††footnotetext: _Key words and phrases:_ Elliptic curves, endomorphism rings,
distribution of primes, sieve methods††footnotetext: _2010 Mathematics Subject
Classification:_ 11G05, 11G20, 11N05 (Primary), 11N36, 11N37, 11N56
(Secondary)

#  Bounds for the distribution of the Frobenius traces associated to products
of non-CM elliptic curves

Alina Carmen Cojocaru and Tian Wang  - Department of Mathematics, Statistics
and Computer Science, University of Illinois at Chicago, 851 S Morgan St, 322
SEO, Chicago, 60607, IL, USA; - Institute of Mathematics “Simion Stoilow” of
the Romanian Academy, 21 Calea Grivitei St, Bucharest, 010702, Sector 1,
Romania<EMAIL_ADDRESS>- Department of Mathematics, Statistics and Computer
Science, University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago,
60607, IL, USA<EMAIL_ADDRESS>
###### Abstract.

Let $g\geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that
is isogenous over $\mathbb{Q}$ to a product of $g$ elliptic curves defined
over $\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and
each without complex multiplication. For an integer $t$ and a positive real
number $x$, denote by $\pi_{A}(x,t)$ the number of primes $p\leq x$, of good
reduction for $A$, for which the Frobenius trace $a_{1,p}(A)$ associated to
the reduction of $A$ modulo $p$ equals $t$. Assuming the Generalized Riemann
Hypothesis for Dedekind zeta functions, we prove that
$\pi_{A}(x,0)\ll_{A}x^{1-\frac{1}{3g+1}}/(\operatorname{log}x)^{1-\frac{2}{3g+1}}$
and
$\pi_{A}(x,t)\ll_{A}x^{1-\frac{1}{3g+2}}/(\operatorname{log}x)^{1-\frac{2}{3g+2}}$
if $t\neq 0$. These bounds largely improve upon recent ones obtained for $g=2$
by H. Chen, N. Jones, and V. Serban, and may be viewed as generalizations to
arbitrary $g$ of the bounds obtained for $g=1$ by M.R. Murty, V.K. Murty, and
N. Saradha, combined with a refinement in the power of $\operatorname{log}x$
by D. Zywina. Under the same assumptions, we also prove the existence of a
density one set of primes $p$ satisfying
$|a_{1,p}(A)|>p^{\frac{1}{3g+1}-\varepsilon}$ for any fixed $\varepsilon>0$.

A.C.C. was partially supported by a Collaboration Grant for Mathematicians
from the Simons Foundation under Award No. 709008.

February 9, 2022

##  1\. Introduction

A prominent open problem in arithmetic geometry, formulated by S. Lang and H.
Trotter in the 1970s ([LaTr76, Part I]), concerns the distribution of the
Frobenius traces associated to the reductions modulo primes of an elliptic
curve defined over $\mathbb{Q}$ and without complex multiplication. In recent
years, this problem has been formulated in broader settings, such as that of
abelian varieties (e.g., [CoDaSiSt17], [ChJoSe20], and [Ka09]). The goal of
the present article is to provide upper bounds related to the distribution of
the Frobenius traces defined by the product of non-isogenous elliptic curves
defined over $\mathbb{Q}$ and having no complex multiplication, as explained
below.

Let $g\geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that
is isogenous over $\mathbb{Q}$ to a product of $g$ elliptic curves defined
over $\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and
each without complex multiplication. Denote by $N_{A}$ the conductor of $A$.
For a prime $p\nmid N_{A}$, we write the characteristic polynomial of the
Frobenius endomorphism acting on the reduction of $A$ modulo $p$ as

$P_{A,p}(X)=X^{2g}+a_{1,p}(A)X^{2g-1}+\ldots+a_{2g-1,p}(A)X+a_{2g,p}(A)\in\mathbb{Z}[X].$

Then, for a fixed integer $t$, we study the counting function

$\pi_{A}(x,t):=\\#\left\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\right\\}.$

When $g=1$, the asymptotic behavior of $\pi_{A}(x,t)$ is predicted by the
Lang-Trotter Conjecture on Frobenius traces (see [LaTr76, Part I, p. 33] for
the original and [BaJo09, Conjectures 2-3, p. 435] for refined versions).
Specifically, in [LaTr76], it is conjectured that there exists an explicit
non-negative constant $C(A,t)$, which depends on the arithmetic of $A$ and
$t$, such that, as $x\rightarrow\infty$,

$\pi_{A}(x,t)\sim C(A,t)\frac{\sqrt{x}}{\operatorname{log}x}.$

When $g=2$ and $t\neq 0$, a similar asymptotic formula is predicted by a
conjecture of H. Chen, N. Jones and V. Serban (see [ChJoSe20, Conjecture
1.0.5, p. 3]). An analogous asymptotic behavior is expected to hold for
arbitrary $t$ and $g$ in our given setting.

When $g=1$, M.R. Murty, V.K. Murty and N. Saradha [MuMuSa88, Theorem, p. 254]
built on the work of J-P. Serre [Se81, Section 8, pp. 188-191] and proved
that, under the assumption of a Generalized Riemann Hypothesis,
$\pi_{A}(x,0)\ll_{A}x^{\frac{3}{4}}$ and
$\displaystyle\pi_{A}(x,t)\ll_{A}\frac{x^{\frac{4}{5}}}{(\operatorname{log}x)^{\frac{1}{5}}}$
if $t\neq 0$. More recently, D. Zywina [Zy15, Theorem 1.2, p. 236] obtained
the improvements
$\displaystyle\pi_{A}(x,0)\ll_{A}\frac{x^{\frac{3}{4}}}{(\operatorname{log}x)^{\frac{1}{2}}}$
and
$\displaystyle\pi_{A}(x,t)\ll_{A}\frac{x^{\frac{4}{5}}}{(\operatorname{log}x)^{\frac{3}{5}}}$
if $t\neq 0$. The bound $\pi_{A}(x,0)\ll_{A}x^{\frac{3}{4}}$ was also proven
unconditionally using the interpretation of the condition $a_{1,p}(A)=0$ as
$p$ being a supersingular prime for the elliptic curve $A$ (see [El91, Theorem
B, p. 131]). When $t\neq 0$, only the bound
$\displaystyle\pi_{A}(x,t)\ll_{A}\frac{x(\operatorname{log}\operatorname{log}x)^{2}}{(\operatorname{log}x)^{2}}$
is known unconditionally, thanks to the work of J. Thorner and A. Zaman
[ThZa17, Theorem 1.4, p. 4997], who built on prior results of J-P. Serre
[Se81, Section 8, pp. 188-191], D. Wan [Wan90, Theorem 1.3, p. 250], and V.K.
Murty [Mu97, Theorem 5.1, p. 302].

When $g=2$ and $t\neq 0$, H. Chen, N. Jones and V. Serban [ChJoSe20, Theorem
2.4.1, p. 14] proved that, under the assumption of a Generalized Riemann
Hypothesis, we have
$\displaystyle\pi_{A}(x,t)\ll_{A}\frac{x^{\frac{13}{14}}}{(\operatorname{log}x)^{\frac{5}{7}}}$;
unconditionally, they proved that
$\displaystyle\pi_{A}(x,t)\ll_{A}\frac{x(\operatorname{log}\operatorname{log}x)^{\frac{2}{7}}(\operatorname{log}\operatorname{log}\operatorname{log}x)^{\frac{1}{7}}}{(\operatorname{log}x)^{\frac{8}{7}}}$.

In this paper, we prove conditional upper bounds for $\pi_{A}(x,t)$ that
largely improve upon the conditional bound of H. Chen, N. Jones, and V. Serban
when $g=2$, and recover the conditional bound of M.R. Murty, V.K. Murty and N.
Saradha and D. Zywina when $g=1$.

######  Theorem 1.

Let $t\in\mathbb{Z}$ and let $A/\mathbb{Q}$ be an abelian variety that is
isogenous over $\mathbb{Q}$ to a product of $g$ elliptic curves defined over
$\mathbb{Q}$, pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each
without complex multiplication. Under the assumption of Generalized Riemann
Hypothesis (GRH) for Dedekind zeta functions, we have that, for any
sufficiently large $x$,

$\pi_{A}(x,t)\ll_{A}\left\\{\begin{array}[]{cl}\displaystyle\frac{x^{1-\frac{1}{3g+1}}}{(\operatorname{log}x)^{1-\frac{2}{3g+1}}}&\text{if
$t=0$,}\\\ \\\
\displaystyle\frac{x^{1-\frac{1}{3g+2}}}{(\operatorname{log}x)^{1-\frac{2}{3g+2}}}&\text{if
$t\neq 0$.}\end{array}\right.$

As an immediate application of Theorem 1, we obtain a result about the non-
lacunarity of the sequence $(a_{1,p}(A))_{p}$, that is, about the non-zero
values of $a_{1,p}(A)$. In fact, we obtain a result about any fixed value of
$a_{1,p}(A)$.

######  Corollary 2.

In the setting and under the assumptions of Theorem 1, we have that, as
$x\rightarrow\infty$,

(1) $\\#\left\\{p\leq x:p\nmid N_{A},a_{1,p}(A)\neq t\right\\}\sim\pi(x).$

As another quick application of Theorem 1, we obtain the existence of a
density one set of primes $p$ with large Frobenius traces $a_{1,p}(A)$.
Namely, in the setting and under the assumptions of Theorem 1, we have that,
for any $\varepsilon>0$ and as $x\rightarrow\infty$,

(2) $\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+2}-\varepsilon}\right\\}\sim\pi(x).$

Without applying Theorem 1 directly, but instead using a proof strategy
similar to that of the proof of Theorem 1, we can improve upon (2) to show the
following.

######  Theorem 3.

Let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$
to a product of $g$ elliptic curves defined over $\mathbb{Q}$, pairwise non-
isogenous over $\overline{\mathbb{Q}}$ and each without complex
multiplication. Under the assumption of Generalized Riemann Hypothesis (GRH)
for Dedekind zeta functions, we have that, for any $\varepsilon>0$ and as
$x\rightarrow\infty$,

$\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+1}-\varepsilon}\right\\}\sim\pi(x).$

The above results may be re-written solely in terms of a $g$-tuple of elliptic
curves $E_{1}/\mathbb{Q},\ldots,E_{g}/\mathbb{Q}$, assumed to be pairwise non-
isogenous over $\overline{\mathbb{Q}}$ and each without complex
multiplication, as follows. Denote by $N_{E_{1}},\ldots,N_{E_{g}}$ the
conductors of $E_{1},\ldots,E_{g}$, respectively. For each integer $1\leq
i\leq g$ and for each prime $p\nmid N_{E_{i}}$, denote by $a_{p}(E_{i})$ the
integer defined by $|\overline{E}_{i}(\mathbb{F}_{p})|=p+1-a_{p}(E_{i})$,
where $\overline{E}_{i}$ is the reduction of $E_{i}$ modulo $p$. Taking
$A:=E_{1}\times\ldots\times E_{g}$, we have that
$a_{1,p}(A)=-(a_{p}(E_{1})+\ldots+a_{p}(E_{g}))$ for any prime $p\nmid
N_{E_{1}}\ldots N_{E_{g}}$ (to be explained in Section 6). Now let
$t\in\mathbb{Z}$. Then equivalent formulations of Theorem 1, Corollary 2, and
Theorem 3 are that, under the assumption of GRH, for any sufficiently large
$x$, we have

$\\#\left\\{p\leq x:p\nmid N_{E_{1}}\ldots
N_{E_{g}},a_{p}(E_{1})+\ldots+a_{p}(E_{g})=t\right\\}\ll_{E_{1},\ldots,E_{g}}\left\\{\begin{array}[]{cl}\displaystyle\frac{x^{1-\frac{1}{3g+1}}}{(\operatorname{log}x)^{1-\frac{2}{3g+1}}}&\text{
{\emph{if}} $t=0$,}\\\ \\\
\displaystyle\frac{x^{1-\frac{1}{3g+2}}}{(\operatorname{log}x)^{1-\frac{2}{3g+2}}}&\text{
{\emph{if}} $t\neq 0$,}\end{array}\right.$

and for any $\varepsilon>0$ and as $x\rightarrow\infty$, we have

$\\#\left\\{p\leq x:p\nmid N_{E_{1}}\ldots
N_{E_{g}},a_{p}(E_{1})+\ldots+a_{p}(E_{g})\neq t\right\\}\sim\pi(x),$

$\\#\left\\{p\leq x:p\nmid N_{E_{1}}\ldots
N_{E_{g}},\left|a_{p}(E_{1})+\ldots+a_{p}(E_{g})\right|>p^{\frac{1}{3g+1}-\varepsilon}\right\\}\sim\pi(x).$

The general strategy of proving an upper bound for $\pi_{A}(x,t)$ is to relax
the equality $a_{1,p}(A)=t$ to a congruence
$\operatorname{tr}\bar{\rho}_{A,m}\left(\left(\frac{\mathbb{Q}(A[m])/\mathbb{Q}}{p}\right)\right)\equiv-t(\operatorname{mod}m)$,
where $m$ is an arbitrary integer coprime to $p$, $\bar{\rho}_{A,m}$ is the
Galois representation associated to the $m$-division field $\mathbb{Q}(A[m])$
of $A$, and $\left(\frac{\mathbb{Q}(A[m])/\mathbb{Q}}{p}\right)$ is the Artin
symbol at $p$ in the Galois group
$\operatorname{Gal}(\mathbb{Q}(A[m])/\mathbb{Q})$. An upper bound for
$\pi_{A}(x,t)$ may then be derived by interpreting this congruence as a
Chebotarev condition in the finite Galois extension
$\mathbb{Q}(A[m])/\mathbb{Q}$ and by invoking an effective version of the
Chebotarev Density Theorem.

Inspired by [MuMuSa88], we pursue a refinement of this general strategy and
relax the equality $a_{1,p}(A)=t$ to a Chebotarev condition that holds in a
suitably chosen proper subextension of $\mathbb{Q}(A[m])/\mathbb{Q}$ in which
Artin’s Holomorphy Conjecture is known to hold. When $g\geq 2$, we unravel a
few such possible subextensions and carry out the refined strategy in the
subextension that leads to the best result, in particular the subextension
that leads to a generalization of the best upper bounds known for $g=1$.

When studying the Frobenius traces of an elliptic curve, it is natural to
place a special focus on elliptic curves $A/\mathbb{Q}$ having the property
that $\operatorname{End}_{\overline{\mathbb{Q}}}(A)\simeq\mathbb{Z}$, since
this case is regarded as generic. When studying the Frobenius traces of an
abelian variety $A$ of higher dimension, it is then natural to place an
initial focus on the following two cases: that of an abelian variety
$A/\mathbb{Q}$ with
$\operatorname{End}_{\overline{\mathbb{Q}}}(A)\simeq\mathbb{Z}$, as pursued in
[CoDaSiSt17] and in an upcoming paper by the present authors, and that of an
abelian variety that is $\mathbb{Q}$-isogenous to a product of $g$ elliptic
curves defined over $\mathbb{Q}$, pairwise non-isogenous over
$\overline{\mathbb{Q}}$ and each without complex multiplication, as pursued in
the present paper. Note that, when $A/\mathbb{Q}$ is $\mathbb{Q}$-isogenous to
a product of $g$ elliptic curves defined over $\mathbb{Q}$ that are pairwise
isogenous over $\overline{\mathbb{Q}}$, the study of $\pi_{A}(x,t)$ for $g\geq
2$ reduces to the study of $\pi_{A}(x,t)$ for $g=1$. Other complementary cases
remain to be pursued separately.

Finally, let us note that other variations of questions inspired by the Lang-
Trotter Conjecture have been investigated in the setting of pairs of elliptic
curves in works such as [FoMu95], [AkDaJu04], and [AkPa19].

General notation. Throughout the paper, we use the following notation.

$\bullet$ For a set $S$, we denote its cardinality by $|S|$ or $\\#S$.

$\bullet$ Given suitably defined real functions $h_{1},h_{2}$, we say that
$h_{1}=\operatorname{o}(h_{2})$ if
$\displaystyle\lim_{x\rightarrow\infty}\frac{h_{1}(x)}{h_{2}(x)}=0$; we say
that $h_{1}=\operatorname{O}(h_{2})$ or, equivalently, that $h_{1}\ll h_{2}$,
if $h_{2}$ is positive valued and there exists a positive constant $c$ such
that $|h_{1}(x)|\leq c\ h_{2}(x)$ for all $x$ in the domain of $h_{1}$ and
$h_{2}$; we say that $h_{1}\asymp h_{2}$ if $h_{1}$, $h_{2}$ are positive
valued and $h_{1}\ll h_{2}\ll h_{1}$; we say that
$h_{1}=\operatorname{O}_{D}(h_{2})$ or, equivalently, that
$h_{1}\ll_{D}h_{2}$, if $h_{1}=\operatorname{O}(h_{2})$ and the implied
$\operatorname{O}$-constant $c$ depends on priorly given data $D$; we say that
$h_{1}\asymp_{D}h_{2}$ if the implied constant $c$ in at least one of the
$\ll$-bounds $h_{1}\ll h_{2}\ll h_{1}$ depends on priorly given data $D$;
finally, we say that $h_{1}\sim h_{2}$ if
$\displaystyle\lim_{x\rightarrow\infty}\frac{h_{1}(x)}{h_{2}(x)}=1$.

$\bullet$ We use the letters $p$ and $\ell$ to denote rational primes. We use
$\pi(x)$ to denote the number of primes $p\leq x$, and $\operatorname{li}x$ to
denote the logarithmic integral
$\displaystyle\int_{2}^{x}\frac{1}{\operatorname{log}t}\ dt$. Note that
$\displaystyle\operatorname{li}x\sim\frac{x}{\operatorname{log}x}$ and recall
that the Prime Number Theorem asserts that $\pi(x)\sim\operatorname{li}x$.

$\bullet$ Given a number field $K$, we denote by ${\mathcal{O}}_{K}$ its ring
of integers; we denote by $\overline{K}$ a fixed algebraic closure of $K$; we
denote by $\operatorname{Gal}(\overline{K}/K)$ the absolute Galois group of
$K$; we denote by ${\sum}_{K}$ the set of non-zero prime ideals of $K$; we
denote by $[K:\mathbb{Q}]$ the degree of $K$ over $\mathbb{Q}$; we denote by
$d_{K}\in\mathbb{Z}$ the discriminant of an integral basis of
${\mathcal{O}}_{K}$ and by
$\operatorname{disc}(K/\mathbb{Q})=\mathbb{Z}d_{K}\unlhd\mathbb{Z}$ the
discriminant ideal of $K/\mathbb{Q}$. For a prime ideal
$\mathfrak{p}\in{\sum}_{K}$, we denote by $N_{K/\mathbb{Q}}(\mathfrak{p})$ its
norm in $K/\mathbb{Q}$ and by
$\operatorname{Frob}_{\mathfrak{p}}\in\operatorname{Gal}(\overline{K}/K)$ its
Frobenius class. We say that $K$ satisfies the Generalized Riemann Hypothesis
(GRH) if the Dedekind zeta function $\zeta_{K}$ of $K$ has the property that,
for any $\rho\in\mathbb{C}$ with $0\leq\operatorname{Re}\rho\leq 1$ and
$\zeta_{K}(\rho)=0$, we have $\operatorname{Re}(\rho)=\frac{1}{2}$. When
$K=\mathbb{Q}$, the Dedekind zeta function is the Riemann zeta function, in
which case GRH is typically referred to as RH.

$\bullet$ For a nonzero unitary commutative ring $R$, we denote by
$R^{\times}$ its group of multiplicative units. For an integer $n\geq 1$, we
denote by $I_{n}$ the identity $n\times n$ matrix with entries in $R$. For an
arbitrary $n\times n$ matrix $M$ with entries in $R$, we denote by
$\operatorname{tr}M$ and $\operatorname{det}M$ its trace and determinant. We
define the general linear group $\operatorname{GL}_{n}(R)$ as the collection
of $n\times n$ matrices $M$ with entries in $R$ and with
$\operatorname{det}M\in R^{\times}$.

$\bullet$ For a positive integer $m$, we denote by $\mathbb{Z}/m\mathbb{Z}$
the ring of integers modulo $m$. For an arbitrary prime $\ell$, we denote by
$\mathbb{Z}_{\ell}$ the ring of $\ell$-adic integers. We set
$\hat{\mathbb{Z}}:=\displaystyle\lim_{\leftarrow\atop
m}\mathbb{Z}/m\mathbb{Z}$ and note that there is a ring isomorphism
$\hat{\mathbb{Z}}\simeq\displaystyle\prod_{\ell}\mathbb{Z}_{\ell}$.

Acknowledgments. We thank Jacob Mayle for a careful reading of our manuscript
and for useful remarks. We are deeply grateful to the referees for their
detailed comments and suggestions.

##  2\. The Chebotarev Density Theorem

The proofs of Theorems 1 and 3 are based on multi-step applications of
different effective versions of the Chebotarev Density Theorem, which we now
recall.

Let $L/K$ be a Galois extension of number fields, with
$G:=\operatorname{Gal}(L/K)$, and let ${\mathcal{C}}\subseteq G$ be a union of
conjugacy classes of $G$; throughout the paper, we will assume
${\mathcal{C}}\neq\emptyset$. We denote by $[L:K]$ the degree of $L$ over $K$
and by $\operatorname{disc}(L/K)\unlhd{\mathcal{O}}_{K}$ the discriminant
ideal of $L/K$. For each $\mathfrak{p}\in{\sum}_{K}$, we denote by
$\wp\in{\sum}_{L}$ an arbitrary prime ideal with $\wp\mid\mathfrak{p}$, by
${\mathcal{D}}_{\wp}$ the decomposition group of $\wp$ in $L/K$, and by
${\mathcal{I}}_{\wp}$ the inertia group of $\wp$ in $L/K$. We denote by
$\left(\frac{L/K}{\wp}\right)\in{\mathcal{D}}_{\wp}/{\mathcal{I}}_{\wp}$ the
Artin symbol associated to $\wp$, and by
$\left(\frac{L/K}{\mathfrak{p}}\right):=\left\\{\gamma\in
G:\exists\wp\mid\mathfrak{p}\text{ such that }\gamma\in{\mathcal{D}}_{\wp}\
\text{and}\ \gamma{\mathcal{I}}_{\wp}=\left(\frac{L/K}{\wp}\right)\right\\}$
the Artin symbol associated to $\mathfrak{p}$ in the extension $L/K$. Notice
that, for any integer $m\geq 1$, the set
$\left(\frac{L/K}{\mathfrak{p}}\right)^{m}:=\left\\{\gamma^{m}:\gamma\in\left(\frac{L/K}{\mathfrak{p}}\right)\right\\}\subseteq
G$ is invariant under conjugation by elements of $G$. We set

${\mathcal{P}}(L/K):=\\{p:\>\ \exists\ \mathfrak{p}\in{\sum}_{K}\ \text{such
that}\ \mathfrak{p}\mid p\ \text{and}\
\mathfrak{p}\mid\operatorname{disc}(L/K)\\},$
$M(L/K):=2[L:K]|d_{K}|^{\frac{1}{[K:\mathbb{Q}]}}\displaystyle\prod_{p\in{\mathcal{P}}(L/K)}p,$

and recall that

(3)
$\operatorname{log}\left|N_{K/\mathbb{Q}}(\operatorname{disc}(L/K))\right|\leq([L:\mathbb{Q}]-[K:\mathbb{Q}])\left(\displaystyle\sum_{p\in{\mathcal{P}}(L/K)}\operatorname{log}p\right)+[L:\mathbb{Q}]\operatorname{log}[L:K]$

(see [Se81, Proposition 5, p. 129]).

The extension $L/K$ is said to satisfy the Artin Holomorphy Conjecture (AHC)
if, for any irreducible character $\chi:G\longrightarrow\mathbb{C}$, the Artin
$L$-function $L(s,\chi)$ extends to a function that is analytic on
$\mathbb{C}$ when $\chi\neq 1$ and that is analytic on
$\mathbb{C}\backslash\\{1\\}$ when $\chi=1$. It is known that AHC holds if
$\operatorname{Gal}(L/K)$ is abelian ([Ar37]).

We denote by $\delta_{\mathcal{C}}:G\longrightarrow\\{0,1\\}$ the
characteristic function on ${\mathcal{C}}$. For any prime ideal
$\mathfrak{p}\in{\sum}_{K}$, we choose an arbitrary prime ideal
$\wp\in{\sum}_{L}$ with $\wp\mid\mathfrak{p}$, and then, for any integer
$m\geq 1$, we define

$\delta_{\mathcal{C}}\left(\left(\frac{L/K}{\mathfrak{p}}\right)^{m}\right):=\frac{1}{|{\mathcal{I}}_{\wp}|}\displaystyle\sum_{\gamma\in{\mathcal{D}}_{\wp}\atop{\gamma{\mathcal{I}}_{\wp}=\left(\frac{L/K}{\wp}\right)^{m}}}\delta_{\mathcal{C}}(\gamma).$

Note that this definition is independent of the choice of $\wp$. For an
arbitrary real number $x>0$, we define

$\displaystyle\pi_{\mathcal{C}}(x,L/K)$ $\displaystyle:=$
$\displaystyle\displaystyle\sum_{\mathfrak{p}\in{\sum}_{K}\atop{\mathfrak{p}\nmid\operatorname{disc}(L/K)\atop{N_{K/\mathbb{Q}}(\mathfrak{p})\leq
x}}}\delta_{\mathcal{C}}\left(\left(\frac{L/K}{\mathfrak{p}}\right)\right),$
$\displaystyle\widetilde{\pi}_{\mathcal{C}}(x,L/K)$ $\displaystyle:=$
$\displaystyle\displaystyle\sum_{m\geq
1}\frac{1}{m}\displaystyle\sum_{\mathfrak{p}\in{\sum}_{K}\atop{N_{K/\mathbb{Q}}(\mathfrak{p}^{m})\leq
x}}\delta_{\mathcal{C}}\left(\left(\frac{L/K}{\mathfrak{p}}\right)^{m}\right).$

For simplicity of notation, we use $\pi_{1}(x,L/K)$ and
$\widetilde{\pi}_{1}(x,L/K)$ to denote
$\pi_{\\{\operatorname{id}_{L}\\}}(x,L/K)$ and
$\widetilde{\pi}_{\\{\operatorname{id}_{L}\\}}(x,L/K)$, respectively.

The Chebotarev Density Theorem states that, as $x\rightarrow\infty$,

(4)
$\pi_{\mathcal{C}}(x,L/K)\sim\frac{|{\mathcal{C}}|}{|G|}\operatorname{li}x.$

Since the growth in $x$ of the counting function
$\widetilde{\pi}_{\mathcal{C}}(x,L/K)$ is closely related to that of
$\pi_{\mathcal{C}}(x,L/K)$ thanks to the bound

(5)
$\left|\widetilde{\pi}_{\mathcal{C}}(x,L/K)-\pi_{\mathcal{C}}(x,L/K)\right|\ll[K:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(L/K)\right)$

(see [Se81, Proposition 7, p. 138]), asymptotic (4) is equivalent to

(6)
$\widetilde{\pi}_{\mathcal{C}}(x,L/K)\sim\frac{|{\mathcal{C}}|}{|G|}\operatorname{li}x.$

The advantage of considering $\widetilde{\pi}_{\mathcal{C}}(x,L/K)$ over
$\pi_{\mathcal{C}}(x,L/K)$ is that the former satisfies the following two
functorial properties.

######  Proposition 4.

Let $L/K$ be a Galois extension of number fields with
$G:=\operatorname{Gal}(L/K)$ and let ${\mathcal{C}}\subseteq G$ be a union of
conjugacy classes of $G$.

  1. (i)

Let $H\leq G$ be a subgroup of $G$ such that every element of ${\mathcal{C}}$
is conjugate to some element of $H$. Denote by $L^{H}$ the subfield of $L$
fixed by $H$. Then, for any $x>0$,

$\widetilde{\pi}_{\mathcal{C}}(x,L/K)\leq\widetilde{\pi}_{{\mathcal{C}}\cap
H}(x,L/L^{H}).$

  2. (ii)

Let $N\unlhd G$ be a normal subgroup of $G$ such that
$N{\mathcal{C}}\subseteq{\mathcal{C}}$. Denote by $L^{N}$ the subfield of $L$
fixed by $N$. Denote by $\widehat{\mathcal{C}}$ the image of ${\mathcal{C}}$
in the quotient group $G/N$, viewed as $\operatorname{Gal}(L^{N}/K)$. Then,
for any $x>0$,

$\widetilde{\pi}_{\mathcal{C}}(x,L/K)=\widetilde{\pi}_{\widehat{\mathcal{C}}}(x,L^{N}/K).$

###### Proof.

See [Se81, Proposition 8, pp. 139–140]. ∎

######  Corollary 5.

Let $L/K$ be a Galois extension of number fields with
$G:=\operatorname{Gal}(L/K)$, and let ${\mathcal{C}}\subseteq G$ be a union of
conjugacy classes of $G$. Let $H\leq G$ be a subgroup of $G$ such that every
element of ${\mathcal{C}}$ is conjugate to some element of $H$. Let $N\unlhd
H$ be a normal subgroup of $H$ such that $N({\mathcal{C}}\cap
H)\subseteq{\mathcal{C}}\cap H$. Denote by $\widehat{\mathcal{C}\cap H}$ the
image of ${\mathcal{C}}\cap H$ in the quotient group $H/N$, viewed as
$\operatorname{Gal}(L^{N}/L^{H})$. Then, for any $x>0$,

$\widetilde{\pi}_{\mathcal{C}}(x,L/K)\leq\widetilde{\pi}_{\widehat{\mathcal{C}\cap
H}}\left(x,L^{N}/L^{H}\right).$

Consequently, for any sufficiently large $x$,

(7) $\displaystyle\pi_{\mathcal{C}}(x,L/K)$ $\displaystyle\ll$
$\displaystyle\pi_{\widehat{\mathcal{C}\cap H}}\left(x,L^{N}/L^{H}\right)$
$\displaystyle+[K:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(L/K)\right)+[L^{H}:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(L^{N}/L^{H})\right).$

###### Proof.

The corollary follows by combining parts (i) and (ii) of Proposition 4 with
(5). ∎

In spite of its strength, the asymptotic formula (4) for
$\pi_{{\mathcal{C}}}(x,L/K)$ does not suffice for our purposes. Instead, the
proofs of our main results will require an effective version with an explicit
error term, such as the following conditional result.

######  Theorem 6.

Let $L/K$ be a Galois extension of number fields, with
$G:=\operatorname{Gal}(L/K)$, and let ${\mathcal{C}}\subseteq G$ be a union of
conjugacy classes of $G$. Assume that GRH holds for the number field $L$. Then
there exists an absolute constant $c>0$ such that, for any $x>2$,

$\displaystyle\left|\pi_{\mathcal{C}}(x,L/K)-\frac{|\mathcal{C}|}{|G|}\pi(x)\right|\leq
c\frac{|\mathcal{C}|}{|G|}x^{\frac{1}{2}}\left(\operatorname{log}|d_{L}|+[L:\mathbb{Q}]\operatorname{log}x\right).$

###### Proof.

The original reference is [LaOd77]. For this variation, see [Se81, Théorème 4,
p. 133]. ∎

The proofs of our main results will also use the following conditional upper
bound for $\pi_{\mathcal{C}}(x,L/K)$.

######  Theorem 7.

Let $L/K$ be a Galois extension of number fields with
$G:=\operatorname{Gal}(L/K)$, and let ${\mathcal{C}}\subseteq G$ be a union of
conjugacy classes of $G$. Assume that GRH holds for the number field $L$ and
that AHC holds for the number field extension $L/K$. Then, for any
sufficiently large $x$,

$\pi_{\mathcal{C}}(x,L/K)\ll\frac{|\mathcal{C}|}{|G|}\cdot\frac{x}{\operatorname{log}x}+|\mathcal{C}|^{\frac{1}{2}}[K:\mathbb{Q}]\left(\operatorname{log}M(L/K)\right)\cdot\frac{x^{\frac{1}{2}}}{\operatorname{log}x}.$

###### Proof.

See [Zy15, Theorem 2.3, p. 240]. ∎

######  Corollary 8.

Let $L/K$ be a Galois extension of number fields with
$G:=\operatorname{Gal}(L/K)$, and let ${\mathcal{C}}\subseteq G$ be a union of
conjugacy classes of $G$. Let $H\leq G$ be a subgroup of $G$ such that every
element of ${\mathcal{C}}$ is conjugate to some element of $H$. Let $N\unlhd
H$ be a normal subgroup of $H$ such that $N({\mathcal{C}}\cap
H)\subseteq{\mathcal{C}}\cap H$. Denote by $\widehat{\mathcal{C}\cap H}$ the
image of ${\mathcal{C}}\cap H$ in the quotient group $H/N$, viewed as
$\operatorname{Gal}(L^{N}/L^{H})$. Assume that the group $H/N$ is abelian and
that GRH holds for $L^{N}$. Then, for any sufficiently large $x$,

$\displaystyle\pi_{\mathcal{C}}(x,L/K)$ $\displaystyle\ll$
$\displaystyle\frac{\left|\widehat{\mathcal{C}\cap
H}\right|\cdot|N|}{|H|}\cdot\frac{x}{\operatorname{log}x}$ $\displaystyle+$
$\displaystyle\left|\widehat{{\mathcal{C}}\cap
H}\right|^{\frac{1}{2}}[L^{H}:\mathbb{Q}]\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}M(L^{N}/L^{H})$
$\displaystyle+$
$\displaystyle[K:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(L/K)\right)+[L^{H}:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(L^{N}/L^{H})\right).$

###### Proof.

First use (7), and then Theorem 7 applied to the extension $L^{N}/L^{H}$. ∎

##  3\. An application of the effective version of the Chebotarev Density
Theorem

The proofs of Theorems 1 and 3 will make use of the following application of
Theorem 6.

######  Lemma 9.

Let ${\mathcal{S}}$ be a non-empty set of rational primes, let
$(K_{p})_{p\in{\mathcal{S}}}$ be a family of finite Galois extensions of
$\mathbb{Q}$, and let $(\mathcal{C}_{p})_{p\in\mathcal{S}}$ be a family of
non-empty sets such that each $\mathcal{C}_{p}$ is a union of conjugacy
classes of $\operatorname{Gal}(K_{p}/\mathbb{Q})$. Assume that there exist an
absolute constant $c_{1}>0$ and a function $f:\mathbb{R}\to(0,\infty)$ such
that

(8) $\displaystyle[K_{p}:\mathbb{Q}]\leq c_{1}\quad\text{for any sufficiently
large}\ p,$

(9) $\displaystyle\operatorname{log}|d_{K_{p}}|\leq f(z)\quad\text{for any
sufficiently large }z\text{ and all }p\leq z.$

For each $x>2$, let $y=y(x)>2$, $u=u(x)>2$ be such that

(10) $u\leq y,$

and assume that, for any $\varepsilon>0$,

(11) $u\geq
c_{2}(\varepsilon)y^{\frac{1}{2}}(\operatorname{log}y)^{2+\varepsilon}\
\text{for some constant}\ c_{2}(\varepsilon)>0$

and

(12)
$\displaystyle\lim_{x\rightarrow\infty}\frac{f(x)}{(\operatorname{log}y)^{1+\varepsilon}}=0.$

Assume GRH for Dedekind zeta functions. Then, for any $\varepsilon>0$, there
exists a constant $c(\varepsilon)>0$ such that, for any sufficiently large
$x$,

(13) $\\#\left\\{p\leq x:p\in{\mathcal{S}}\right\\}\leq
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq y+u}\\#\left\\{p\leq
x:p\in{\mathcal{S}},\ell\nmid
d_{K_{p}},\left(\frac{K_{p}/\mathbb{Q}}{\ell}\right)\subseteq\mathcal{C}_{p}\right\\}.$

###### Proof.

The proof follows the strategy of [MuMuSa88, Lemma 4.4, p. 269], adapted to
our general setting. We start by fixing $x>2$, $y=y(x)>2$, $u=u(x)>2$, and
$\varepsilon>0$ such that (10) - (12) hold. Then, we observe that

$\displaystyle(\pi(y+u)-\pi(y))\displaystyle\max_{y\leq\ell\leq
y+u}\\#\left\\{p\leq x:p\in{\mathcal{S}},\ell\nmid
d_{K_{p}},\left(\frac{K_{p}/\mathbb{Q}}{\ell}\right)\subseteq\mathcal{C}_{p}\right\\}$
$\displaystyle\geq$ $\displaystyle\displaystyle\sum_{y\leq\ell\leq
y+u}\\#\left\\{p\leq x:p\in{\mathcal{S}},\ell\nmid
d_{K_{p}},\left(\frac{K_{p}/\mathbb{Q}}{\ell}\right)\subseteq\mathcal{C}_{p}\right\\}$
$\displaystyle=$ $\displaystyle\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid
d_{K_{p}},\left(\frac{K_{p}/\mathbb{Q}}{\ell}\right)\subseteq\mathcal{C}_{p}\right\\}$
$\displaystyle=$ $\displaystyle\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\left(\pi_{\mathcal{C}_{p}}(y+u,K_{p}/\mathbb{Q})-\pi_{\mathcal{C}_{p}}(y,K_{p}/\mathbb{Q})\right).$

To estimate the above difference, we apply Theorem 6 under GRH and deduce that

$\displaystyle\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\left(\pi_{\mathcal{C}_{p}}(y+u,K_{p}/\mathbb{Q})-\pi_{\mathcal{C}_{p}}(y,K_{p}/\mathbb{Q})\right)$
$\displaystyle=$ $\displaystyle\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\frac{|\mathcal{C}_{p}|}{[K_{p}:\mathbb{Q}]}(\pi(y+u)-\pi(y))+\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\left(E_{1}(y,u,\mathcal{C}_{p},K_{p})+E_{2}(y,u,\mathcal{C}_{p},K_{p})\right)$

for some real-valued functions $E_{1}(y,u,\mathcal{C}_{p},K_{p})$,
$E_{2}(y,u,\mathcal{C}_{p},K_{p})$, which depend on $y$, $u$,
$\mathcal{C}_{p}$, and $K_{p}$, and for which there exist absolute constants
$c_{3}>0$ and $c_{4}>0$ such that

$|E_{1}(y,u,\mathcal{C}_{p},K_{p})|\leq
c_{3}(y+u)^{\frac{1}{2}}\left(\frac{|\mathcal{C}_{p}|\operatorname{log}|d_{K_{p}}|}{[K_{p}:\mathbb{Q}]}+|\mathcal{C}_{p}|\operatorname{log}(y+u)\right),$

$|E_{2}(y,u,\mathcal{C}_{p},K_{p})|\leq
c_{4}y^{\frac{1}{2}}\left(\frac{|\mathcal{C}_{p}|\operatorname{log}|d_{K_{p}}|}{[K_{p}:\mathbb{Q}]}+|\mathcal{C}_{p}|\operatorname{log}y\right).$

Recalling assumptions (8) - (9), we obtain the upper bound

(16) $\displaystyle\left|\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\left(E_{1}(y,u,\mathcal{C}_{p},K_{p})+E_{2}(y,u,\mathcal{C}_{p},K_{p})\right)\right|$
$\displaystyle\leq$ $\displaystyle
c_{5}(y+u)^{\frac{1}{2}}(f(x)+\operatorname{log}(y+u))\ \\#\left\\{p\leq
x:p\in{\mathcal{S}}\right\\}$

for some absolute constant $c_{5}>0$.

Thanks to (11) and the assumption of GRH (specifically, RH in this case),
there exists some constant $c_{6}(\varepsilon)>0$ such that

(17) $\pi(y+u)-\pi(y)\geq c_{6}(\varepsilon)\frac{u}{\operatorname{log}u}.$

In particular, $\pi(y+u)-\pi(y)>0$.

Putting together (3) - (17) and dividing the resulting inequality by
$\pi(y+u)-\pi(y)$, we obtain that

(18) $\displaystyle\displaystyle\max_{y\leq\ell\leq y+u}\\#\left\\{p\leq
x:p\in{\mathcal{S}},\ell\nmid
d_{K_{p}},\left(\frac{K_{p}/\mathbb{Q}}{\ell}\right)\subseteq\mathcal{C}_{p}\right\\}$
$\displaystyle\geq$ $\displaystyle\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\frac{|\mathcal{C}_{p}|}{[K_{p}:\mathbb{Q}]}+\operatorname{O}_{\varepsilon}\left(\frac{\operatorname{log}u}{u}(y+u)^{\frac{1}{2}}(f(x)+\operatorname{log}(y+u))\
\\#\\{p\leq x:p\in{\mathcal{S}}\\}\right).$

Invoking assumptions (10) - (11), we deduce that the above
$\operatorname{O}_{\varepsilon}$-term becomes

$\operatorname{O}_{\varepsilon}\left(\frac{f(x)+\operatorname{log}y}{(\operatorname{log}y)^{1+\varepsilon}}\
\\#\\{p\leq x:p\in{\mathcal{S}}\\}\right).$

Invoking assumption (12), we deduce that the latter
$\operatorname{O}_{\varepsilon}$-term is $\operatorname{o}(\\#\\{p\leq
x:p\in{\mathcal{S}}\\})$. Finally, recalling (8) and that each
$\mathcal{C}_{p}$ is non-empty, we deduce that

$\displaystyle\sum_{p\leq
x\atop{p\in{\mathcal{S}}}}\frac{|\mathcal{C}_{p}|}{[K_{p}:\mathbb{Q}]}\geq\frac{1}{c_{1}}\
\\#\\{p\leq x:p\in{\mathcal{S}}\\}.$

Then (18) gives

$\displaystyle\max_{y\leq\ell\leq y+u}\\#\left\\{p\leq
x:p\in{\mathcal{S}},\ell\nmid
d_{K_{p}},\left(\frac{K_{p}/\mathbb{Q}}{\ell}\right)\subseteq\mathcal{C}_{p}\right\\}\gg_{\varepsilon}\\#\\{p\leq
x:p\in{\mathcal{S}}\\},$

as desired. ∎

######  Remark 10.

_In the proof of Lemma 9, instead of hypotheses (8) - (9), it suffices to make
the weaker hypotheses that there exist absolute constants
$c^{\prime}_{1},c^{\prime}_{2},c^{\prime}_{3}>0$ such that, for any
sufficiently large $z$,_

$\displaystyle\sum_{p\leq
z\atop{p\in{\mathcal{S}}}}\frac{\left|\mathcal{C}_{p}\right|}{[K_{p}:\mathbb{Q}]}\geq
c^{\prime}_{1}\ \\#\\{p\leq z:p\in{\mathcal{S}}\\},$ $\displaystyle\sum_{p\leq
z\atop{p\in{\mathcal{S}}}}\frac{\left|\mathcal{C}_{p}\right|\operatorname{log}|d_{K_{p}}|}{[K_{p}:\mathbb{Q}]}\leq
c^{\prime}_{2}f(z)\ \\#\\{p\leq z:p\in{\mathcal{S}}\\},$
$\displaystyle\sum_{p\leq
z\atop{p\in{\mathcal{S}}}}\left|\mathcal{C}_{p}\right|\leq c^{\prime}_{3}\
\\#\\{p\leq z:p\in{\mathcal{S}}\\}.$

_._

##  4\. Subgroups of $\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})^{g}$

Our main contribution to generalizing the method of M.R. Murty, V.K. Murty,
and N. Saradha [MuMuSa88] to the case of a product of $g$ non-isogenous
elliptic curves without complex multiplication consists of unravelling
suitable Galois extensions $L/K$ of number fields with
$\operatorname{Gal}(L/K)<\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})^{g}$
for some well-chosen prime $\ell$, and of unravelling suitable conjugacy
classes $\mathcal{C}\subseteq\operatorname{Gal}(L/K)$ for which we can invoke
the version of the Chebotarev Density Theorem given in Corollary 8. With this
goal in mind, we devote Section 4 to investigations of particular subgroups of
$\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})^{g}$.

For a fixed rational prime $\ell$, our main focus will be on the subgroup

(19)
$G(\ell):=\\{(M_{1},\ldots,M_{g})\in\mathbb{G}(\ell):\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}\\}$

of the product group

$\mathbb{G}(\ell):=\operatorname{GL}_{2}(\ell)^{g},$

where

$\operatorname{GL}_{2}(\ell):=\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z}).$

Denoting by ${\mathcal{G}}_{m}(\ell)$ the multiplicative group
$(\mathbb{Z}/\ell\mathbb{Z})^{\times}$ and by ${\mathcal{G}}_{a}(\ell)$ the
additive group $\mathbb{Z}/\ell\mathbb{Z}$, we set

$\mathcal{B}_{\operatorname{GL}_{2}}(\ell):=\left\\{\begin{pmatrix}a_{1}&b_{1}\\\
0&a_{2}\end{pmatrix}\in\operatorname{GL}_{2}(\ell):a_{1},a_{2}\in\mathcal{G}_{m}(\ell),b_{1}\in\mathcal{G}_{a}(\ell)\right\\},$

$\mathcal{U}_{\operatorname{GL}_{2}}(\ell):=\left\\{\begin{pmatrix}1&b\\\
0&1\end{pmatrix}\in\operatorname{GL}_{2}(\ell):b\in\mathcal{G}_{a}(\ell)\right\\},$

$\mathcal{T}_{\operatorname{GL}_{2}}(\ell):=\left\\{\begin{pmatrix}a_{1}&0\\\
0&a_{2}\end{pmatrix}\in\operatorname{GL}_{2}(\ell):a_{1},a_{2}\in\mathcal{G}_{m}(\ell)\right\\},$

$B(\ell):=\\{(M_{1},\ldots,M_{g})\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}:\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}\\},$

$U(\ell):=\mathcal{U}_{\operatorname{GL}_{2}}(\ell)^{g},$

$U^{\prime}(\ell):=(\mathbb{Z}/\ell\mathbb{Z})^{\times}\cdot U(\ell),$

$T(\ell):=\\{(M_{1},\ldots,M_{g})\in\mathcal{T}_{\operatorname{GL}_{2}}(\ell)^{g}:\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}\\}.$

The following lemmas will be crucial in the proof of our main theorems.

######  Lemma 11.

For any prime $\ell$, we have that $U(\ell)$ and $U^{\prime}(\ell)$ are normal
subgroups of $B(\ell)$, and that $B(\ell)/U(\ell)$ are
$B(\ell)/U^{\prime}(\ell)$ are abelian groups.

###### Proof.

The subgroup $\mathcal{U}_{\operatorname{GL}_{2}}(\ell)$ is normal in
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)$, and the quotient
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)/\mathcal{U}_{\operatorname{GL}_{2}}(\ell)$
is abelian, since it satisfies a group isomorphism
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)/\mathcal{U}_{\operatorname{GL}_{2}}(\ell)\simeq\mathcal{T}_{\operatorname{GL}_{2}}(\ell)$.
Consequently, the subgroup $U(\ell)$ is normal in
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}$, with
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}/U(\ell)$ an abelian group. By
the inclusion of subgroups $U(\ell)\subseteq U^{\prime}(\ell)\subseteq
B(\ell)\subseteq\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g},$ we deduce that
both $U(\ell)$ and $U^{\prime}(\ell)$ are normal subgroups in $B(\ell)$ and
that both $B(\ell)/U(\ell)$ and $B(\ell)/U^{\prime}(\ell)$ are abelian. ∎

######  Lemma 12.

For any prime $\ell$, we have

$\displaystyle|\operatorname{GL}_{2}(\ell)|$ $\displaystyle=$
$\displaystyle(\ell-1)\ell(\ell^{2}-1),$ $\displaystyle|\mathbb{G}(\ell)|$
$\displaystyle=$ $\displaystyle(\ell-1)^{g}\ell^{g}(\ell^{2}-1)^{g},$
$\displaystyle|\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}|$
$\displaystyle=$ $\displaystyle(\ell-1)^{2g}\ell^{g},$
$\displaystyle|G(\ell)|$ $\displaystyle=$
$\displaystyle(\ell-1)\ell^{g}(\ell^{2}-1)^{g},$ $\displaystyle|B(\ell)|$
$\displaystyle=$ $\displaystyle(\ell-1)^{g+1}\ell^{g},$
$\displaystyle|U(\ell)|$ $\displaystyle=$ $\displaystyle\ell^{g},$
$\displaystyle\left|U^{\prime}(\ell)\right|$ $\displaystyle=$
$\displaystyle(\ell-1)\ell^{g},$ $\displaystyle|T(\ell)|$ $\displaystyle=$
$\displaystyle(\ell-1)^{g+1}.$

###### Proof.

The formula for the order of $\operatorname{GL}_{2}(\ell)$ is well-known and
easy to derive. The formulae for the orders of $\mathbb{G}(\ell)$,
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}$, $U(\ell)$,
$U^{\prime}(\ell)$, and $T(\ell)$ are clear from the definitions of the
groups. To find the orders of $G(\ell)$ and $B(\ell)$, observe that there are
short exact sequences of finite groups

$1\to
G(\ell)\to\mathbb{G}(\ell)\xrightarrow{\operatorname{det}_{1,\ldots,g-1}}\mathcal{G}_{m}(\ell)^{g-1}\to
1,$ $1\to
B(\ell)\to\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}\xrightarrow{\operatorname{det}_{1,\ldots,g-1}}\mathcal{G}_{m}(\ell)^{g-1}\to
1,$

where
$\operatorname{det}_{1,\ldots,g-1}(M_{1},\ldots,M_{g}):=((\operatorname{det}M_{1})(\operatorname{det}M_{2})^{-1},\ldots,(\operatorname{det}M_{1})(\operatorname{det}M_{g})^{-1})$.
We deduce that

$|G(\ell)|=|\mathbb{G}(\ell)|/|\mathcal{G}_{m}(\ell)|^{g-1}=(\ell-1)\ell^{g}(\ell^{2}-1)^{g},$
$|B(\ell)|=|\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}|/|\mathcal{G}_{m}(\ell)|^{g-1}=(\ell-1)^{g+1}\ell^{g}.$

∎

######  Lemma 13.

For any prime $\ell$, we have a group isomorphism $B(\ell)/U(\ell)\simeq
T(\ell)$.

###### Proof.

Consider the group homomorphisms $T(\ell)\hookrightarrow
B(\ell)\twoheadrightarrow B(\ell)/U(\ell).$ The composition has kernel
$T(\ell)\cap U(\ell)=1$, hence it is injective. Since, by Lemma 12, each of
the groups $B(\ell)/U(\ell)$ and $T(\ell)$ has size $(\ell-1)^{g+1}$, the
composition must be an isomorphism. ∎

##  5\. Conjugacy classes of subgroups of $G(\ell)$

Let $\ell\nmid 2g$ be a fixed prime and let $t\in\mathbb{Z}$ be a fixed
integer. We devote Section 5 to investigations of particular unions of
conjugacy classes of $G(\ell)$, $B(\ell)$, and $T(\ell)$. As usual, when we
regard $\mathbb{Z}/\ell\mathbb{Z}$ as a field, we use the notation
$\mathbb{F}_{\ell}$. In this case, we denote by $\overline{\mathbb{F}}_{\ell}$
a fixed algebraic closure of $\mathbb{F}_{\ell}$.

Before introducing the unions of conjugacy classes we are interested in
investigating, let us observe that if $M=(M_{1},\ldots,M_{g})\in G(\ell)$,
then the characteristic polynomial of $M$ relates to the characteristic
polynomials of $M_{1},\ldots,M_{g}\in\operatorname{GL}_{2}(\ell)$ through the
equation

(20)
$\operatorname{char}_{M}(X)=\operatorname{char}_{M_{1}}(X)\cdot\ldots\cdot\operatorname{char}_{M_{g}}(X)\in\mathbb{F}_{\ell}[X].$

When working over the algebraic closure $\overline{\mathbb{F}}_{\ell}$ of the
field $\mathbb{F}_{\ell}$, for each $1\leq i\leq g$ we may write the linear
factorization of $\operatorname{char}_{M_{i}}(X)$ as

(21)
$\operatorname{char}_{M_{i}}(X)=(X-\lambda_{1}(M_{i}))(X-\lambda_{2}(M_{i}))\in\overline{\mathbb{F}}_{\ell}[X].$

Then, by putting together (20) and (21), we see that the linear factorization
of $\operatorname{char}_{M}(X)$ over $\overline{\mathbb{F}}_{\ell}$ is

(22) $\operatorname{char}_{M}(X)=\displaystyle\prod_{1\leq i\leq
g}(X-\lambda_{1}(M_{i}))(X-\lambda_{2}(M_{i}))\in\overline{\mathbb{F}}_{\ell}[X].$

Using the above notation and fixing $z>0$, we now introduce the sets to be
investigated in this section:

$\displaystyle\mathcal{C}(\ell,t)$ $\displaystyle:=$
$\displaystyle\left\\{(M_{1},\ldots,M_{g})\in
G(\ell):\lambda_{1}(M_{i}),\lambda_{2}(M_{i})\in\mathbb{F}_{\ell}^{\times}\
\forall 1\leq i\leq g,\displaystyle\sum_{1\leq i\leq
g}\operatorname{tr}M_{i}\equiv-t(\operatorname{mod}\ell)\right\\},$
$\displaystyle\mathcal{C}_{\text{Borel}}(\ell,t)$ $\displaystyle:=$
$\displaystyle{\mathcal{C}}(\ell,t)\cap B(\ell),$
$\displaystyle\widehat{\mathcal{C}}_{\text{Borel}}(\ell,t)$ $\displaystyle:=$
$\displaystyle\;\text{the image of $\mathcal{C}_{\text{Borel}}(\ell,t)$ in
$B(\ell)/U(\ell)$},$
$\displaystyle\widehat{\mathcal{C}}^{\prime}_{\text{Borel}}(\ell,t)$
$\displaystyle:=$ $\displaystyle\;\text{the image of
$\mathcal{C}_{\text{Borel}}(\ell,t)$ in $B(\ell)/U^{\prime}(\ell)$},$
$\displaystyle{\mathcal{C}}_{\text{Torus}}(\ell,t)$ $\displaystyle:=$
$\displaystyle{\mathcal{C}}(\ell,t)\cap T(\ell),$
$\displaystyle\mathcal{C}(\ell,|t|\leq z)$ $\displaystyle:=$
$\displaystyle\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\mathcal{C}(\ell,t),$
$\displaystyle\mathcal{C}_{\text{Borel}}(\ell,|t|\leq z)$ $\displaystyle:=$
$\displaystyle\mathcal{C}(\ell,|t|\leq z)\cap B(\ell),$
$\displaystyle\widehat{\mathcal{C}}_{\text{Borel}}(\ell,|t|\leq z)$
$\displaystyle:=$ $\displaystyle\text{the image of
}\mathcal{C}_{\text{Borel}}(\ell,|t|\leq z)\text{ in }B(\ell)/U(\ell).$

######  Lemma 14.

  1. (i)

$\emptyset\neq\mathcal{C}_{\text{Torus}}(\ell,t)\subseteq{\mathcal{C}}_{\text{Borel}}(\ell,t)\subseteq{\mathcal{C}}(\ell,t)$
and $\mathcal{C}_{\text{Borel}}(\ell,|t|\leq z)\neq\emptyset$.

  2. (ii)

$\mathcal{C}(\ell,t)$ is a union of conjugacy classes of $G(\ell)$.

  3. (iii)

$\mathcal{C}_{\text{Borel}}(\ell,t)$ is a union of conjugacy classes of
$B(\ell)$.

  4. (iv)

$\mathcal{C}_{\text{Torus}}(\ell,t)$ is a union of conjugacy classes of
$T(\ell)$.

  5. (v)

$U(\ell)\
{\mathcal{C}}_{\text{Borel}}(\ell,t)\subseteq{\mathcal{C}}_{\text{Borel}}(\ell,t)$
and $U(\ell)\ \mathcal{C}_{\text{Borel}}(\ell,|t|\leq
z)\subseteq\mathcal{C}_{\text{Borel}}(\ell,|t|\leq z)$.

  6. (vi)

$U^{\prime}(\ell)\
\mathcal{C}_{\text{Borel}}(\ell,0)\subseteq\mathcal{C}_{\text{Borel}}(\ell,0)$.

###### Proof.

(i) Since $\ell$ is a prime such $\ell\nmid 2g$, we know that
$(2g)^{-1}(\operatorname{mod}\ell)$ exists. Now we consider each of the cases
$t\neq 0$ and $t=0$. When $t\neq 0$, for each $1\leq i\leq g$, we take

$M_{i}:=\begin{pmatrix}-t(2g)^{-1}(\operatorname{mod}\ell)&0\\\
0&-t(2g)^{-1}(\operatorname{mod}\ell)\\\ \end{pmatrix}.$

Then

$\lambda_{1}(M_{i})\equiv\lambda_{2}(M_{i})\equiv-t(2g)^{-1}(\operatorname{mod}\ell)\in\mathbb{F}_{\ell}^{\times}\
\forall 1\leq i\leq g,\quad\sum_{1\leq i\leq
g}\operatorname{tr}M_{i}\equiv-t(\operatorname{mod}\ell),$

and so $(M_{1},\ldots,M_{g})\in\mathcal{C}_{\text{Torus}}(\ell,t)$. When
$t=0$, for each $1\leq i\leq g$, we take

$M_{i}:=\begin{pmatrix}(2g)^{-1}(\operatorname{mod}\ell)&0\\\
0&-(2g)^{-1}(\operatorname{mod}\ell)\end{pmatrix}.$

Then

$\lambda_{1}(M_{i})\equiv(2g)^{-1}(\operatorname{mod}\ell)\ \text{and}\
\lambda_{2}(M_{i})\equiv-(2g)^{-1}(\operatorname{mod}\ell)\in\mathbb{F}_{\ell}^{\times}\
\forall 1\leq i\leq g,\quad\sum_{1\leq i\leq g}\operatorname{tr}M_{i}\equiv
0(\operatorname{mod}\ell),$

and so $(M_{1},\ldots,M_{g})\in\mathcal{C}_{\text{Torus}}(\ell,0)$. We
conclude that $\mathcal{C}_{\text{Torus}}(\ell,t)\neq\emptyset$.

The inclusions
$\mathcal{C}_{\text{Torus}}(\ell,t)\subseteq{\mathcal{C}}_{\text{Borel}}(\ell,t)\subseteq{\mathcal{C}}(\ell,t)$
are clear from the definitions of the sets.

Finally, observing that $\displaystyle\mathcal{C}_{\text{Borel}}(\ell,|t|\leq
z)=\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\mathcal{C}_{\text{Borel}}(\ell,t)$, the claim that
$\mathcal{C}_{\text{Borel}}(\ell,|t|\leq z)\neq\emptyset$ follows from the
previous inclusions and the non-emptiness of
$\mathcal{C}_{\text{Borel}}(\ell,t)$.

(ii) $\mathcal{C}(\ell,t)$ is a subset of $G(\ell)$ characterized by some
condition on eigenvalues of $g$ invertible $2\times 2$ matrices. Since
eigenvalues are invariant under conjugation, $\mathcal{C}(\ell,t)$ is a union
of conjugacy classes of $G(\ell).$

(iii) By (ii), $\mathcal{C}(\ell,t)$ is invariant under conjugation by
$B(\ell)$. Since $B(\ell)$ is obviously invariant under conjugation by
$B(\ell)$, so is the intersection ${\mathcal{C}}(\ell,t)\cap B(\ell)$. This
implies that $\mathcal{C}_{\text{Borel}}(\ell,t)$ is a union of conjugacy
classes of $B(\ell)$.

(iv) Since ${\mathcal{C}}(\ell,t)$ and $T(\ell)$ are invariant under
conjugation by $T(\ell)$, the set $\mathcal{C}_{\text{Torus}}(\ell,t)$ is a
union of conjugacy classes of $T(\ell)$.

(v) Let $M^{\prime}\in U(\ell)$, $M\in{\mathcal{C}}_{\text{Borel}}(\ell,t)$.
Then $M^{\prime}M\in B(\ell)$, with $M^{\prime}M$ having the same diagonal
entries as $M$. Since $\operatorname{tr}M=-t$, we obtain that
$\operatorname{tr}(M^{\prime}M)=-t$. Thus $U(\ell)\
{\mathcal{C}}_{\text{Borel}}(\ell,t)\subseteq{\mathcal{C}}_{\text{Borel}}(\ell,t)$.
Upon recalling that $\displaystyle\mathcal{C}_{\text{Borel}}(\ell,|t|\leq
z)=\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\mathcal{C}_{\text{Borel}}(\ell,t)$, it also follows that
$U(\ell)\ \mathcal{C}_{\text{Borel}}(\ell,|t|\leq
z)\subseteq\mathcal{C}_{\text{Borel}}(\ell,|t|\leq z)$.

(vi) Let $M^{\prime}\in U^{\prime}(\ell)$ be such that its diagonal elements
are all equal to some $a\in(\mathbb{Z}/\ell\mathbb{Z})^{\times}$. Let
$M\in{\mathcal{C}}_{\text{Borel}}(\ell,0)$. Then $M^{\prime}M\in B(\ell)$ and
$\operatorname{tr}(M^{\prime}M)=a\operatorname{tr}M=0$, which implies that
$U^{\prime}(\ell)\
\mathcal{C}_{\text{Borel}}(\ell,0)\subseteq\mathcal{C}_{\text{Borel}}(\ell,0)$.
∎

######  Lemma 15.

Any element of $\operatorname{GL}_{2}(\ell)$ with a split characteristic
polynomial in $\mathbb{F}_{\ell}[X]$ is conjugate to an element of
$\mathcal{B}_{\operatorname{GL}_{2}}(\ell)$ by an element of
$\operatorname{SL}_{2}(\ell):=\\{M\in\operatorname{GL}_{2}(\ell):\operatorname{det}M=1\\}$.

###### Proof.

Let $M\in\operatorname{GL}_{2}(\ell)$. Observe that
$\lambda_{1}(M),\lambda_{2}(M)\in\mathbb{F}_{\ell}^{\times}$. This means that
there exists $N\in\operatorname{GL}_{2}(\ell)$ such that

$NMN^{-1}=\begin{pmatrix}\lambda_{1}(M)&\ast\\\
0&\lambda_{2}(M)\end{pmatrix}\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell).$

If $\operatorname{det}N=1$, we are done. Otherwise, by taking

$N^{\prime}:=\begin{pmatrix}(\operatorname{det}N)^{-1}&0\\\
0&1\end{pmatrix}N,$

we have that
$N^{\prime}MN^{\prime-1}\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell)$, as well
as that $\operatorname{det}N^{\prime}=1$, and again we are done. ∎

######  Lemma 16.

Every conjugacy class in $\mathcal{C}(\ell,t)$ or $\mathcal{C}(\ell,|t|\leq
z)$ contains an element of $B(\ell)$.

###### Proof.

Let $M=(M_{1},\ldots,M_{g})\in{\mathcal{C}}(\ell,t)$. Upon fixing $1\leq i\leq
g$, we observe that, by Lemma 15, there exists
$N_{i}\in\operatorname{SL}_{2}(\ell)$ such that
$N_{i}M_{i}N_{i}^{-1}\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell)$. Let
$N:=(N_{1},\ldots,N_{g})$. Then $N\in G(\ell)$ and
$NMN^{-1}\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell)^{g}\cap
G(\ell)=B(\ell)$. We deduce that $M$ is conjugate to an element of $B(\ell)$.
Thus, every conjugacy class in $\mathcal{C}(\ell,t)$ contains an element of
$B(\ell)$. The similar statement about $\mathcal{C}(\ell,|t|\leq z)$ follows
upon noting that every conjugacy class in this set is a conjugacy class in
$\mathcal{C}(\ell,t)$ for some integer $t$ with $|t|\leq z$. ∎

######  Lemma 17.

  1. (i)

$|{\mathcal{C}}_{\text{Torus}}(\ell,t)|\leq 2(\ell-1)^{g}$.

  2. (ii)

$|{\mathcal{C}}_{\text{Borel}}(\ell,t)|=\ell^{g}|{\mathcal{C}}_{\text{Torus}}(\ell,t)|\leq
2(\ell-1)^{g}\ell^{g}.$

  3. (iii)

$|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,t)|=|{\mathcal{C}}_{\text{Torus}}(\ell,t)|\leq
2(\ell-1)^{g}.$

  4. (iv)

$|\widehat{\mathcal{C}}^{\prime}_{\text{Borel}}(\ell,0)|=\displaystyle\frac{|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,0)|}{\ell-1}\leq
2(\ell-1)^{g-1}.$

  5. (v)

$|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,|t|\leq z)|<5(\ell-1)^{g}z$.

###### Proof.

(i) Observe that

${\mathcal{C}}_{\text{Torus}}(\ell,t)=\\{(M_{1},\ldots,M_{g})\in
T(\ell):\sum_{1\leq i\leq
g}\operatorname{tr}M_{i}\equiv-t(\operatorname{mod}\ell)\\}.$

Then

$\displaystyle|{\mathcal{C}}_{\text{Torus}}(\ell,t)|$
$\displaystyle=\displaystyle\sum_{d\in(\mathbb{Z}/\ell\mathbb{Z})^{\times}}\quad\displaystyle\sum_{M_{1},\ldots,M_{g-1}\in\mathcal{T}_{\operatorname{GL}_{2}}(\ell)\atop{\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}=d}}\\#\\{M_{g}\in\mathcal{T}_{\operatorname{GL}_{2}}(\ell):\operatorname{det}M_{g}=d,\operatorname{tr}M_{g}\equiv-t-\sum_{1\leq
i\leq g-1}\operatorname{tr}M_{i}(\operatorname{mod}\ell)\\}$
$\displaystyle\leq\displaystyle\sum_{d\in(\mathbb{Z}/\ell\mathbb{Z})^{\times}}\quad\displaystyle\sum_{M_{1},\ldots,M_{g-1}\in\mathcal{T}_{\operatorname{GL}_{2}}(\ell)\atop{\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}=d}}2$
$\displaystyle=2(\ell-1)^{g}.$

(ii) Observe that

${\mathcal{C}}_{\text{Borel}}(\ell,t)=\\{(M_{1},\ldots,M_{g})\in
B(\ell):\sum_{1\leq i\leq
g}\operatorname{tr}M_{i}\equiv-t(\operatorname{mod}\ell)\\}.$

Then

$\displaystyle|{\mathcal{C}}_{\text{Borel}}(\ell,t)|$
$\displaystyle=\displaystyle\sum_{d\in(\mathbb{Z}/\ell\mathbb{Z})^{\times}}\quad\displaystyle\sum_{M_{1},\ldots,M_{g-1}\in\mathcal{T}_{\operatorname{GL}_{2}}(\ell)\atop{\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}=d}}\\#\\{M_{g}\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell):\operatorname{det}M_{g}=d,\operatorname{tr}M_{g}\equiv-t-\sum_{1\leq
i\leq g-1}\operatorname{tr}M_{i}(\operatorname{mod}\ell)\\}.$

Since we have the equality

$\displaystyle\\#\\{M_{g}\in\mathcal{B}_{\operatorname{GL}_{2}}(\ell):\operatorname{det}M_{g}=d,\operatorname{tr}M_{g}\equiv-t-\sum_{1\leq
i\leq g-1}\operatorname{tr}M_{i}(\operatorname{mod}\ell)\\}$ $\displaystyle=$
$\displaystyle\ \ell^{g}\
\\#\\{M_{g}\in\mathcal{T}_{\operatorname{GL}_{2}}(\ell):\operatorname{det}M_{g}=d,\operatorname{tr}M_{g}\equiv-t-\sum_{1\leq
i\leq g-1}\operatorname{tr}M_{i}(\operatorname{mod}\ell)\\},$

we obtain that
$|{\mathcal{C}}_{\text{Borel}}(\ell,t)|=\ell^{g}|{\mathcal{C}}_{\text{Torus}}(\ell,t)|$.

(iii) Observe that, upon identifying $B(\ell)/U(\ell)$ with $T(\ell)$ via
Lemma 13, there is a bijection between the sets
$\widehat{\mathcal{C}}_{\text{Borel}}(\ell,t)$ and
$\mathcal{C}_{\text{Torus}}(\ell,t)$. Thus
$|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,t)|=|{\mathcal{C}}_{\text{Torus}}(\ell,t)|.$

(iv) Since $U(\ell)\leq U^{\prime}(\ell)$ and
$U^{\prime}(\ell)/U(\ell)\simeq(\mathbb{Z}/\ell\mathbb{Z})^{\times}$, there is
a natural surjection $B(\ell)/U(\ell)\to B(\ell)/U^{\prime}(\ell)$ whose
kernel $U^{\prime}(\ell)/U(\ell)$ has cardinality $\ell-1$. We deduce that
$\widehat{\mathcal{C}}^{\prime}_{\text{Borel}}(\ell,0)$ is the image of
$\widehat{\mathcal{C}}_{\text{Borel}}(\ell,0)$ in $B(\ell)/U^{\prime}(\ell)$.

By a similar argument as that in the proof of part (vi) of Lemma 14, we obtain
that
$U^{\prime}(\ell)\widehat{\mathcal{C}}_{\text{Borel}}(\ell,0)=\widehat{\mathcal{C}}_{\text{Borel}}(\ell,0)$.
Thus $\widehat{\mathcal{C}}_{\text{Borel}}(\ell,0)$ is the full inverse image
of $\widehat{\mathcal{C}}^{\prime}_{\text{Borel}}(\ell,0)$ in
$B(\ell)/U(\ell)$.

From the above, we infer that
$|\widehat{\mathcal{C}}^{\prime}_{\text{Borel}}(\ell,0)|=\displaystyle\frac{|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,0)|}{\ell-1}\leq
2(\ell-1)^{g-1}\ell^{g}$, as claimed.

(v) Recall that

$\displaystyle\mathcal{C}_{\text{Borel}}(\ell,|t|\leq
z)=\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\mathcal{C}_{\text{Borel}}(\ell,t).$

Then, by part (v) of Lemma 14, we deduce that

$\widehat{\mathcal{C}}_{\text{Borel}}(\ell,|t|\leq
z)=\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\widehat{\mathcal{C}}_{\text{Borel}}(\ell,t).$

Using part (iii) of the present lemma, we deduce further that

$|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,|t|\leq z)|\leq 2\
\\#\\{t\in\mathbb{Z}:|t|\leq z\\}(\ell-1)^{g}<5z(\ell-1)^{g}.$

∎

##  6\. Proof of Theorem 1

###  6.1. Setting and basic properties

Let $A$ be an abelian variety defined over $\mathbb{Q}$ that is isogenous over
$\mathbb{Q}$ to a product of $g$ elliptic curves defined over $\mathbb{Q}$,
pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each without complex
multiplication. In what follows, we record a few properties of $A$ that we
will need in the proofs of our main results. We keep all the associated
notation introduced in the prior sections, some of which we recall below. For
background on abelian varieties, we refer the reader to [Fa83], [Ho68],
[La83], [Oo08], [SeTa68], and [Wat69].

We fix a $g$-tuple of elliptic curves, $E_{1}/\mathbb{Q}$, $\ldots$,
$E_{g}/\mathbb{Q}$, equipped with a $\mathbb{Q}$-isogeny between $A$ and
$E_{1}\times\ldots\times E_{g}$ of minimal degree, denoted $d_{A}$, among all
such choices. We denote by $N_{E_{1}},\ldots,N_{E_{g}}$ the conductors of
$E_{1},\ldots,E_{g}$, respectively, and remark that the conductor $N_{A}$ of
$A$ and the product $N_{E_{1}}\ldots N_{E_{g}}$ have the same prime factors.

We denote by

$\rho_{A}:\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\longrightarrow\operatorname{GL}_{2g}(\widehat{\mathbb{Z}})$

the adelic Galois representation of $A$, defined by the inverse limit of the
residual representations

$\bar{\rho}_{A,m}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\operatorname{GL}_{2g}(\mathbb{Z}/m\mathbb{Z})$

of the absolute Galois group
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acting on the
$m$-torsion group $A[m]\subset A(\overline{\mathbb{Q}})$, with $m$ a positive
integer. For any prime $\ell$, we denote by

$\rho_{A,\ell}:\operatorname{Gal}\left(\overline{\mathbb{Q}}/\mathbb{Q}\right)\longrightarrow\operatorname{GL}_{2g}(\mathbb{Z}_{\ell})$

the $\ell$-adic representation of
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the $\ell$-adic Tate
module ${\displaystyle\lim_{\leftarrow\atop{n}}A[\ell^{n}}]$.

Similarly to notation (19) introduced in Section 4, we set

$G(\widehat{\mathbb{Z}}):=\\{(M_{1},\ldots,M_{g})\in\operatorname{GL}_{2}(\widehat{\mathbb{Z}})^{g}:\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}\\},$
$G(m):=\\{(M_{1},\ldots,M_{g})\in\operatorname{GL}_{2}(\mathbb{Z}/m\mathbb{Z})^{g}:\operatorname{det}M_{1}=\ldots=\operatorname{det}M_{g}\\},$

and recall that

$\operatorname{Im}\rho_{A}\subseteq G(\widehat{\mathbb{Z}}),$
$\operatorname{Im}\bar{\rho}_{A,m}\subseteq G(m),$

where $m$ is an arbitrary positive integer.

For any prime $p\nmid N_{A}$, we denote by $\operatorname{Frob}_{p}$ the
Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$,
and for any prime $\ell\neq p$, we denote by

$P_{A,p}(X):=\operatorname{det}(XI_{2g}-\rho_{A,\ell}(\operatorname{Frob}_{p}))$

the characteristic polynomial of $\rho_{A,\ell}(\operatorname{Frob}_{p})$ in
$\operatorname{GL}_{2g}(\mathbb{Z}_{\ell})$.

We recall that $P_{A,p}(X)$ is the characteristic polynomial of the Frobenius
endomorphism acting on the reduction of $A$ modulo $p$. We also recall that
$P_{A,p}(X)$ is a $p$-Weil polynomial, has integer coefficients independent of
$\ell$, and satisfies the congruence

(23)
$P_{A,p}(X)\equiv\operatorname{det}\left(XI_{2g}-\bar{\rho}_{A,m}\left(\left(\frac{\mathbb{Q}(A[m])/\mathbb{Q}}{p}\right)\right)\right)(\operatorname{mod}m)$

for any integer $m$ coprime to $p$.

In analogy with the discussion and notation for $A$, for each $1\leq i\leq g$
we denote by $\rho_{E_{i}}$, $\overline{\rho}_{E_{i},m}$, and
$\rho_{E_{i},\ell}$ the adelic, residual, and $\ell$-adic Galois
representations of $E_{i}$, and for each prime $p\nmid N_{E_{i}}$ we denote by
$P_{E_{i},p}(X)$ the associated $p$-Weil polynomial. We recall the congruence
relation

(24)
$P_{E_{i},p}(X)\equiv\operatorname{det}\left(XI_{2}-\bar{\rho}_{E_{i},m}\left(\frac{\mathbb{Q}(E_{i}[m])/\mathbb{Q}}{p}\right)\right)(\operatorname{mod}m),$

which holds for any integer $m$ coprime to $p$.

Now let us fix a prime $p\nmid N_{A}d_{A}$. We write the polynomials
$P_{A,p}(X)$, $P_{E_{i},p}(X)$ explicitly as

$\displaystyle P_{A,p}(X)$ $\displaystyle=$ $\displaystyle
X^{2g}+a_{1,p}(A)X^{2g-1}+\ldots+a_{2g-1,p}(A)X+a_{2g,p}(A)\in\mathbb{Z}[X],$
$\displaystyle P_{E_{i},p}(X)$ $\displaystyle=$ $\displaystyle
X^{2}-a_{p}(E_{i})X+p\in\mathbb{Z}[X].$

On one hand, we have the polynomial relation

(25) $P_{A,p}(X)=P_{E_{1},p}(X)\cdot\ldots\cdot P_{E_{g},p}(X),$

from which we obtain that

(26) $a_{1,p}(A)=-(a_{p}(E_{1})+\ldots+a_{p}(E_{g})).$

On the other hand, by restricting the residual representations
$\overline{\rho}_{A,m}$, $\overline{\rho}_{E_{i},m}$ to the division fields
$\mathbb{Q}(A[m])$, $\mathbb{Q}(E_{i}[m])$, respectively, we deduce from (23)
- (24) that, for any integer $m$ coprime to $p$, we have

(27)
$\displaystyle\operatorname{tr}\bar{\rho}_{A,m}\left(\left(\frac{\mathbb{Q}(A[m])/\mathbb{Q}}{p}\right)\right)$
$\displaystyle\equiv$ $\displaystyle-a_{1,p}(A)(\operatorname{mod}m),$ (28)
$\displaystyle\operatorname{tr}\bar{\rho}_{E_{i},m}\left(\left(\frac{\mathbb{Q}(E_{i}[m])/\mathbb{Q}}{p}\right)\right)$
$\displaystyle\equiv$ $\displaystyle a_{p}(E_{i})(\operatorname{mod}m).$

From (26), (27) and (28), we deduce that, for any integer $m$ coprime to $p$,
we have

(29)
$\operatorname{tr}\bar{\rho}_{A,m}\left(\left(\frac{\mathbb{Q}(A[m])/\mathbb{Q}}{p}\right)\right)\equiv-\displaystyle\sum_{1\leq
i\leq
g}\operatorname{tr}\bar{\rho}_{E_{i},m}\left(\left(\frac{\mathbb{Q}(E_{i}[m])/\mathbb{Q}}{p}\right)\right)(\operatorname{mod}m).$

###  6.2. A conditional reduction step

For an abelian variety $A/\mathbb{Q}$, a prime $p\nmid N_{A}$, and an integer
$t$, we remark that, congruence (27) already allows us to relax the equality
$a_{1,p}(A)=t$ to the congruence
$\operatorname{tr}\bar{\rho}_{A,\ell}\left(\left(\frac{\mathbb{Q}(A[\ell])/\mathbb{Q}}{p}\right)\right)\equiv-t(\operatorname{mod}\ell)$
for some prime $\ell$ and then to apply an effective version of the Chebotarev
Density Theorem in the extension $\mathbb{Q}(A[\ell])/\mathbb{Q}$. However, we
can obtain better results by relaxing the equality $a_{1,p}(A)=t$ to a
Chebotarev condition that takes place in the extension
$\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}$. The key
ingredient is as follows.

######  Lemma 18.

Let $t\in\mathbb{Z}$. Let $A/\mathbb{Q}$ be an abelian variety defined over
$\mathbb{Q}$ that is isogenous over $\mathbb{Q}$ to a product
$E_{1}\times\ldots\times E_{g}$ of elliptic curves $E_{1}/\mathbb{Q}$,
$\ldots$, $E_{g}/\mathbb{Q}$, without complex multiplication and pairwise non-
isogenous over $\overline{\mathbb{Q}}$. For each $x>2$, let $y=y(x)>2$,
$u=u(x)>2$ be such that

(30) $u\leq y.$

Assume that, for any $\varepsilon>0$,

(31) $u\geq
c^{\prime}(\varepsilon)y^{\frac{1}{2}}(\operatorname{log}y)^{2+\varepsilon}\
\text{for some}\ c^{\prime}(\varepsilon)>0$

and

(32)
$\displaystyle\lim_{x\rightarrow\infty}\frac{\operatorname{log}x}{(\operatorname{log}y)^{1+\varepsilon}}=0.$

Assume GRH for Dedekind zeta functions. Then, for any $\varepsilon>0$, there
exists a constant $c(\varepsilon)>0$ such that, for any positive real number
$z$ and for any sufficiently large $x$, we have

(33) $\pi_{A}(x,t)\leq c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\pi_{A}(x,\ell,t)$

and

(34) $\sum_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq z\end{subarray}}\
\pi_{A}(x,t)\leq c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\sum_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq z\end{subarray}}\
\pi_{A}(x,\ell,t),$

where, as before,

$\pi_{A}(x,t)=\\#\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\\},$

and where

$\displaystyle\pi_{A}(x,\ell,t)$ $\displaystyle:=$
$\displaystyle\\#\left\\{p\leq x:p\nmid\ell N_{A},a_{1,p}(A)=t,\right.$
$\displaystyle\hskip 45.52458pt\left.\ell\ \text{splits completely in each of
the fields}\
\mathbb{Q}(\pi_{p}(E_{1})),\ldots,\mathbb{Q}(\pi_{p}(E_{g}))\right\\}.$

###### Proof.

We apply Lemma 9 to the set ${\mathcal{S}}=\left\\{p:p\nmid
N_{A},a_{1,p}(A)=t\right\\}$, the field $K_{p}=$ the compositum of the fields
$\mathbb{Q}(\pi_{p}(E_{1})),\ldots,\mathbb{Q}(\pi_{p}(E_{g}))$, the conjugacy
class ${\mathcal{C}}_{p}=\left\\{\operatorname{id}_{K_{p}}\right\\}$, and the
function $f(v)=\operatorname{log}v$.

Note that $K_{p}/\mathbb{Q}$ is Galois and that $1\neq[K_{p}:\mathbb{Q}]\mid
2^{g}$. Also note that if a prime $\ell$ is ramified in $K_{p}$, then
$\ell\mid\left(4p-a_{p}(E_{1})^{2}\right)\ldots\left(4p-a_{p}(E_{g})^{2}\right)$
(see [To55, Theorem, p. 43]). Therefore, upon recalling (3), we have that
$\displaystyle\operatorname{log}|d_{K_{p}}|\ll_{E_{1},\ldots,E_{g}}\operatorname{log}p$.
Since all hypotheses of Lemma 9 are satisfied, bound (33) follows. Bound (34)
follows from a similar argument, except for taking
${\mathcal{S}}=\left\\{p:p\nmid N_{A},|a_{1,p}(A)|\leq z\right\\}$, while
keeping $K_{p}$, ${\mathcal{C}}_{p}$, and $f$ as before.

∎

###  6.3. Proof of Theorem 1 for arbitrary $t$

Let $A$ be an abelian variety defined over $\mathbb{Q}$ that is isogenous over
$\mathbb{Q}$ to a product of $g$ elliptic curves defined over $\mathbb{Q}$,
pairwise non-isogenous over $\overline{\mathbb{Q}}$ and each without complex
multiplication. As in Subsection 6.1, we fix a $g$-tuple of elliptic curves,
$E_{1}/\mathbb{Q}$, $\ldots$, $E_{g}/\mathbb{Q}$, equipped with a
$\mathbb{Q}$-isogeny between $A$ and $E_{1}\times\ldots\times E_{g}$ of
minimal degree, $d_{A}$, among all such choices. We keep all the associated
notation introduced so far.

Let $t\in\mathbb{Z}$ and $x>2$. By Lemma 18, we can bound $\pi_{A}(x,t)$ from
above if we succeed in bounding $\pi_{A}(x,\ell,t)$ from above for some
suitably chosen prime $\ell=\ell(x)$.

For now, we fix a prime $\ell\nmid 2g$ such that

(35) $\operatorname{Im}\overline{\rho}_{A,\ell}=G(\ell).$

The existence of infinitely many such primes $\ell$ is ensured by [Se72,
Théorème 6, p. 324] and [Lo16, Theorem 1.1, p. 387] under our hypotheses that
$E_{1}/\mathbb{Q},\ldots,E_{g}/\mathbb{Q}$ are without complex multiplication
and pairwise non-isogenous over $\overline{\mathbb{Q}}$. Indeed, from [Lo16,
Theorem 1.1, p. 387], we infer that there exists a least positive integer
$m_{A}$ such that $\operatorname{Im}\rho_{A}$ equals the inverse image under
the canonical projection $G(\widehat{\mathbb{Z}})\longrightarrow G(m_{A})$ of
$\operatorname{Im}\overline{\rho}_{A,m_{A}}$. In particular, this means that
$\operatorname{Im}\bar{\rho}_{A,\ell}=G(\ell)$ for any prime $\ell\nmid
m_{A}$. Later on, we will choose $\ell$ in an explicit interval that depends
on $x$.

The choice of a prime $\ell$ such that (35) holds allows us to consider the
subextensions of $\mathbb{Q}(A[\ell])$ fixed by the groups $U(\ell)$ and
$B(\ell)$, namely

$\mathbb{Q}\subset\mathbb{Q}(A[\ell])^{B(\ell)}\subset\mathbb{Q}(A[\ell])^{U(\ell)}\subset\mathbb{Q}(A[\ell]),$

for which we immediately obtain the following Galois group structures:

$\displaystyle\operatorname{Gal}(\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\simeq$ $\displaystyle G(\ell),$
$\displaystyle\operatorname{Gal}\left(\mathbb{Q}(A[\ell])/\mathbb{Q}(A[\ell])^{B(\ell)}\right)$
$\displaystyle\simeq$ $\displaystyle B(\ell),$
$\displaystyle\operatorname{Gal}\left(\mathbb{Q}(A[\ell])/\mathbb{Q}(A[\ell])^{U(\ell)}\right)$
$\displaystyle\simeq$ $\displaystyle U(\ell),$
$\displaystyle\operatorname{Gal}\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)$
$\displaystyle\simeq$ $\displaystyle B(\ell)/U(\ell).$

In relation to these Galois groups, we recall from part (v) of Lemma 11 that
$U(\ell)\unlhd B(\ell)$ and that $B(\ell)/U(\ell)$ is abelian.

Using (25), (29), and (35), we deduce that every prime $p\nmid d_{A}$ counted
by $\pi_{A}(x,\ell,t)$ is also counted by
$\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$, where
$\mathcal{C}(\ell,t)$ is the union of conjugacy classes of $G(\ell)$ studied
in Lemma 14 of Section 5. Thus

(36)
$\pi_{A}(x,\ell,t)\ll_{A}\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q}).$

To estimate $\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$, we
invoke part (ii) of Corollary 8 of Section 2 with $K=\mathbb{Q}$,
$L=\mathbb{Q}(A[\ell])$, $G=G(\ell)$, $H=B(\ell)$, $N=U(\ell)$, and
${\mathcal{C}}={\mathcal{C}}(\ell,t)$. The hypotheses of this corollary hold
thanks to our assumption of GRH and to part (v) of Lemma 11, part (v) of Lemma
14, and Lemma 16. We deduce that

$\displaystyle\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{\left|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,t)\right|\cdot|U(\ell)|}{|B(\ell)|}\cdot\frac{x}{\operatorname{log}x}$
$\displaystyle+$
$\displaystyle\left|\widehat{{\mathcal{C}}}_{\text{Borel}}(\ell,t)\right|^{\frac{1}{2}}[\mathbb{Q}(A[\ell])^{B(\ell)}:\mathbb{Q}]\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)$
$\displaystyle+$
$\displaystyle\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M\left(\mathbb{Q}(A[\ell])/\mathbb{Q}\right)\right)$
$\displaystyle+$
$\displaystyle[\mathbb{Q}(A[\ell])^{B(\ell)}:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)\right).$

Using Lemma 12 and Lemma 17, we deduce from the above that

(37)
$\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})\ll\frac{x}{\ell\operatorname{log}x}+(\ell-1)^{\frac{g}{2}}(\ell+1)^{g}\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right).$

We estimate
$\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)$
by putting together (3), Lemma 12, and the Néron-Ogg-Shafarevich criterion for
$A/\mathbb{Q}$, which states that the extension
$\mathbb{Q}(A[\ell])/\mathbb{Q}$ is unramified outside of $\ell N_{A}$ (see
[SeTa68, Theorem 1, p. 493]). We obtain that

(38)
$\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)\leq
2\operatorname{log}|B(\ell)/U(\ell)|+2\operatorname{log}(\ell
N_{A})+\operatorname{log}2\ll g\operatorname{log}(\ell N_{A}).$

Using this bound in the upper-estimate (37) for
$\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$, we deduce that

(39)
$\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})\ll\frac{x}{\ell\operatorname{log}x}+g\
\ell^{\frac{3g}{2}}\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}(\ell
N_{A}).$

Finally, we apply Lemma 18 with

$y\asymp\frac{x^{\frac{1}{3g+2}}}{(\operatorname{log}x)^{\frac{2}{3g+2}}}$

and

$u\asymp y^{\frac{1}{2}}(\operatorname{log}y)^{2+\varepsilon}$

for an arbitrarily chosen $\varepsilon>0$, and conclude that

$\pi_{A}(x,t)\leq c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\pi_{A}(x,\ell,t)\ll_{\varepsilon,A}\frac{x^{1-\frac{1}{3g+2}}}{(\operatorname{log}x)^{1-\frac{2}{3g+2}}}.$

###  6.4. Proof of Theorem 1 for $t=0$

We keep the setting of Subsection 6.3. Let $x>2$. As before, we take
$\ell\nmid 2g$ to be a sufficiently large prime such that (35) holds. Then
taking $t=0$ in (36), we have that

(40)
$\pi_{A}(x,\ell,0)\ll_{A}\pi_{\mathcal{C}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q}).$

To estimate $\pi_{\mathcal{C}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$, we
follow the strategy of the previous subsection, the main difference being that
we will now work with the subextensions of $\mathbb{Q}(A[\ell])$ fixed by the
groups $U^{\prime}(\ell)$ and $B(\ell)$, instead of by $U(\ell)$ and
$B(\ell)$. More precisely, we will work with the extensions

$\mathbb{Q}\subset\mathbb{Q}(A[\ell])^{B(\ell)}\subset\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}\subset\mathbb{Q}(A[\ell]).$

Observe that
$\operatorname{Gal}\left(\mathbb{Q}(A[\ell])/\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}\right)\simeq
U^{\prime}(\ell)$ and
$\operatorname{Gal}\left(\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)\simeq
B(\ell)/U^{\prime}(\ell)$, and recall from part (vi) of Lemma 11 that
$U^{\prime}(\ell)\unlhd B(\ell)$ and that $B(\ell)/U^{\prime}(\ell)$ is
abelian. Upon also recalling part (vi) of Lemma 14 and Lemma 16, we apply part
(ii) of Corollary 8 of Section 2 with $K=\mathbb{Q}$, $L=\mathbb{Q}(A[\ell])$,
$G=G(\ell)$, $H=B(\ell)$, $N=U^{\prime}(\ell)$, and
${\mathcal{C}}={\mathcal{C}}(\ell,0)$, and deduce that

$\displaystyle\pi_{\mathcal{C}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{\left|\widehat{\mathcal{C}}^{\prime}_{\text{Borel}}(\ell,0)\right|\cdot|U^{\prime}(\ell)|}{|B(\ell)|}\cdot\frac{x}{\operatorname{log}x}$
$\displaystyle+$
$\displaystyle\left|\widehat{{\mathcal{C}}}^{\prime}_{\text{Borel}}(\ell,0)\right|^{\frac{1}{2}}[\mathbb{Q}(A[\ell])^{B(\ell)}:\mathbb{Q}]\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)$
$\displaystyle+$
$\displaystyle\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M\left(\mathbb{Q}(A[\ell])/\mathbb{Q}\right)\right)$
$\displaystyle+$
$\displaystyle[\mathbb{Q}(A[\ell])^{B(\ell)}:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)\right).$

Using Lemma 12 and Lemma 17, we infer from the above that

$\displaystyle\pi_{\mathcal{C}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{x}{\ell\operatorname{log}x}+\ell^{\frac{g-1}{2}}(\ell+1)^{g}\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}M(\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}).$

Similarly to how we deduced (38), we obtain that

(41)
$\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U^{\prime}(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)\leq
2\operatorname{log}|B(\ell)/U^{\prime}(\ell)|+2\operatorname{log}(\ell
N_{A})+\operatorname{log}2\ll g\operatorname{log}(\ell N_{A}).$

Thus,

$\displaystyle\pi_{\mathcal{C}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$ $\displaystyle\frac{x}{\ell\operatorname{log}x}+g\
\ell^{\frac{3g-1}{2}}\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}(\ell
N_{A}).$

Now we apply Lemma 18 with

$y\asymp\frac{x^{\frac{1}{3g+1}}}{(\operatorname{log}x)^{\frac{2}{3g+1}}}$

and

$u\asymp y^{\frac{1}{2}}(\operatorname{log}y)^{2+\varepsilon}$

for an arbitrarily chosen $\varepsilon>0$. We conclude that

$\pi_{A}(x,0)\leq c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\pi_{A}(x,\ell,0)\ll_{\varepsilon,A}\frac{x^{1-\frac{1}{3g+1}}}{(\operatorname{log}x)^{1-\frac{2}{3g+1}}}.$

This completes the proof of Theorem 1.

###  6.5. Proof of applications (1) and (2)

We keep the setting of Subsection 6.3. Let $t\in\mathbb{Z}$ and $x>2$. By
applying Theorem 1, we deduce right away that

$\displaystyle\\#\left\\{p\leq x:p\nmid N_{A},a_{1,p}(A)\neq
t\right\\}=\pi(x)-\\#\left\\{p\leq x:p\nmid
N_{A},a_{1,p}(A)=t\right\\}-\\#\left\\{p\leq x:p\mid
N_{A}\right\\}\sim\pi(x),$

which confirms (1). To confirm (2), we fix $\varepsilon>0$ and make the
following observations, the third of which uses Theorem 1:

$\displaystyle\pi(x)$ $\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+2}-\varepsilon}\right\\}+\\#\left\\{p\leq
x:p\nmid N_{A},|a_{1,p}(A)|\leq
p^{\frac{1}{3g+2}-\varepsilon}\right\\}+\\#\left\\{p\leq x:p\mid
N_{A}\right\\}$ $\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+2}-\varepsilon}\right\\}+\operatorname{O}\left(\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq
x^{\frac{1}{3g+2}-\varepsilon}}}\pi_{A}(x,t)\right)+\operatorname{O}_{A}(1)$
$\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+2}-\varepsilon}\right\\}+\operatorname{O}_{A,\varepsilon}\left(x^{\frac{1}{3g+2}-\varepsilon}\cdot\displaystyle\frac{x^{1-\frac{1}{3g+2}}}{(\operatorname{log}x)^{1-\frac{2}{3g+2}}}\right)$
$\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+2}-\varepsilon}\right\\}+\operatorname{O}_{A,\varepsilon}\left(\displaystyle\frac{x^{1-\varepsilon}}{(\operatorname{log}x)^{1-\frac{2}{3g+2}}}\right)$
$\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+2}-\varepsilon}\right\\}+\operatorname{o}\left(\pi(x)\right).$

###  6.6. Proof of Theorem 3

We keep the setting of Subsection 6.3. Let $x>2$ and $z=z(x)>1$. Note that,
for any prime $\ell$, we have

$\displaystyle\sum_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\ \pi_{A}(x,\ell,t)$ $\displaystyle:=\\#\left\\{p\leq
x:p\nmid\ell N_{A},|a_{1,p}(A)|\leq z,\right.$ (42) $\displaystyle\hskip
45.52458pt\left.\ell\ \text{splits completely in each of the fields}\
\mathbb{Q}(\pi_{p}(E_{1})),\ldots,\mathbb{Q}(\pi_{p}(E_{g}))\right\\}.$

As before, observe that every prime $p\nmid d_{A}$ which is counted on the
right-hand side of (6.6) is also counted by $\pi_{\mathcal{C}(\ell,|t|\leq
z)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$. Thus

$\displaystyle\sum_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\pi_{A}(x,\ell,t)\ll_{A}\pi_{\mathcal{C}(\ell,|t|\leq
z)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q}).$

We take $\ell\nmid 2g$ to be a sufficiently large prime such that (35) holds.
Recalling part (vi) of Lemma 14 and Lemma 16, we apply part (ii) of Corollary
8 of Section 2 with $K=\mathbb{Q}$, $L=\mathbb{Q}(A[\ell])$, $G=G(\ell)$,
$H=B(\ell)$, $N=U(\ell)$, and ${\mathcal{C}}={\mathcal{C}}(\ell,|t|\leq z)$,
and obtain

$\displaystyle\pi_{\mathcal{C}(\ell,|t|\leq
z)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$ $\displaystyle\ll$
$\displaystyle\frac{\left|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,|t|\leq
z)\right|}{\left|B(\ell)/U(\ell)\right|}\cdot\frac{x}{\operatorname{log}x}$
$\displaystyle+$
$\displaystyle\left|\widehat{\mathcal{C}}_{\text{Borel}}(\ell,|t|\leq
z)\right|^{\frac{1}{2}}\left[\mathbb{Q}(A[\ell])^{B(\ell)}:\mathbb{Q}\right]\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\left(\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U(\ell)}/\mathbb{Q}(A[\ell])^{B(\ell)}\right)\right).$

Using Proposition 12 and Lemma 17, we infer from the above that

$\pi_{\mathcal{C}(\ell,|t|\leq
z)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})\ll\frac{z}{\ell}\cdot\frac{x}{\operatorname{log}x}+g\
z^{\frac{1}{2}}\ell^{\frac{3g}{2}}\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\operatorname{log}(\ell
N_{A}).$

Applying (34) in Lemma 18 with $y\asymp z(\operatorname{log}x)^{\eta}$ for
some fixed arbitrary $\eta>0$, we obtain that

$\displaystyle\sum_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\pi_{A}(x,t)$
$\displaystyle\ll_{A}\frac{x}{(\operatorname{log}x)^{1+\eta}}+x^{\frac{1}{2}}(\operatorname{log}x)^{\frac{3g\eta}{2}-1}z^{\frac{3g+1}{2}}(\operatorname{log}z+\operatorname{log}\operatorname{log}x).$

Now fix an arbitrary $\varepsilon>0$ and set
$z:=x^{\frac{1}{3g+1}-\varepsilon}.$ Then

$\sum_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
x^{\frac{1}{3g+1}-\varepsilon}\end{subarray}}\pi_{A}(x,t)\ll_{A,\varepsilon}\frac{x}{(\operatorname{log}x)^{1+\eta}}=\operatorname{o}(\pi(x)).$

Theorem 3 follows from the above upper bound and the following observations:

$\displaystyle\pi(x)$ $\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+1}-\varepsilon}\right\\}+\\#\left\\{p\leq
x:p\nmid N_{A},|a_{1,p}(A)|\leq
p^{\frac{1}{3g+1}-\varepsilon}\right\\}+\\#\left\\{p\leq x:p\mid
N_{A}\right\\}$ $\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+1}-\varepsilon}\right\\}+\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq
x^{\frac{1}{3g+1}-\varepsilon}}}\pi_{A}(x,t)+\operatorname{O}_{A}(1)$
$\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>p^{\frac{1}{3g+1}-\varepsilon}\right\\}+\operatorname{o}\left(\pi(x)\right).$

##  7\. Final remarks

It is a natural question to ask whether one can find subgroups and conjugacy
classes other than those chosen in Sections 4 - 5 and use them, following a
strategy similar to the one of Theorem 1, to obtain a better upper bound for
$\pi_{A}(x,t)$. We relegate this question to future research. Below are
candidates of subgroups and conjugacy classes which, used mutatis mutandis in
our proof, do not improve on Theorem 1.

We keep the setting of Subsection 6.3. We could replace $B(\ell)$ with
$T(\ell)$, $U(\ell)$ with $\\{I_{2g}\\}$, and
$\widehat{{\mathcal{C}}}_{\text{Borel}}(\ell,t)$ with
$\mathcal{C}_{\text{Torus}}(\ell,t)$. For every $t\in\mathbb{Z}$, this would
lead to the upper bound

(43)
$\pi_{A}(x,t)\ll_{A}\frac{x^{1-\frac{1}{5g+2}}}{(\operatorname{log}x)^{1-\frac{2}{5g+2}}}.$

Alternatively, we could replace $B(\ell)$ with $C^{ns}_{\xi}(\ell)^{g}\cap
G(\ell)$, $U(\ell)$ with $\\{I_{2g}\\}$, and
$\widehat{{\mathcal{C}}}_{\text{Borel}}(\ell,t)$ with
$\mathcal{C}_{ns}(\ell,\xi,t)$, where $C^{ns}_{\xi}(\ell)$ is the non-split
Cartan subgroup of $\operatorname{GL}_{2}(\ell)$ defined by

$C^{ns}_{\xi}(\ell):=\left\\{\begin{pmatrix}a&\xi b\\\
b&a\end{pmatrix}:a,b\in\mathbb{F}_{\ell},(a,b)\neq(0,0)\right\\}$

for some non-square element $\xi$ of $\mathbb{F}_{\ell}^{\times}$, and where

$\mathcal{C}_{ns}(\ell,\xi,t):=\left\\{(M_{1},\ldots,M_{g})\in
C^{ns}_{\xi}(\ell)^{g}\cap G(\ell):\displaystyle\sum_{1\leq i\leq
g}\operatorname{tr}M_{i}\equiv-t(\operatorname{mod}\ell)\right\\}.$

Note that $C^{ns}_{\xi}(\ell)^{g}\simeq(\mathbb{F}_{\ell^{2}}^{\times})^{g}$,
where $\mathbb{F}_{\ell^{2}}$ denotes the finite field with $\ell^{2}$
elements. Hence $C^{ns}_{\xi}(\ell)^{g}$ is an abelian group. With the above
choices, instead of considering the counting function $\pi_{A}(x,\ell,t)$, we
would consider

$\pi^{ns}_{A}(x,\ell,t):=\\#\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t,\ \ell\
\text{inert in each of the fields}\
\mathbb{Q}(\pi_{p}(E_{1})),\ldots,\mathbb{Q}(\pi_{p}(E_{g}))\\}$

and prove an upper bound similar to that of Lemma 18. Then, applying Corollary
8, we would obtain the previous bound (43), which is weaker than that of
Theorem 1.

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