sum_{1\leq m\leq
M}\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{ splits in
}K\end{subarray}}\frac{1}{m}\cos(m\theta_{E}(p)).$

Hence by Lemma 5.4 (1), we have

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\equiv a\bmod q\\\ p\text{
splits in
}K\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))-\frac{|I|}{\pi}\frac{\pi(x)}{2\varphi(q)}\Big{(}1+\chi_{K}(a)\delta(K,q)\Big{)}$
$\displaystyle\ll\frac{x}{\log
x}\Big{(}\frac{x^{-\Cr{451}/\sqrt{q}}}{\varphi(q)}+\exp\Big{(}\frac{-\Cr{452}\log
x}{\sqrt{\log x}+3\log q}\Big{)}(\log
xq)^{4}\Big{)}+\frac{\pi(x)}{M\varphi(q)}$ $\displaystyle+\sum_{1\leq m\leq
M}\frac{x}{m\log x}\Big{(}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}qm)}\Big{)}(\log(xN_{E}qm)^{4}\Big{)}.$

Note that since there are finitely many possibilities for $K$, we may drop the
$|D_{K}|$ contribution to the error term. Similarly, by Lemma 5.4 (2) we may
estimate the contribution from the $p$ which remain inert in $K$ as

$\displaystyle\mathbf{1}_{\pi/2\in I}\sum_{\begin{subarray}{c}p\leq x\\\
p\equiv a\bmod q\\\ p\text{ inert in }K\end{subarray}}1$
$\displaystyle=\mathbf{1}_{\pi/2\in
I}\Bigg{(}\frac{\pi(x)}{2\varphi(q)}\left(1-\chi_{K}(a)\delta(K,q)\right)$
$\displaystyle+O\Big{(}\pi(x)\Big{(}\frac{x^{-\Cr{451}/\sqrt{q}}}{\varphi(q)}+\exp\Big{(}\frac{-\Cr{452}\log
x}{\sqrt{\log x}+3\log q}\Big{)}(\log xq)^{4}\Big{)}\Big{)}\Bigg{)}.$

Now, collating the bounds in the previous two equations gives

$\displaystyle\pi_{I}(x;q,a)-\frac{|I|}{\pi}\frac{\pi(x)}{2\varphi(q)}\Big{(}1+\chi_{K}(a)\delta(K,q)\Big{)}-\mathbf{1}_{\pi/2\in
I}\Big{(}\frac{\pi(x)}{2\varphi(q)}\Big{(}1-\chi_{K}(a)\delta(K,q)\Big{)}\Big{)}$
$\displaystyle\ll\frac{x}{\log
x}\Big{(}\frac{x^{-\Cr{451}/\sqrt{q}}}{\varphi(q)}+\exp\Big{(}\frac{-\Cr{452}\log
x}{\sqrt{\log x}+3\log q}\Big{)}(\log
xq)^{4}\Big{)}+\frac{\pi(x)}{M\varphi(q)}$ $\displaystyle+\frac{x\log M}{\log
x}\Big{(}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}qM)}\Big{)}(\log(xN_{E}qM)^{4}\Big{)}.$

Setting $M=\exp(\sqrt{\log x})$ now gives us that there exists an absolute
constant $\Cl[abcon]{457}$ such that

$\displaystyle\bigg{|}\pi_{I}(x;q,a)-\frac{\pi(x)}{2\varphi(q)}\bigg{(}\frac{|I|}{\pi}+\mathbf{1}_{\pi/2\in
I}+\chi_{K}(a)\delta(K,q)\Big{(}\frac{|I|}{\pi}-\mathbf{1}_{\pi/2\in
I}\Big{)}\bigg{)}\bigg{|}$
$\displaystyle\ll\Big{(}\pi(x)\Big{(}\frac{x^{-\Cr{451}/\sqrt{q}}}{\varphi(q)}+(\log
x)^{9/2}\exp\Big{(}\frac{-\Cr{457}\log x}{\sqrt{\log
x}+\log(N_{E}q)}\Big{)}\Big{)}\Big{)},$

as desired. ∎

##  8\. Proof of Theorem 1.7 (1.b)

Throughout this section, we will fix the notation from Sections 1, 3, and 4.3.
In particular, we will consider two elliptic curves $E,E^{\prime}$, the former
having complex multiplication over an imaginary quadratic field $K$, the
latter without complex multiplication over any imaginary quadratic field.
Moreover, let $\xi$ denote the primitive Grössencharacter which satisfies
$L(s,E)=L(s,\xi)$, and let $g_{m}$ denote the cusp form which induces
$\xi_{m}$ (where $\xi_{m}$ is as defined in Section 3.3).

###### Proof of Theorem 1.7 (1.b).

We have

(8.1) $\displaystyle\pi_{I,I^{\prime}}(x)=\sum_{\begin{subarray}{c}p\leq x\\\
p\text{ splits in
}K\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))+\mathbf{1}_{\frac{\pi}{2}\in
I}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ inert in
}K\end{subarray}}\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p)).$

We first estimate the contribution from the splitting primes to (8.1).
Applying Lemma 3.2, we obtain

(8.2) $\displaystyle\bigg{|}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{
splits}\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))-\frac{|I|}{\pi}\mu_{\text{ST}}(I^{\prime})\sum_{\begin{subarray}{c}p\leq
x\\\ p\text{ splits}\end{subarray}}$ $\displaystyle 1\bigg{|}$
$\displaystyle\ll\frac{\pi(x)}{M}+\sum_{\begin{subarray}{c}p\leq x\\\ p\text{
splits}\end{subarray}}\bigg{(}\sum_{m=1}^{M}\frac{\cos(m\theta_{E}(p))+U_{m}(\cos\theta_{E^{\prime}}(p))}{m}$
$\displaystyle+\sum_{1\leq m,m^{\prime}\leq
M}\frac{\cos(m\theta_{E}(p))U_{m^{\prime}}(\cos\theta_{E^{\prime}}(p))}{mm^{\prime}}\bigg{)}.$

We first bound the third term on the right side of (8.2). By Lemma 5.5, there
exists an absolute constant $\Cr{161}>0$ such that if
$M=\frac{\Cr{161}\sqrt{\log x}}{\log(N_{E}N_{E^{\prime}}\log x)}$, then for
all $1\leq m,m^{\prime}\leq M$,

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\nmid
N_{E}N_{E^{\prime}}\end{subarray}}\cos(m\theta_{E}(p))U_{m}(\cos\theta_{E^{\prime}}(p))\log
p$ $\displaystyle\ll m^{\prime
2}x\Big{(}x^{-\frac{1}{32\Cr{161}M^{2}}}+\exp\Big{(}-\frac{\Cr{461}\log
x}{4M^{2}\log(N_{E}{N_{E^{\prime}}}M)}\Big{)}+\exp\Big{(}-\frac{\sqrt{\Cr{461}\log
x}}{2M}\Big{)}\Big{)}.$

Now, by partial summation, we have

$\displaystyle\sum_{p\leq x}\sum_{1\leq m,m^{\prime}\leq
M}\frac{\cos(m\theta_{E}(p))U_{m^{\prime}}(\cos\theta_{E^{\prime}}(p))}{mm^{\prime}}$
$\displaystyle\ll M^{2}\log M\Big{(}\frac{x}{\log
x}\Big{)}\Big{(}x^{-\frac{1}{32\Cr{161}M^{2}}}+\exp\Big{(}-\frac{\Cr{461}\log
x}{4M^{2}\log(N_{E}{N_{E^{\prime}}}M)}\Big{)}+\exp\Big{(}-\frac{\sqrt{\Cr{461}\log
x}}{2M}\Big{)}\Big{)}.$

Hence, we may estimate the first and third terms on the right side of (8.2) as
follows:

(8.3) $\displaystyle\frac{\pi(x)}{M}+\sum_{\begin{subarray}{c}p\leq x\\\
p\text{ splits}\end{subarray}}\sum_{1\leq m,m^{\prime}\leq
M}\frac{\cos(m\theta_{E}(p))U_{m^{\prime}}(\cos\theta_{E^{\prime}}(p))}{mm^{\prime}}$
$\displaystyle\ll\pi(x)\bigg{(}\frac{1}{M}+M^{2}\log
M\Big{(}x^{-\frac{1}{32\Cr{43}M^{2}}}+\exp\Big{(}-\frac{\Cr{461}\log
x}{4M^{2}\log(N_{E}{N_{E^{\prime}}}M)}\Big{)}+\exp\Big{(}-\frac{\sqrt{\Cr{461}\log
x}}{2M}\Big{)}\Big{)}\bigg{)}$
$\displaystyle\ll\pi(x)\frac{\log(N_{E}N_{E^{\prime}}\log x)}{\sqrt{\log x}}.$

We now bound the second term on the right side of (8.2). Note that

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\text{
splits}\end{subarray}}\sum_{m=1}^{M}\frac{\cos(m\theta_{E}(p))+U_{m}(\cos\theta_{E^{\prime}}(p))}{m}$
$\displaystyle=\sum_{\begin{subarray}{c}p\leq
x\end{subarray}}\bigg{(}\frac{1+\chi_{K}(p)}{2}\bigg{)}\sum_{m=1}^{M}\frac{\cos(m\theta_{E}(p))+U_{m}(\cos\theta_{E^{\prime}}(p))}{m}.$

Now, by Lemma 5.2 and partial summation, we have that

$\displaystyle\sum_{\begin{subarray}{c}p\leq
x\end{subarray}}\bigg{(}\frac{1+\chi_{K}(p)}{2}\bigg{)}U_{m}(\cos\theta_{E^{\prime}}(p))$
$\displaystyle\ll\frac{m^{2}x}{\log x}\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}m)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)}.$

Note that since the possibilities for $\chi_{K}$ are finite, we may omit both
the contribution from $q$ and the possible Siegel zero. Moreover, by using
(5.1) and applying partial summation, we may obtain

$\displaystyle\sum_{\begin{subarray}{c}p\leq
x\end{subarray}}\bigg{(}\frac{1+\chi_{K}(p)}{2}\bigg{)}\cos(m\theta_{E}(p))$
$\displaystyle\ll\frac{x}{\log x}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}m)}\Big{)}\log(xN_{E}m)^{4}.$

Note that the contribution from the prime powers to (5.1) can be absorbed into
the other error terms (by the same reasoning as in the proof to Lemma 5.3),
and hence has been neglected above. Now, collating the above bounds and
substituting in the value of $M$, we have that

(8.4) $\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\text{
splits}\end{subarray}}\sum_{m=1}^{M}\frac{\cos(m\theta_{E}(p))+U_{m}(\cos\theta_{E^{\prime}}(p))}{m}$
$\displaystyle\ll\sum_{m=1}^{M}\frac{mx}{\log
x}\Big{(}\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2m^{2}\log(N_{E}m)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{m}}\Big{)}\Big{)}\Big{)}$ $\displaystyle+\frac{x}{m\log
x}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}m)}\Big{)}\log(xN_{E}m)^{4}$ $\displaystyle\ll\frac{M^{2}x}{\log
x}\Big{(}\Big{(}\exp\Big{(}-\frac{\Cr{42}\log
x}{2M^{2}\log(N_{E}M)}\Big{)}+\exp\Big{(}-\frac{\Cr{42}\sqrt{\log
x}}{2\sqrt{M}}\Big{)}\Big{)}\Big{)}$ $\displaystyle+\frac{x\log M}{\log
x}\exp\Big{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}M)}\Big{)}\log(xN_{E}M)^{4}$
$\displaystyle\ll\pi(x)\frac{\log(N_{E})^{4}\log(N_{E^{\prime}}\log
x)}{\sqrt{\log x}}.$

We now estimate the contribution from inert primes to (8.1). However, note
that if $D_{K}$ is the conductor of $\chi_{K}$, then exactly half of the
residues in $(\mathbb{Z}/D_{K}\mathbb{Z})^{\times}$ correspond to inert
primes. Hence, by Theorem 1.7 (2.a) we have

(8.5) $\displaystyle\mathbf{1}_{\frac{\pi}{2}\in
I}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{
inert}\end{subarray}}\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))$
$\displaystyle=\mathbf{1}_{\frac{\pi}{2}\in
I}\Bigg{(}\frac{1}{2}\mu_{\text{ST}}(I^{\prime})\pi(x)+O\bigg{(}\pi(x)\frac{\log(N_{E^{\prime}}\log
x)}{\sqrt{\log x}}\bigg{)}\Bigg{)}.$

Combining (8.3), (8.4), and (8.5) now concludes the proof. ∎

##  9\. Proof of Theorem 1.7 (1.c)

Throughout this section, we will fix the notation from Sections 1, 3, and 4.4.
In particular, we will consider two elliptic curves $E,E^{\prime}$ having
complex multiplication over two imaginary quadratic fields $K,K^{\prime}$
(respectively).

###### Proof of Theorem 1.7 (1.c).

The proof is trivial when $x<N_{E}N_{E^{\prime}}$; hence assume $x\geq
N_{E}N_{E^{\prime}}$. First, recall that by Lemmas 5.3 and 5.6 and applying
partial summation, we have that for all positive integers $m,m^{\prime}$,

(9.1) $\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in
}K\end{subarray}}\cos(m\theta_{p})\ll\bigg{(}\frac{x}{\log
x}\bigg{)}\exp\bigg{(}\frac{-\Cr{44}\log x}{\sqrt{\log
x}+3\log(N_{E}m)}\bigg{)}\log(xN_{E}m)^{4}$

and

(9.2) $\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in
}K\end{subarray}}\cos(m\theta_{E}(p))\cos(m^{\prime}\theta_{E^{\prime}}(p))\ll\bigg{(}\frac{x}{\log
x}\bigg{)}\exp\bigg{(}\frac{-\Cr{471}\log x}{\sqrt{\log
x}+3\log\mathfrak{q}}\bigg{)}(\log x\mathfrak{q})^{4}\bigg{)}.$

Now we consider

$\displaystyle\pi_{I,I^{\prime}}(x)$ $\displaystyle=\sum_{p\leq
x}\mathbf{1}_{I}(\theta_{E}(p))\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))$
(9.3) $\displaystyle=\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in
}K\\\ p\text{ splits in
}K^{\prime}\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))+\mathbf{1}_{\frac{\pi}{2}\in
I^{\prime}}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in }K\\\ p\text{
inert in }K^{\prime}\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))$
$\displaystyle+\mathbf{1}_{\frac{\pi}{2}\in I}\sum_{\begin{subarray}{c}p\leq
x\\\ p\text{ inert in }K\\\ p\text{ splits in
}K^{\prime}\end{subarray}}\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))+\mathbf{1}_{\frac{\pi}{2}\in
I}\mathbf{1}_{\frac{\pi}{2}\in I^{\prime}}\sum_{\begin{subarray}{c}p\leq x\\\
p\text{ inert in }K\\\ p\text{ inert in
}K^{\prime}\end{subarray}}1+O(N_{E}N_{E^{\prime}}).$

Here $O(N_{E}N_{E^{\prime}})$ comes from the contribution of the omitted
primes $p$ which divide $N_{E}N_{E^{\prime}}$. Now, (9.3) splits
$\pi_{I,I^{\prime}}(x)$ into 4 terms. We now simplify the first $3$ terms.

Term 1. By Lemma 3.2, we have

(9.4) $\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in
}K\\\ p\text{ splits in
}K^{\prime}\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))\mathbf{1}_{I^{\prime}}(\theta_{E^{\prime}}(p))$

is equal to

$\displaystyle\frac{|I||I^{\prime}|}{\pi^{2}}\sum_{\begin{subarray}{c}p\leq
x\\\ p\text{ splits in }K\\\ p\text{ splits in }K^{\prime}\end{subarray}}1$
$\displaystyle+O\bigg{(}\frac{\pi(x)}{M}+\sum_{m=1}^{M}\frac{1}{m}\bigg{(}\cos(m\theta_{E}(p))+\cos(m^{\prime}\theta_{E^{\prime}}(p))\bigg{)}$
$\displaystyle+\sum_{1\leq m,m^{\prime}\leq
M}\frac{1}{mm^{\prime}}\cos(m\theta_{E}(p))\cos(m^{\prime}\theta_{E^{\prime}}(p))\bigg{)}.$

By (9.1) and (9.2), the error term above is bounded by

(9.5) $\displaystyle O\bigg{(}\frac{x}{\log x}\bigg{(}\frac{1}{M}+\log
M\exp\bigg{(}\frac{-\Cr{44}\log x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}M}\bigg{)}\log(xN_{E}N_{E^{\prime}}M)^{4}$
$\displaystyle+(\log M)^{2}\exp\bigg{(}\frac{-\Cr{471}\log x}{\sqrt{\log
x}+3\log
N_{E}N_{E^{\prime}}M}\bigg{)}\log(xN_{E}N_{E^{\prime}}M)^{4}\bigg{)}\bigg{)}.$

Hence, there exists an absolute constant $\Cl[abcon]{473}$ such that the term
in (9.5) is bounded by

$\displaystyle O\bigg{(}\frac{x}{\log x}\bigg{(}\frac{1}{M}+(\log
M)^{2}\exp\bigg{(}\frac{-\Cr{473}\log x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}M}\bigg{)}\log(xN_{E}N_{E^{\prime}}M)^{4}\bigg{)}\bigg{)}.$

Taking $M=\exp(\sqrt{\log x})$, we obtain that there exists an absolute
constant $\Cl[abcon]{474}$ such that (9.4) is equal to

$\displaystyle\frac{|I||I^{\prime}|}{\pi^{2}}\bigg{(}\sum_{\begin{subarray}{c}p\leq
x\\\ p\text{ splits in }K\\\ p\text{ splits in
}K^{\prime}\end{subarray}}1\bigg{)}+O\bigg{(}\frac{x}{\log
x}\bigg{(}\exp\bigg{(}\frac{-\Cr{474}\log x}{\sqrt{\log x}+3\log
N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}\bigg{)}.$

Term 2. When $\frac{\pi}{2}$ is not in $I^{\prime}$ or $K\sim K^{\prime}$, the
second term is automatically 0. Hence, assume $\frac{\pi}{2}\in I^{\prime}$
and $K\not\sim K^{\prime}$. In this case,

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in }K\\\
p\text{ inert in }K^{\prime}\end{subarray}}\mathbf{1}_{I}(\theta_{E}(p))$
$\displaystyle=\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in
}K\end{subarray}}\frac{1-\chi_{K^{\prime}}(p)}{2}\mathbf{1}_{I}(\theta_{E}(p)).$

By the proof of Theorem 1.7 (2.b), we have that there exists an absolute
constant $\Cl[abcon]{475}$ for which the above equals

$\displaystyle\frac{|I|}{2\pi}\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\
p\text{ splits in }K\end{subarray}}1\bigg{)}+O\bigg{(}\frac{x}{\log
x}\exp\bigg{(}\frac{-\Cr{475}\log x}{\sqrt{\log x}+\log
N_{E}}\bigg{)}\bigg{)}.$

Here the Siegel zero contribution is omitted as there are only finitely many
possible moduli for $\chi_{K}$.

Now, term 3 can be dealt with in the same way as term 2. Term 4 can be
estimated using (5.2). Collating the above results, we have that when
$K\not\sim K^{\prime}$, there exists an absolute constant $\Cl[abcon]{476}$
such that

$\displaystyle\pi_{I,I^{\prime}}(x)$
$\displaystyle=\frac{|I||I^{\prime}|}{\pi^{2}}\bigg{(}\sum_{\begin{subarray}{c}p\leq
x\\\ p\text{ splits in }K\\\ p\text{ splits in
}K^{\prime}\end{subarray}}1\bigg{)}+\frac{|I|\mathbf{1}_{\frac{\pi}{2}\in
I^{\prime}}}{2\pi}\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits
in }K\end{subarray}}1\bigg{)}$
$\displaystyle+\frac{|I^{\prime}|\mathbf{1}_{\frac{\pi}{2}\in
I}}{2\pi}\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ splits in
}K^{\prime}\end{subarray}}1\bigg{)}+\mathbf{1}_{\frac{\pi}{2}\in
I}\mathbf{1}_{\frac{\pi}{2}\in
I^{\prime}}\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ inert in }K\\\
p\text{ inert in }K^{\prime}\end{subarray}}1\bigg{)}$
$\displaystyle+O\bigg{(}x(\log x)^{4}\exp\bigg{(}\frac{-\Cr{474}\log
x}{\sqrt{\log x}+\log N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}$
$\displaystyle=\mu_{\text{ST}}^{1}(I)\mu_{\text{ST}}^{2}(I^{\prime})\pi(x)+O\bigg{(}x(\log
x)^{4}\exp\bigg{(}\frac{-\Cr{476}\log x}{\sqrt{\log x}+\log
N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}.$

When $K\sim K^{\prime}$, there exists an absolute constant $\Cl[abcon]{477}$
such that

$\displaystyle\pi_{I,I^{\prime}}(x)$
$\displaystyle=\frac{|I||I^{\prime}|}{\pi^{2}}\bigg{(}\sum_{\begin{subarray}{c}p\leq
x\\\ p\text{ splits in }K\sim
K^{\prime}\end{subarray}}1\bigg{)}+\mathbf{1}_{\frac{\pi}{2}\in
I}\mathbf{1}_{\frac{\pi}{2}\in
I^{\prime}}\bigg{(}\sum_{\begin{subarray}{c}p\leq x\\\ p\text{ inert in }K\sim
K^{\prime}\end{subarray}}1\bigg{)}$ $\displaystyle+O\bigg{(}x(\log
x)^{4}\exp\bigg{(}\frac{-\Cr{474}\log x}{\sqrt{\log x}+\log
N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}$
$\displaystyle=\frac{1}{2}\bigg{(}\frac{|I||I^{\prime}|}{\pi^{2}}+\mathbf{1}_{\frac{\pi}{2}\in
I}\mathbf{1}_{\frac{\pi}{2}\in I^{\prime}}\bigg{)}\pi(x)+O\bigg{(}x(\log
x)^{4}\exp\bigg{(}\frac{-\Cr{477}\log x}{\sqrt{\log x}+\log
N_{E}N_{E^{\prime}}}\bigg{)}\bigg{)}.$

Hence both the desired bounds have been proved. ∎

##  Appendix A The Sato–Tate distributions of Double Quadric and K3 Surfaces

Figure 1. Graph of $a_{X_{\lambda}}^{\ast}$, $\lambda=2,p\leq 5\times 10^{5}$.
We overlay this with the “Batman” distribution. Figure 2. Graph of
$a_{X_{\lambda}}^{\ast}$, $\lambda=3,p\leq 2\times 10^{5}$. We overlay this
with the graph of $\frac{1}{2\pi}\sqrt{\frac{3-x}{1+x}}$. Figure 3. Graph of
$a_{X_{\lambda}}^{\ast}$, $\lambda=1,p\leq 5\times 10^{5}$. We overlay this
with the “flying Batman” distribution.

Figure 4. Graph of $a_{X_{\lambda}}^{\ast}$, $\lambda=-4,p\leq 5\times
10^{5}$. We overlay this with the graph of $\frac{1}{2\pi\sqrt{3+2x-x^{2}}}$.
Figure 5. Graph of $a_{X_{\lambda}}^{\ast}$, $\lambda=8,p\leq 5\times 10^{5}$.
We overlay this with the graph of $\frac{1}{2\pi\sqrt{3+2x-x^{2}}}$. Figure 6.
Graph of $a_{\mathcal{Z}}^{\ast}$, where the two elliptic curves are
$x_{1}^{3}+7x_{1}z_{1}^{2}+13z_{1}^{3}$ and
$x_{1}^{3}+7x_{1}z_{1}^{2}+17z_{1}^{3}$, and $p\leq 5\times 10^{5}$. We
overlay this with the graph of $C_{1}$.

Figure 7. Graph of $a_{\mathcal{Z}}^{\ast}$, where the two elliptic curves are
$x_{1}^{3}+7x_{1}z_{1}^{2}+13z_{1}^{3}$ and
$x_{1}^{3}-99x_{1}z_{1}^{2}+378z_{1}^{3}$, and $p\leq 5\times 10^{5}$. We
overlay this with the graph of $C_{2}$. Figure 8. Graph of
$a_{\mathcal{Z}}^{\ast}$, where the two elliptic curves are
$x_{1}^{3}-11x_{1}z_{1}^{2}-14z_{1}^{3}$ and
$x_{1}^{3}-120x_{1}z_{1}^{2}+506z_{1}^{3}$, and $p\leq 2\times 10^{6}$. We
overlay this with the graph of $C_{3}$.

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