††footnotetext: _Key words and phrases:_ abelian varieties, endomorphism
rings, Galois representations, distribution of primes, density theorems
††footnotetext: _2010 Mathematics Subject Classification:_ 11G05, 11G20, 11N05
(Primary), 11N36, 11N37, 11N56 (Secondary)

#  Bounds for the distribution of the Frobenius traces associated to a generic
abelian variety

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 $A$ be an abelian variety defined over
$\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large
prime $\ell$, the image of the residual modulo $\ell$ Galois representation of
$A$ is isomorphic to $\operatorname{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$. 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
$\pi_{A}(x,0)\ll_{A}x^{1-\frac{1}{2g^{2}+g+1}}/(\operatorname{log}x)^{1-\frac{2}{2g^{2}+g+1}}$
and that
$\pi_{A}(x,t)\ll_{A}x^{1-\frac{1}{2g^{2}+g+2}}/(\operatorname{log}x)^{1-\frac{2}{2g^{2}+g+2}}$
if $t\neq 0$. These bounds largely improve upon those obtained by J-P. Serre
in 1981 for $g=1$ and by A.C. Cojocaru, R. Davis, A. Silverberg, and K.E.
Stange for $g\geq 2$ in 2017. They 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 in 1988, combined with a refinement in the power of
$\operatorname{log}x$ obtained by D. Zywina in 2015. Under the assumptions
stated above, we also prove the existence of a density one set of primes $p$
satisfying
$|a_{1,p}(A)|>p^{\frac{1}{2g^{2}+g+1}}/(\operatorname{log}p)^{\varepsilon}$
for any fixed $\varepsilon>0$. Assuming, in addition to the Generalized
Riemann Hypothesis for Dedekind zeta functions, Artin’s Holomorphy Conjecture
and a Pair Correlation Conjecture for Artin L-functions, we prove that
$\pi_{A}(x,0)\ll_{A}x^{1-\frac{1}{g+1}}/(\operatorname{log}x)^{1-\frac{4}{g+1}}$
and that
$\pi_{A}(x,t)\ll_{A}x^{1-\frac{1}{g+2}}/(\operatorname{log}x)^{1-\frac{4}{g+2}}$
if $t\neq 0$, and we deduce the existence of a density one set of primes $p$
satisfying $|a_{1,p}(A)|>p^{\frac{1}{g+2}-\varepsilon}$ for any fixed
$\varepsilon>0$. These are currently the best known conditional upper bounds
for $\pi_{A}(x,t)$ and the best known conditional lower bounds for
$|a_{1,p}(A)|$, for most primes $p$.

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

##  1\. Introduction

In 1976, Lang and Trotter [LaTr76] proposed a prime distribution problem,
which remains open, in the setting of elliptic curves. Known as the Lang-
Trotter Conjecture on Frobenius traces, the problem aims to understand the
reduction type of a given elliptic curve defined over the field of rational
numbers. It may be described in terms of the distribution of Frobenius
elements in an infinite family of Galois number fields derived from the given
elliptic curve. This description may then be generalized to prime distribution
problems related to compatible systems of Galois representations, such as
those associated to modular forms and those associated to higher dimensional
abelian varieties. For example, such generalizations were proposed in
[AkPa19], [BaGo97], [ChJoSe20], [CoDaSiSt17], [Ka09], and [Mu99]. The goal of
our paper is to prove results related to the generalization of the Lang-
Trotter Conjecture on Frobenius traces formulated in [CoDaSiSt17] in the
setting of generic abelian varieties defined over $\mathbb{Q}$, as explained
below.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $g\geq
1$, and of conductor $N_{A}$. For a rational prime $\ell$, consider the group
representation obtained from the action of the absolute Galois group
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on the $\ell$-adic Tate
module $T_{\ell}(A)$ of $A$. Recalling that there is an isomorphism of
$\mathbb{Z}_{\ell}$-modules
$T_{\ell}(A)\simeq_{\mathbb{Z}_{\ell}}(\mathbb{Z}_{\ell})^{2g}$ and fixing a
$\mathbb{Z}_{\ell}$-basis of $T_{\ell}(A)$, we obtain a Galois representation

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

which we call the $\ell$-adic Galois representation of $A$. Taking the
projection of $\operatorname{GL}_{2g}(\mathbb{Z}_{\ell})$ onto
$\operatorname{GL}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$, we obtain a Galois
representation

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

which we call the residual modulo $\ell$ Galois representation of $A$.

For a rational prime $p\nmid\ell N_{A}$, we denote by $a_{1,p}(A)$ the trace
of $\rho_{A,\ell}(\operatorname{Frob}_{p})$, where
$\operatorname{Frob}_{p}\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
is a fixed Frobenius element at $p$. It is known that $a_{1,p}(A)$ is an
integer which does not depend on $\ell$ and which satisfies the Hasse-Weil
bound $|a_{1,p}(A)|<2gp^{\frac{1}{2}}$.

In [LaTr76], Lang and Trotter considered the case of an abelian variety over
$\mathbb{Q}$ of dimension 1 and having a trivial
$\overline{\mathbb{Q}}$-endomorphism ring, and proposed the investigation of
the asymptotic behaviour of the function counting primes $p$ for which
$a_{1,p}(A)$ equals a given integer. This problem may also be considered when
$g>1$, as follows.

Let $t\in\mathbb{Z}$ and $x\in\mathbb{R}$ with $x>0$, and define

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

Assume, for simplicity, that $A$ is principally polarized, in which case the
Galois representation $\rho_{A}$, defined as the product of the $\ell$-adic
Galois representations $\rho_{A,\ell}$, has its image contained in
$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. If $\rho_{A}$ has open image in
$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$ and if the normalized traces
$\frac{a_{1,p}(A)}{\sqrt{p}}$ are equidistributed on the interval $[-2g,2g]$
with respect to the projection by the trace map of the normalized Haar measure
on the unitary symplectic group $\operatorname{USp}_{2g}$, then Cojocaru,
Davis, Silverberg and Stange [CoDaSiSt17] conjectured that, provided $t\neq 0$
when $g\geq 2$, there exists an explicit constant $c(A,t)\geq 0$ such that, as
$x\rightarrow\infty$,

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

In the case of an abelian variety of dimension $g=1$ and having a trivial
$\overline{\mathbb{Q}}$-endomorphism ring, both the open image assumption and
the equidistribution assumption are known to hold and the asymptotic formula
(2) coincides with the Lang-Trotter Conjecture on Frobenius traces proposed in
[LaTr76]. In the case of an abelian variety of dimension $g\geq 2$, the open
image assumption is known to hold if $g=2,6,$ or odd, and the
$\overline{\mathbb{Q}}$-endomorphism ring of $A$ is trivial (see [Se85],
[Se86]); in contrast, the equidistribution assumption – a special case of a
broad conjecture that generalizes the Sato-Tate Conjecture (see [Se94] and
[KaSa99]) – remains open.

The growth of the function $\pi_{A}(x,t)$ is intimately connected to the
distribution of Frobenius elements in extensions of $\mathbb{Q}$ derived from
$A$. For example, the Chebotarev Density Theorem, applied in extensions of
$\mathbb{Q}$ associated to the division fields of $A$, plays a fundamental
role in the study of $\pi_{A}(x,t)$. Since the best effective versions of this
theorem depend on the assumption of a Generalized Riemann Hypothesis (denoted
GRH, for short), it is natural to investigate the growth of $\pi_{A}(x,t)$
under such an assumption.

Currently, the best results about the growth of $\pi_{A}(x,t)$, obtained under
some GRH, are as follows: provided that $g=1$ and $A$ has a trivial
$\overline{\mathbb{Q}}$-endomorphism ring, it was proven in [MuMuSa88] and
[Zy15] that

(3)
$\pi_{A}(x,t)\ll_{A}\left\\{\begin{array}[]{cl}\frac{x^{1-\frac{1}{5}}}{(\operatorname{log}x)^{1-\frac{4}{5}}}&\text{if
$t\neq 0$,}\\\ \\\
\frac{x^{1-\frac{1}{4}}}{(\operatorname{log}x)^{1-\frac{1}{2}}}&\text{if
$t=0$};\end{array}\right.$

provided that $g\geq 2$ and $\rho_{A}$ has open image in
$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$, it was proven in [CoDaSiSt17]
that, for any $\varepsilon>0$,

(4)
$\pi_{A}(x,t)\ll_{A,\varepsilon}\left\\{\begin{array}[]{cl}x^{1-\frac{1}{2(2g^{2}-g+3)}+\varepsilon}&\text{if
$t\neq\pm 2g,0$,}\\\ \\\ x^{1-\frac{1}{2(2g^{2}+g+1)}+\varepsilon}&\text{if
$t=\pm 2g$,}\end{array}\right.$

and

(5)
$\pi_{A}(x,0)\ll_{A,\varepsilon}\left\\{\begin{array}[]{cl}x^{1-\frac{1}{16}+\varepsilon}&\text{if
$g=2$,}\\\ \\\ x^{1-\frac{1}{2(2g^{2}-g+1)}+\varepsilon}&\text{if $g\geq
3$.}\end{array}\right.$

Our main goal in this paper is to improve upon the above results when $g\geq
2$. Specifically, under a GRH assumption, we prove the following upper bounds
for $\pi_{A}(x,t)$.

######  Theorem 1.

Let $t\in\mathbb{Z}$ and let $A$ be an abelian variety defined over
$\mathbb{Q}$ and of dimension $g$. Assume that, for any sufficiently large
prime $\ell$, the image of the residual modulo $\ell$ Galois representation
$\overline{\rho}_{A,\ell}$ of $A$ is isomorphic to
$\operatorname{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$. Assume the Generalized
Riemann Hypothesis for Dedekind zeta functions. Then, for any sufficiently
large $x$, we have

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

Note that the bounds of Theorem 1 recover (3) when $g=1$ and largely improve
upon (4) - (5) when $g\geq 2$. When $g=2$, Theorem 1 also matches a recent
upper bound presented in [KuKuWe22] which, by [BoCaGePi21], is applicable to a
modular abelian surface defined over $\mathbb{Q}$ and having a trivial
endomorphism ring.

We remark that the assumption made in Theorem 1 about the image of the
residual modulo $\ell$ Galois representation $\overline{\rho}_{A,\ell}$ of $A$
being isomorphic to $\operatorname{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$ is
weaker than, but implied by, the open image assumption of (2). As mentioned
earlier, this latter assumption is known to hold for all abelian varieties
over $\mathbb{Q}$ of dimension $g=1,2,6,$ or odd and having a trivial
$\overline{\mathbb{Q}}$-endomorphism ring. Moreover, it is known to hold for
any abelian $g$-fold that arises as the Jacobian of a hyperelliptic curve
defined by $Y^{2}=f(X)$ for some monic polynomial $f(X)\in\mathbb{Z}[X]$ of
degree $\operatorname{deg}f\in\\{2g+1,2g+2\\}$ for some $g\geq 2$ and having
the property that either the Galois group of $f$ is the permutation group on
$\operatorname{deg}f$ elements, or there exists a rational prime $p$ for which
$f(X)(\operatorname{mod}p)$ has $\operatorname{deg}f-1$ distinct roots over an
algebraic closure of $\mathbb{F}_{p}$, one of which is a double root (see the
results of Hall and Kowalski in [Ha11] and of Zarhin in [Za00]). Consequently,
the strongest assumption of Theorem 1 is that of a GRH.

Our secondary goal in this paper is to improve the bounds of Theorem 1 under
additional hypotheses. It is known that if, in addition to a GRH, one assumes
further hypotheses on the Artin L-functions associated to a given Galois
extension, then one may improve the error term in the effective version of the
Chebotarev Density Theorem pertaining to that extension. The impact of such
improvements on the growth of $\pi_{A}(x,t)$ motivated the bounds (3) of
[MuMuSa88] and [Zy15] when $g=1$, which, in turn, motivated the following
results of Bellaïche [Be16] when $g\geq 2$: assume that, for any sufficiently
large prime $\ell$, the image of the $\ell$-adic Galois representation
$\rho_{A,\ell}$ of $A$ is isomorphic to
$\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell})$; assume GRH for Dedekind zeta
functions; assume Artin’s Holomorphy Conjecture for Artin L-functions; then,
for any $\varepsilon>0$,

(6)
$\pi_{A}(x,t)\ll_{A,\varepsilon}\left\\{\begin{array}[]{cl}x^{1-\frac{2}{2g^{2}-g+4}+\varepsilon}&\text{if
$t\neq 0$,}\\\ \\\ x^{1-\frac{2}{2g^{2}+g+3}+\varepsilon}&\text{if
$t=0$.}\end{array}\right.$

In [MuMuWo18], Murty, Murty, and Wong improved the error term in the effective
version of the Chebotarev Density Theorem even further by assuming, in
addition to GRH and Artin’s Holomorphy Conjecture, a Pair Correlation
Conjecture for the Artin L-functions associated to the given Galois extension.
As an application, they obtained the following improvement to (3): assume that
$g=1$ and that $A$ has a trivial $\overline{\mathbb{Q}}$-endomorphism ring;
assume GRH for Dedekind zeta functions; assume Artin’s Holomorphy Conjecture
and a Pair Correlation Conjecture for Artin L-functions; then

(7)
$\pi_{A}(x,t)\ll_{A}\left\\{\begin{array}[]{cl}x^{1-\frac{1}{3}}(\operatorname{log}x)^{\frac{1}{3}}&\text{if
$t\neq 0$,}\\\ \\\ x^{1-\frac{1}{2}}\operatorname{log}x&\text{if
$t=0$.}\end{array}\right.$

In this paper, we generalize (7) to the case of an abelian variety of higher
dimensions and obtain the following upper bounds for $\pi_{A}(x,t)$.

######  Theorem 2.

Let $t\in\mathbb{Z}$ and let $A$ be an abelian variety defined over
$\mathbb{Q}$ and of dimension $g$. Assume that, for any sufficiently large
prime $\ell$, the image of the residual modulo $\ell$ Galois representation
$\overline{\rho}_{A,\ell}$ of $A$ is isomorphic to
$\operatorname{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$. Assume the Generalized
Riemann Hypothesis for Dedekind zeta functions, Artin’s Holomorphy Conjecture
for Artin L-functions, and a Pair Correlation Conjecture for Artin
L-functions. Then, for any sufficiently large $x$, we have

$\pi_{A}(x,t)\ll_{A}\left\\{\begin{array}[]{cl}\frac{x^{1-\frac{1}{g+2}}}{(\operatorname{log}x)^{1-\frac{4}{g+2}}}&\text{if
$t\neq 0$,}\\\ \\\
\frac{x^{1-\frac{1}{g+1}}}{(\operatorname{log}x)^{1-\frac{4}{g+1}}}&\text{if
$t=0$.}\end{array}\right.$

By using Theorems 1 and 2, we obtain the following non-trivial lower bounds
for $|a_{1,p}(A)|$ for almost all $p$.

######  Theorem 3.

Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$.
Assume that, for any sufficiently large prime $\ell$, the image of the
residual modulo $\ell$ Galois representation $\overline{\rho}_{A,\ell}$ of $A$
is isomorphic to $\operatorname{GSp}_{2g}(\mathbb{Z}/\ell\mathbb{Z})$.

  1. (i)

Assume the Generalized Riemann Hypothesis for Dedekind zeta functions. Then,
for any $\varepsilon>0$, the inequality

$|a_{1,p}(A)|>\frac{p^{\frac{1}{2g^{2}+g+1}}}{(\operatorname{log}p)^{\varepsilon}}$

holds for a set of primes $p$ of density 1.

  2. (ii)

Assume the Generalized Riemann Hypothesis for Dedekind zeta functions, Artin’s
Holomorphy Conjecture for Artin L-functions, and a Pair Correlation Conjecture
for Artin L-functions. Then, for any $\varepsilon>0$, the inequality

$|a_{1,p}(A)|>p^{\frac{1}{g+2}-\varepsilon}$

holds for a set of primes $p$ of density 1.

The motivation for Theorem 3 comes from a conjecture of Atkin and Serre on a
lower bound for $\tau(p)$ for the Ramanujan $\tau$-function (see [Se76]). To
see this, recall that, when $g=1$, the integer $a_{1,p}(A)$ coincides with the
$p$-th Fourier coefficient of the weight two normalized newform, of level
$N_{A}$, associated to $A$ under the Shimura-Taniyama-Weil Conjecture. Using
this point of view, Theorem 3 relates to results on the growth of the $p$-th
Fourier coefficients of newforms pursued in [MuMu84] and [MuMuSa88], which had
been motivated by the aforementioned conjecture of Atkin and Serre.

Before concluding the introduction, let us comment briefly on the proofs of
our first two main results. The proofs of Theorems 1 \- 2 have their roots in
the observation that if $a_{1,p}(A)=t$, then $a_{1,p}(A)\equiv
t(\operatorname{mod}\ell)$ for any prime $\ell$, and in the interpretation of
$a_{1,p}(A)(\operatorname{mod}\ell)$ as the trace of the image of
$\operatorname{Frob}_{p}$ under the residual modulo $\ell$ Galois
representation of $A$. An application of some version of the Chebotarev
Density Theorem should then lead to some non-trivial upper bound for
$\pi_{A}(x,t)$. For example, when $g=1$ and $A$ has a trivial
$\overline{\mathbb{Q}}$-endomorphism ring, by assuming GRH and by using the
effective version of the Chebotarev Density Theorem of Lagarias and Odlyzko
[LaOd77], one obtains the upper bound
$\pi_{A}(x,t)\ll_{A,\varepsilon}x^{1-\frac{1}{8}+\varepsilon}$. While this mod
$\ell$ approach easily leads to a non-trivial upper bound for $\pi_{A}(x,t)$
when $g=1$, refining it to improve upon this bound or generalizing it to
obtain a non-trivial upper bound for $\pi_{A}(x,t)$ when $g\geq 2$ requires
further tools and new ideas.

One strategy of refining the mod $\ell$ approach is to consider,
simultaneously, $\operatorname{mod}\ell^{k}$ congruences, for some fixed prime
$\ell$ and varying positive integers $k\geq 1$. For example, in [Se81], Serre
considered the case $g=1$ and $A$ has a trivial
$\overline{\mathbb{Q}}$-endomorphism ring, and related the problem of
estimating $\pi_{A}(x,t)$ to a prime counting problem in the setting of an
infinite Galois extension of $\mathbb{Q}$ having the $\ell$-adic Lie group
$\operatorname{GL}_{2}(\mathbb{Z}_{\ell})$ as its Galois group, where $\ell$
is some sufficiently large prime defined in terms of $x$. Then, he used an
$\ell$-adic version of the Chebotarev Density Theorem to estimate, from above,
the emerging set of primes. In particular, by assuming GRH, Serre obtained the
upper bound $\pi_{A}(x,t)\ll_{A,\varepsilon}x^{1-\frac{1}{6}+\varepsilon}$.
This $\ell$-adic approach was generalized to $g\geq 2$ by Cojocaru, Davis,
Silverberg, and Stange in [CoDaSiSt17], who proved the upper bounds (4) and
(5), under GRH.

Another strategy of refining the mod $\ell$ approach is to improve the error
term in the effective version of the Chebotarev Density Theorem. For example,
in [MuMuSa88], by assuming, in addition to GRH, Artin’s Holomorphy Conjecture
(denoted AHC, for short), Murty, Murty, and Saradha proved a refined version
of the effective Chebotarev Density Theorem of [LaOd77]. Considering the case
$g=1$ and $A$ has a trivial $\overline{\mathbb{Q}}$-endomorphism ring, a
direct application of this theorem in the mod $\ell$ approach would lead to
the upper bound
$\pi_{A}(x,t)\ll_{A,\varepsilon}x^{1-\frac{1}{5}+\varepsilon}$, under GRH and
AHC. To circumvent the assumption of AHC and still obtain this upper bound,
Murty, Murty, and Saradha related the problem of estimating $\pi_{A}(x,t)$ to
a prime counting problem in the setting of a particular subextension $L/K$ of
number fields, having its Galois group isomorphic to an abelian and
sufficiently large quotient of $\operatorname{GL}_{2}(\mathbb{F}_{\ell})$,
where $\ell$ is some sufficiently large prime defined in terms of $x$. Then
they estimated the emerging set of primes by appealing to their improved
effective version of the Chebotarev Density Theorem, assuming only GRH, since
AHC is known in an abelian setting. Overall, their refined mod $\ell$
approach, together with an improvement of [Zy15], led to the upper bounds (3).
In [Be16], Bellaïche refined the aforementioned improved version of the
effective Chebotarev Density Theorem of Murty, Murty, and Saradha in certain
cases and succeeded in generalizing the proof of (3) in the case $g=2$ and $A$
has a trivial $\overline{\mathbb{Q}}$-endomorphism ring, proving, under both
GRH and AHC, that
$\pi_{A}(x,0)\ll_{A}\frac{x^{1-\frac{1}{10}}}{(\operatorname{log}x)^{1-\frac{2}{5}}}$.
Unlike the final bound of Murty, Murty, and Saradha, which only assumes GRH,
that of Bellaïche assumes both GRH and AHC.

One could also relate mod $\ell$ approach of estimating $\pi_{A}(x,t)$ to a
sieve, such as the large sieve, and then use an improved version of the
effective Chebotarev Density Theorem to execute the sieve application. This
approach was elaborated on in [Be16], leading to the improved upper bounds
(6), which hold for $g\geq 1$ and which assume both GRH and AHC.

In the case $g=1$ and $A$ has a trivial $\overline{\mathbb{Q}}$-endomorphism
ring, the mod $\ell$ approach witnessed yet another refinement in [MuMuWo18].
Therein, by assuming, in addition to GRH and AHC, a Pair Correlation
Conjecture (denoted PCC, for short), Murty, Murty, and Wong improved the
effective Chebotarev Density Theorem of [MuMuSa88]. Then, they used this
theorem directly in the mod $\ell$ approach to prove the upper bounds (7),
under GRH, AHC, and PCC. Via this strategy, in order to take advantage of the
power of the PCC assumption, one cannot circumvent the assumption of AHC by
working in an abelian extension of number fields.

Our proof of Theorem 1 generalizes the mod $\ell$ approach of [MuMuSa88],
under GRH, to the case $g\geq 2$, overcoming prior obstacles faced in [Be16]
and [CoDaSiSt17]. The main challenges were those of unraveling, for some
sufficiently large prime $\ell$, a suitable distinguished pair of subfields of
a finite Galois extension of $\mathbb{Q}$ having its Galois group isomorphic
to an abelian and sufficiently large quotient of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$, and of finding a suitable
conjugacy class in this group for which the effective version of the
Chebotarev Density Theorem of [MuMuSa88] could be applied successfully.

Our proof of Theorem 2 generalizes the mod $\ell$ approach of [MuMuWo18],
under GRH, AHC, and PCC, to the case $g\geq 2$, overcoming obstacles faced in
[CoDaSiSt17]. The main challenge was that of unraveling, for some sufficiently
large prime $\ell$, a suitable conjugacy class in
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ for which the effective version
of the Chebotarev Density Theorem of [MuMuWo18] could be applied successfully.

The statements of the Generalized Riemann Hypothesis, Artin’s Holomorphy
Conjecture, and the Pair Correlation Conjecture, as well as the main notation
used in the paper, are given in Section 2. The effective versions of the
Chebotarev Density Theorem needed for our proofs are presented in Section 3.
The distinguished subgroups and conjugacy classes of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ needed in our approaches are
discussed in Sections 4 - 5. The proofs of Theorems 1 \- 3 are given in
Section 6, with Propositions 18 \- 19 playing the role of crucial ingredients.
Following these proofs, brief remarks about possible other approaches towards
obtaining conditional bounds for $\pi_{A}(x,t)$ are included in Section 7.

Acknowledgments. We thank Professor Kiran S. Kedlaya and Professor Peter
Sarnak for stimulating discussions related to this work.

##  2\. Notation

Throughout the paper, we use the following notation.

$\bullet$ Given a finite set $S$, we denote its cardinality by $\\#S$.

$\bullet$ 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 common domain of $h_{1}$
and $h_{2}$; 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
$h_{1}\sim h_{2}$ 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
denote by $\pi(x)$ the number of primes $p\leq x$ and recall that, by the
Prime Number Theorem, $\pi(x)\sim\frac{x}{\operatorname{log}x}$.

$\bullet$ Given a positive integer $m$, we denote by $\mathbb{Z}/m\mathbb{Z}$
the ring of integers modulo $m$. When $m$ is a prime $\ell$, we denote
$\mathbb{Z}/\ell\mathbb{Z}$ by $\mathbb{F}_{\ell}$ to emphasize its field
structure. For an integer $a$, we denote by $a(\operatorname{mod}\ell)$ its
residue class modulo $\ell$.

$\bullet$ Given a 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 recall that there exists a ring isomorphism
$\hat{\mathbb{Z}}\simeq\displaystyle\prod_{\ell}\mathbb{Z}_{\ell}$.

$\bullet$ Given a number field $K$, we denote by ${\mathcal{O}}_{K}$ its ring
of integers, by ${\sum}_{K}$ the set of non-zero prime ideals of
${\mathcal{O}}_{K}$, by $[K:\mathbb{Q}]$ the degree of $K$ over $\mathbb{Q}$,
by $d_{K}\in\mathbb{Z}\backslash\\{0\\}$ 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 $K/\mathbb{Q}$. 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 we refer to GRH as the
Riemann Hypothesis (RH).

$\bullet$ Given a finite Galois extension $L/K$ of number fields and given an
irreducible character $\chi$ of the Galois group of $L/K$, we denote by
$\mathfrak{f}(\chi)\unlhd{\mathcal{O}}_{K}$ the global Artin conductor of
$\chi$, by
$A_{\chi}:=|d_{L}|^{\chi(1)}N_{K/\mathbb{Q}}(\mathfrak{f}(\chi))\in\mathbb{Z}$
the conductor of $\chi$, and by ${\mathcal{A}}_{\chi}(T)$ the function of a
positive real variable $T>3$ defined by the relation

$\operatorname{log}{\mathcal{A}}_{\chi}(T)=\operatorname{log}A_{\chi}+\chi(1)[K:\mathbb{Q}]\operatorname{log}T.$

$\bullet$ Given a finite Galois extension $L/K$ of number fields, we say that
it satisfies Artin’s Holomorphy Conjecture (AHC) if, for any irreducible
character $\chi$ of the Galois group of $L/K$, the Artin L-function
$L(s,\chi,L/K)$ extends to a function that is analytic on the whole
$\mathbb{C}$, except at $s=1$ when $\chi=1$. We recall that, if we assume GRH
for $L$ and AHC for $L/K$, then, given any irreducible character $\chi$ of the
Galois group of $L/K$, and given any non-trivial zero $\rho$ of
$L(s,\chi,L/K)$, the real part $\operatorname{Re}\rho$ of $\rho$ satisfies
$\operatorname{Re}\rho=\frac{1}{2}$. In this case, we write
$\rho=\frac{1}{2}+i\gamma$, where $\gamma$ denotes the imaginary part of
$\rho$.

$\bullet$ Given a finite Galois extension $L/K$ of number fields, let us
assume GRH for $L$ and AHC for $L/K$. For an irreducible character $\chi$ of
the Galois group of $L/K$ and an arbitrary $T>0$, we define the pair
correlation function of $L(s,\chi,L/K)$ by

${\mathcal{P}}_{T}(X,\chi):=\displaystyle\sum_{-T\leq\gamma_{1}\leq
T}\displaystyle\sum_{-T\leq\gamma_{2}\leq
T}w(\gamma_{1}-\gamma_{2})e((\gamma_{1}-\gamma_{2})X),$

where $\gamma_{1}$ and $\gamma_{2}$ range over all the imaginary parts of the
non-trivial zeroes $\rho=\frac{1}{2}+\gamma$ of $L(s,\chi,L/K)$, counted with
multiplicity, and where, for an arbitrary real number $u$,
$e(u):=\operatorname{exp}(2\pi iu)$ and $w(u):=\frac{4}{4+u^{2}}$. We say that
the extension $L/K$ satisfies the Pair Correlation Conjecture (PCC) if, for
any irreducible character $\chi$ of the Galois group of $L/K$ and for any
$A>0$ and $T>3$, provided $0\leq Y\leq
A\chi(1)[K:\mathbb{Q}]\operatorname{log}T$, we have

${\mathcal{P}}_{T}(Y,\chi)\ll_{A}\chi(1)^{-1}T\operatorname{log}{\mathcal{A}}_{\chi}(T).$

$\bullet$ Given a field $F$, we denote by $\operatorname{char}{F}$ its field
characteristic, by $\overline{F}$ a fixed algebraic closure of $F$, by
$F^{\text{sep}}$ a fixed separable closure of $F$ in $\overline{F}$, and by
$\operatorname{Gal}(F^{\text{sep}}/F)$ the absolute Galois group of $F$.

$\bullet$ Given a non-zero unitary commutative ring $R$, we denote by
$R^{\times}$ its group of multiplicative units.

$\bullet$ Given a non-zero unitary commutative ring $R$ and a free module
${\mathcal{M}}$ over $R$, of finite rank $n\geq 1$, endowed with a non-
degenerate alternating bilinear form
$e:{\mathcal{M}}\times{\mathcal{M}}\rightarrow R$, we denote by
$\operatorname{GSp}({\mathcal{M}},e)$ the group of simplectic similitudes of
${\mathcal{M}}$ with respect to $e$, that is, the group of $R$-automorphisms
$\sigma\in\operatorname{Aut}({\mathcal{M}})$ such that there exists $\mu\in
R^{\times}$ with the property that $e(\sigma(v),\sigma(w))=\mu\ e(v,w)$ for
all $v,w\in{\mathcal{M}}$. Since ${\mathcal{M}}$ has a unique (up to
isomorphism) non-degenerate alternating bilinear form, we may write
$\operatorname{GSp}_{n}(R)$ for $\operatorname{GSp}({\mathcal{M}},e)$.

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

$\text{diag}(a_{1},\ldots,a_{n}):=\begin{pmatrix}a_{1}&&\\\ &\ddots&\\\
&&a_{n}\end{pmatrix}\in M_{n}(R).$

When $n=2g$ for some integer $g\geq 1$, we define

$J_{2g}:=\begin{pmatrix}0&I_{g}\\\ -I_{g}&0\end{pmatrix}\in M_{2g}(R).$

The general symplectic group over $R$ may then be described as

$\operatorname{GSp}_{2g}(R)=\left\\{M\in\operatorname{GL}_{2g}(R):M^{t}J_{2g}M=\mu
J_{2g}\ \text{for some}\ \mu\in R^{\times}\right\\}.$

Associated to $\operatorname{GSp}_{2g}(R)$, we have the character
$\operatorname{GSp}_{2g}(R)\longrightarrow R^{\times},$ $M\mapsto\mu$. The
scalar $\mu$ corresponding to $M$ is called the multiplicator of $M$. We write
the group of scalar matrices in $\operatorname{GSp}_{2g}(R)$ as
$\Lambda_{2g}(R)$ and note that it is isomorphic to $R^{\times}$. We define
the projective general symplectic group $\operatorname{PGSp}_{2g}(R)$ as the
quotient $\operatorname{GSp}_{2g}(R)/\Lambda_{2g}(R)$.

##  3\. Frobenius distributions in number field extensions

In this section, we record results on the theme of Frobenius distributions in
a number field Galois extension; these results will play a crucial role in the
proofs of Theorems 1 \- 3.

Let $L/K$ be a finite Galois extension of number fields. Denote by
$\operatorname{Gal}(L/K)$ the Galois group of $L/K$ and keep the number field
notation introduced in Section 2. Denote by $\operatorname{Gal}(L/K)^{\\#}$
the set of conjugacy classes of $\operatorname{Gal}(L/K)$. Additionally,
denote by $[L:K]$ the degree of $L/K$ and by $\operatorname{disc}(L/K)$ the
discriminant ideal of $L/K$.

We define

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

and we recall from [Se81, Prop. 5, p. 129] that

(8)
$\operatorname{log}\left|N_{K/\mathbb{Q}}(\operatorname{disc}(L/K))\right|\leq([L:\mathbb{Q}]-[K:\mathbb{Q}])\left(\displaystyle\sum_{p\in
P(L/K)}\operatorname{log}p\right)+[L:\mathbb{Q}]\operatorname{log}[L:K].$

For a place $\wp\in\Sigma_{L}$, we denote by ${\mathcal{D}}_{\wp}$ its
decomposition group in $L/K$, by ${\mathcal{I}}_{\wp}$ its inertia group in
$L/K$, and by
$\left(\frac{L/K}{\wp}\right)\in{\mathcal{D}}_{\wp}/{\mathcal{I}}_{\wp}$ its
Frobenius element in $L/K$.

For a place $\mathfrak{p}\in\Sigma_{K}$, we set

$\left(\frac{L/K}{\mathfrak{p}}\right):=\left\\{\left(\frac{L/K}{\wp}\right):\wp\in\Sigma_{L},\wp\
\text{lies over}\ \mathfrak{p}\right\\}.$

For a non-empty set ${\mathcal{C}}\subseteq\operatorname{Gal}(L/K)$, stable
under conjugation by elements of $\operatorname{Gal}(L/K)$, we denote by
$\delta_{\mathcal{C}}:\operatorname{Gal}(L/K)\longrightarrow\\{0,1\\}$ its
characteristic function and, for an arbitrary place
$\mathfrak{p}\in\Sigma_{K}$ and an arbitrary integer $m\geq 1$, we set

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

where $\wp\in\Sigma_{L}$ is an arbitrary place above $\mathfrak{p}$, whose
choice leaves the above definition unchanged. For a real number $x\geq 2$, we
set

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

and we recall from [Se81, Proposition 7, p. 138] and [Zy15, Lemma 2.7, p. 8]
that

(9)
$\tilde{\pi}_{\mathcal{C}}(x,L/K)=\pi_{\mathcal{C}}(x,L/K)+\operatorname{O}\left([K:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(L/K)\right)\right).$

The variations of the effective version of the Chebotarev Density Theorem of
[LaOd77] that relate to the proofs of our main theorems are as follows.

######  Theorem 4.

Let $L/K$ be a Galois extension of number fields. Let
$\emptyset\neq\mathcal{C}\subseteq\operatorname{Gal}(L/K)$ be a subset stable
under conjugation by elements of $\operatorname{Gal}(L/K)$.

  1. (i)

Assume GRH for the Dedekind zeta function of $L$. Then

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

  2. (ii)

Assume GRH for the Dedekind zeta function of $L$ and AHC for the extension
$L/K$. Then

$\pi_{\mathcal{C}}(x,L/K)=\frac{\\#\mathcal{C}}{[L:K]}\pi(x)+\operatorname{O}\left((\\#\mathcal{C})^{\frac{1}{2}}\
x^{\frac{1}{2}}\ [K:\mathbb{Q}]\operatorname{log}(M(L/K)x)\right).$

  3. (iii)

Assume GRH for the Dedekind zeta function of $L$, and AHC and PCC for the
extension $L/K$. Then

$\pi_{\mathcal{C}}(x,L/K)=\frac{\\#\mathcal{C}}{[L:K]}\pi(x)+\operatorname{O}\left((\\#\mathcal{C})^{\frac{1}{2}}\left(\frac{\\#\operatorname{Gal}(L/K)^{\\#}}{[L:K]}\right)^{\frac{1}{2}}x^{\frac{1}{2}}\
[K:\mathbb{Q}]^{\frac{1}{2}}\operatorname{log}(M(L/K)x)\right).$

###### Proof.

Part (i) is [Se81, Théorème 4, p. 133]. Part (ii) is [MuMuSa88, Corollary 3.7,
p. 265]. Part (iii) is [MuMuWo18, Theorem 1.2, p. 402]. ∎

In the proofs of our main results, we do not always need the full strength of
the effective asymptotic formulae in Theorem 4. For example, we do not use
part (ii) of this theorem, which we included for clarity and for comparison
with parts (i) and (iii). Instead of part (ii), we will use Theorem 5, stated
below. We will also use Theorem 7, stated below, which is a consequence of
part (iii).

######  Theorem 5.

Let $L/K$ be a Galois extension of number fields. Let
$\emptyset\neq\mathcal{C}\subseteq\operatorname{Gal}(L/K)$ be a subset stable
under conjugation by elements of $\operatorname{Gal}(L/K)$. Assume GRH for the
Dedekind zeta function of $L$ and AHC for the extension $L/K$. Then

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

###### Proof.

This is [Zy15, Theorem 2.3, p. 5]. ∎

######  Theorem 6.

Let $L/K$ be a Galois extension of number fields. Let
$\emptyset\neq\mathcal{C}_{1}\subseteq\mathcal{C}_{2}\subseteq\operatorname{Gal}(L/K)$
be subsets stable under conjugation by elements of $\operatorname{Gal}(L/K)$.
Let $H\leq\operatorname{Gal}(L/K)$ be a subgroup of $\operatorname{Gal}(L/K)$
and let $N\unlhd H$ be a normal subgroup of $H$. Assume that:

  1. (i)

every element of $\mathcal{C}_{1}$ is conjugate to some element of $H$;

  2. (ii)

$N(\mathcal{C}_{2}\cap H)\subseteq\mathcal{C}_{2}\cap H$;

  3. (iii)

$H/N$ is an abelian group;

  4. (iv)

GRH holds for the Dedekind zeta function of $L^{N}$.

Then

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

where $\widehat{{\mathcal{C}_{2}}\cap H}$ denotes the image of
$\mathcal{C}_{2}\cap H$ in $\frac{H}{N}$ under the canonical projection
$H\twoheadrightarrow\frac{H}{N}$.

###### Proof.

Using assumption (i) and [Zy15, Lemma 2.6 (i)], we obtain that
${\tilde{\pi}}_{{\mathcal{C}}_{1}}(x,L/K)\leq\tilde{\pi}_{{\mathcal{C}}_{1}\cap
H}\left(x,L/L^{H}\right)$. Since
${\mathcal{C}}_{1}\subseteq{\mathcal{C}}_{2}$, we obviously have
$\tilde{\pi}_{{\mathcal{C}}_{1}\cap
H}\left(x,L/L^{H}\right)\leq\tilde{\pi}_{{\mathcal{C}}_{2}\cap
H}\left(x,L/L^{H}\right)$. Using assumption (ii), [Zy15, Lemma 2.6 (ii)], and
asymptotic formula (9), we obtain that

$\tilde{\pi}_{{\mathcal{C}}_{2}\cap
H}\left(x,L/L^{H}\right)=\tilde{\pi}_{\widehat{{\mathcal{C}}_{2}\cap
H}}\left(x,L^{N}/L^{H}\right)=\pi_{\widehat{{\mathcal{C}}_{2}\cap
H}}\left(x,L^{N}/L^{H}\right)+\operatorname{O}\left([L^{H}:\mathbb{Q}]\left(\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M\left(L^{N}/L^{H}\right)\right)\right).$

Using assumption (iii) and [Ar27], we deduce that AHC holds for the extension
$L^{N}/L^{H}$. Then, using assumption (iv) and Theorem 5, we deduce that

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

Finally, putting all these observations together and using (9) again to relate
$\pi_{{\mathcal{C}}_{1}}(x,L/K)$ to $\tilde{\pi}_{{\mathcal{C}}_{1}}(x,L/K)$,
we infer that

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

∎

######  Theorem 7.

Let $L/K$ be a Galois extension of number fields. Let
$\emptyset\neq\mathcal{C}\subseteq\operatorname{Gal}(L/K)$ be a subset stable
under conjugation by elements of $\operatorname{Gal}(L/K)$. Let
$H\leq\operatorname{Gal}(L/K)$ be a subgroup of $\operatorname{Gal}(L/K)$ and
let $N\unlhd H$ be a normal subgroup of $H$. Assume that:

  1. (i)

every element of $\mathcal{C}$ is conjugate to some element of $H$;

  2. (ii)

$N(\mathcal{C}\cap H)\subseteq\mathcal{C}\cap H$;

  3. (iii)

GRH holds for the Dedekind zeta function of $L^{N}$;

  4. (iv)

AHC and PCC hold for the number field extension $L^{N}/L^{H}$.

Then

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

where $\widehat{{\mathcal{C}}\cap H}$ denotes the image of $\mathcal{C}\cap H$
in $\frac{H}{N}$ under the canonical projection
$H\twoheadrightarrow\frac{H}{N}$.

###### Proof.

Similarly to the proof of the previous theorem, we obtain that

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

Then, using assumptions (iii)-(iv) and part (iii) of Theorem 4, we deduce that

$\pi_{\widehat{{\mathcal{C}}\cap
H}}(x,L^{N}/L^{H})=\frac{\\#(\widehat{{\mathcal{C}}\cap
H})}{[H:N]}\pi(x)+\operatorname{O}\left(\\#(\widehat{{\mathcal{C}}\cap
H})^{\frac{1}{2}}\
\left(\frac{\\#\operatorname{Gal}(L^{N}/L^{H})^{\\#}}{[L^{N}:L^{H}]}\right)^{\frac{1}{2}}\
x^{\frac{1}{2}}\
[L^{H}:\mathbb{Q}]^{\frac{1}{2}}\operatorname{log}\left(M(L^{N}/L^{H})x\right)\right).$

Putting all these observations together, we infer that

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

∎

In Section 6, we will apply Theorems 4, 6, and 7 to number field extensions
associated to an abelian variety defined over $\mathbb{Q}$.

##  4\. Distinguished subgroups in the general symplectic groups over a finite
prime field

In this section, we fix an arbitrary integer $g\geq 1$ and an arbitrary
rational prime $\ell$, and turn our attention to particular subgroups of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$. As we will see in Section 6,
these subgroups give rise to particular subextensions of the $\ell$-division
field of an abelian variety defined over $\mathbb{Q}$, of dimension $g$.

We will view the elements of $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$
using the following block matrix description

$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})=\left\\{\begin{array}[]{lll}&&-C^{t}A+A^{t}C=0\\\
M=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\operatorname{GL}_{2g}(\mathbb{F}_{\ell}):&A,B,C,D\in
M_{g}(\mathbb{F}_{\ell}),&-C^{t}B+A^{t}D=\mu I_{g}\ \text{for some}\
\mu\in\mathbb{F}_{\ell}^{\times}\\\ &&-D^{t}B+B^{t}D=0\end{array}\right\\},$

wher the element $\mu$ above is the multiplicator of the matrix $M$.

Before introducing the subgroups of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ needed in the proofs of our main
results, we introduce several subgroups of
$\operatorname{GL}_{g}(\mathbb{F}_{\ell})$, as follows:

$\displaystyle\mathcal{B}_{g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}a_{11}&\cdots&a_{1g}\\\ &\ddots&\vdots\\\
&&a_{gg}\end{pmatrix}\in\operatorname{GL}_{g}(\mathbb{F}_{\ell}):a_{ii}\in\mathbb{F}_{\ell}^{\times}\
\forall 1\leq i\leq g,a_{ij}\in\mathbb{F}_{\ell}\ \forall 1\leq i<j\leq
g\right\\};$ $\displaystyle\mathcal{U}_{g}(\mathbb{F}_{\ell})$
$\displaystyle:=$ $\displaystyle\left\\{\begin{pmatrix}1&\cdots&a_{1g}\\\
&\ddots&\vdots\\\
&&1\end{pmatrix}\in\operatorname{GL}_{g}(\mathbb{F}_{\ell}):a_{ij}\in\mathbb{F}_{\ell}\
\forall 1\leq i<j\leq g\right\\};$
$\displaystyle\mathcal{U}^{\prime}_{g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}\lambda&\cdots&a_{1g}\\\
&\ddots&\vdots\\\
&&\lambda\end{pmatrix}\in\operatorname{GL}_{g}(\mathbb{F}_{\ell}):\lambda\in\mathbb{F}_{\ell}^{\times},a_{ij}\in\mathbb{F}_{\ell}\
\forall 1\leq i<j\leq g\right\\};$
$\displaystyle\mathcal{T}_{g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}a_{11}&&\\\ &\ddots&\\\
&&a_{gg}\end{pmatrix}\in\operatorname{GL}_{g}(\mathbb{F}_{\ell}):a_{ii}\in\mathbb{F}_{\ell}^{\times}\
\forall 1\leq i\leq g\right\\}.$

For a matrix $A=\begin{pmatrix}\lambda&\cdots&a_{1g}\\\ &\ddots&\vdots\\\
&&\lambda\end{pmatrix}\in\mathcal{U}^{\prime}_{g}(\mathbb{F}_{\ell})$, we set
$d(A):=\lambda.$

Now we introduce four distinguished subgroups of
$\operatorname{GL}_{2g}(\mathbb{F}_{\ell})$:

$\displaystyle B_{2g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}A&\mu^{-1}AS\\\
0&\mu(A^{t})^{-1}\end{pmatrix}\in\operatorname{GL}_{2g}(\mathbb{F}_{\ell}):A\in\mathcal{B}_{g}(\mathbb{F}_{\ell}),\mu\in\mathbb{F}_{\ell}^{\times},S\in
M_{g}(\mathbb{F}_{\ell})\ \text{symmetric}\right\\};$ $\displaystyle
U_{2g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}A&AS\\\ 0&(A^{t})^{-1}\\\
\end{pmatrix}\in\operatorname{GL}_{2g}(\mathbb{F}_{\ell}):A\in\mathcal{U}_{g}(\mathbb{F}_{\ell}),S\in
M_{g}(\mathbb{F}_{\ell})\ \text{symmetric}\right\\};$ $\displaystyle
U^{\prime}_{2g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}A&\mu^{-1}AS\\\
0&\mu(A^{t})^{-1}\end{pmatrix}\in\operatorname{GL}_{2g}(\mathbb{F}_{\ell}):A\in\mathcal{U}^{\prime}_{g}(\mathbb{F}_{\ell}),\mu=d(A)^{2},S\in
M_{g}(\mathbb{F}_{\ell})\ \text{symmetric}\right\\};$ $\displaystyle
T_{2g}(\mathbb{F}_{\ell})$ $\displaystyle:=$
$\displaystyle\left\\{\begin{pmatrix}A&0\\\
0&\mu(A^{t})^{-1}\end{pmatrix}\in\operatorname{GL}_{2g}(\mathbb{F}_{\ell}):A\in\mathcal{T}_{g}(\mathbb{F}_{\ell}),\mu\in\mathbb{F}_{\ell}^{\times}\right\\}.$

For the rest of the section, we focus on properties of these subgroups that
will be used in the proofs of Theorems 1 \- 3.

######  Proposition 8.

  1. (i)

$B_{2g}(\mathbb{F}_{\ell})$ is a subgroup of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

  2. (ii)

$U_{2g}(\mathbb{F}_{\ell})$ is a normal subgroup of
$B_{2g}(\mathbb{F}_{\ell})$.

  3. (iii)

$U^{\prime}_{2g}(\mathbb{F}_{\ell})$ is a normal subgroup of
$B_{2g}(\mathbb{F}_{\ell})$.

###### Proof.

(i) Let

$M=\begin{pmatrix}A^{\prime}&\mu^{-1}A^{\prime}S\\\ 0&\mu(A^{\prime
t})^{-1}\end{pmatrix}\in B_{2g}(\mathbb{F}_{\ell}).$

Using the block matrix description of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ and taking

$A=A^{\prime},\ B=\mu^{-1}A^{\prime}S,\ C=0,\ D=\mu(A^{\prime t})^{-1},$

we can verify that

$-C^{t}A+A^{t}C=0,\ -C^{t}B+A^{t}D=\mu I_{g},\ -D^{t}B+B^{t}D=0.$

This shows that $M\in\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

Note that for any matrices

$M_{1}=\begin{pmatrix}A_{1}&\mu_{1}^{-1}A_{1}S_{1}\\\
0&\mu_{1}(A_{1}^{t})^{-1}\end{pmatrix},\
M_{2}=\begin{pmatrix}A_{2}&\mu_{2}^{-1}A_{2}S_{2}\\\
0&\mu_{2}(A_{2}^{t})^{-1}\end{pmatrix}\in B_{2g}(\mathbb{F}_{\ell}),$

we have

$\displaystyle M_{1}M_{2}^{-1}$
$\displaystyle=\begin{pmatrix}A_{1}&\mu_{1}^{-1}A_{1}S_{1}\\\
0&\mu_{1}(A_{1}^{t})^{-1}\end{pmatrix}\begin{pmatrix}A_{2}^{-1}&-\mu_{2}^{-2}S_{2}A_{2}^{t}\\\
0&\mu_{2}^{-1}A_{2}^{t}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}A_{1}A_{2}^{-1}&-\mu_{2}^{-2}A_{1}S_{2}A_{2}^{t}+\mu_{1}^{-1}\mu_{2}^{-1}A_{1}SA_{2}^{t}\\\
0&\mu_{1}\mu_{2}^{-1}((A_{1}A_{2}^{-1})^{t})^{-1}\end{pmatrix}.$

Then, by setting

$\mu_{3}:=\mu_{1}\mu_{2}^{-1},\ A_{3}:=A_{1}A_{2}^{-1},\
S_{3}:=(-\mu_{1}\mu_{2}^{-3}+\mu_{2}^{-2})A_{2}S_{2}A_{2}^{t},$

we see that

$\displaystyle M_{1}M_{2}^{-1}=\begin{pmatrix}A_{3}&\mu_{3}^{-1}A_{3}S_{3}\\\
0&\mu_{3}(A_{3}^{t})^{-1}\end{pmatrix}\in B_{2g}(\mathbb{F}_{\ell}).$

This shows that $B_{2g}(\mathbb{F}_{\ell})$ is a group, hence a subgroup of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

(ii), (iii) It follows from an argument similar to the one used in the proof
of (i) that both $U_{2g}(\mathbb{F}_{\ell})$ and
$U^{\prime}_{2g}(\mathbb{F}_{\ell})$ are subgroups of
$B_{2g}(\mathbb{F}_{\ell})$. Note that, for any matrices

$M_{1}=\begin{pmatrix}A_{1}&\mu^{-1}A_{1}S_{1}\\\
0&\mu(A_{1}^{t})^{-1}\end{pmatrix}\in B_{2g}(\mathbb{F}_{\ell}),\quad
M_{2}=\begin{pmatrix}A_{2}&A_{2}S_{2}\\\ 0&(A_{2}^{t})^{-1}\\\
\end{pmatrix}\in U_{2g}(\mathbb{F}_{\ell}),$

we have

$\displaystyle M_{1}^{-1}M_{2}M_{1}$
$\displaystyle=\begin{pmatrix}A_{1}^{-1}&-\mu_{1}^{-2}S_{1}A_{1}^{t}\\\
0&\mu_{1}^{-1}A_{1}^{t}\end{pmatrix}\begin{pmatrix}A_{2}&A_{2}S_{2}\\\
0&(A_{2}^{t})^{-1}\\\
\end{pmatrix}\begin{pmatrix}A_{1}&\mu_{1}^{-1}A_{1}S_{1}\\\
0&\mu_{1}(A_{1}^{t})^{-1}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}A_{1}^{-1}A_{2}A_{1}&\mu_{1}^{-1}A_{1}^{-1}A_{2}A_{1}S_{1}+\mu_{1}A_{1}^{-1}A_{2}S_{2}(A_{1}^{t})^{-1}-\mu_{1}^{-1}S_{1}A_{1}^{t}(A_{2}^{t})^{-1}(A_{1}^{t})^{-1}\\\
0&A_{1}^{t}(A_{2}^{t})^{-1}(A_{1}^{t})^{-1}\end{pmatrix}.$

Since $S_{1}$ is symmetric and since
$\mathcal{U}_{g}(\ell)\unlhd\mathcal{B}_{g}(\mathbb{F}_{\ell})$, we obtain
that $A_{1}^{-1}A_{2}A_{1}\in\mathcal{U}_{g}(\mathbb{F}_{\ell})$. Then, by
setting

$A_{3}:=A_{1}^{-1}A_{2}A_{1},\
S_{3}:=\mu_{1}^{-1}S_{1}+\mu_{1}A_{1}^{-1}S_{2}(A_{1}^{t})^{-1}-\mu_{1}A_{2}^{-1}S_{1}(A_{2}^{t})^{-1},$

we obtain that

$M_{1}^{-1}M_{2}M_{1}=\begin{pmatrix}A_{3}&A_{3}S_{3}\\\
0&(A_{3}^{t})^{-1}\end{pmatrix}\in U_{2g}(\mathbb{F}_{\ell}).$

This proves that $U_{2g}(\mathbb{F}_{\ell})$ is a normal subgroup of
$B_{2g}(\mathbb{F}_{\ell})$. Since $U_{2g}(\mathbb{F}_{\ell})\leq
U^{\prime}_{2g}(\mathbb{F}_{\ell})$, we deduce that
$U^{\prime}_{2g}(\mathbb{F}_{\ell})$ is a normal subgroup of
$B_{2g}(\mathbb{F}_{\ell})$. ∎

######  Proposition 9.

The following formulae hold:

$\displaystyle\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle(\ell-1)\prod_{1\leq i\leq g}(\ell^{2i}-1)\ell^{2i-1};$
$\displaystyle\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\prod_{1\leq i\leq g}(\ell^{2i}-1)\ell^{2i-1};$
$\displaystyle\\#\mathcal{B}_{g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\ell^{\frac{g(g-1)}{2}}(\ell-1)^{g};$
$\displaystyle\\#\mathcal{U}_{g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\ell^{\frac{g(g-1)}{2}};$
$\displaystyle\\#\mathcal{U}^{\prime}_{g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\ell^{\frac{g(g-1)}{2}}(\ell-1);$
$\displaystyle\\#\mathcal{T}_{g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle(\ell-1)^{g};$ $\displaystyle\\#B_{2g}(\mathbb{F}_{\ell})$
$\displaystyle=$ $\displaystyle\ell^{g^{2}}(\ell-1)^{g+1};$
$\displaystyle\\#U_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\ell^{g^{2}};$
$\displaystyle\\#U^{\prime}_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\ell^{g^{2}}(\ell-1);$
$\displaystyle\\#T_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle(\ell-1)^{g+1}.$

###### Proof.

The formulae for $\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ and
$\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})$ are well-known (for example,
see [O’Me78, Theorem 3.1.2, p. 35]). The formulae for the orders of
$\mathcal{B}_{g}(\mathbb{F}_{\ell})$, $\mathcal{U}_{g}(\mathbb{F}_{\ell})$,
$\mathcal{U}^{\prime}_{g}(\mathbb{F}_{\ell})$ and
$\mathcal{T}_{g}(\mathbb{F}_{\ell})$ are clear from the definitions of the
groups. The formulae for the orders of $B_{2g}(\mathbb{F}_{\ell})$,
$U_{2g}(\mathbb{F}_{\ell})$, $U^{\prime}_{2g}(\mathbb{F}_{\ell})$ and
$T_{2g}(\mathbb{F}_{\ell})$ can be obtained from the definitions of the groups
by counting the number of each block matrices, as follows:

$\displaystyle\\#B_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\\#\mathcal{B}_{g}(\mathbb{F}_{\ell})\cdot\\#(\mathbb{Z}/\ell\mathbb{Z})^{\times}\cdot\ell^{\frac{g(g+1)}{2}}=\ell^{g^{2}}(\ell-1)^{g+1};$
$\displaystyle\\#U_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\\#\mathcal{U}_{g}(\mathbb{F}_{\ell})\cdot\ell^{\frac{g(g+1)}{2}}=\ell^{g^{2}};$
$\displaystyle\\#U^{\prime}_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\\#\mathcal{U}_{g}(\mathbb{F}_{\ell})\cdot\\#(\mathbb{Z}/\ell\mathbb{Z})^{\times}\cdot\ell^{\frac{g(g+1)}{2}}=\ell^{g^{2}}(\ell-1);$
$\displaystyle\\#T_{2g}(\mathbb{F}_{\ell})$ $\displaystyle=$
$\displaystyle\\#\mathcal{T}_{g}(\mathbb{F}_{\ell})\cdot\\#(\mathbb{Z}/\ell\mathbb{Z})^{\times}=(\ell-1)^{g+1}.$

∎

######  Proposition 10.

The following upper bounds hold:

$\displaystyle\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$
$\displaystyle\ll_{g}$ $\displaystyle\ell^{g+1},$
$\displaystyle\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$
$\displaystyle\ll_{g}$ $\displaystyle\ell^{g}.$

###### Proof.

The upper bound for $\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$ can
be obtained from [Wa63, (iii), p. 36] and [Ga70, (2), p.1]. To obtain an upper
bound for $\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})$, we proceed as
follows. Note that each conjugacy class in
$\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})$ may be viewed as the equivalence
class of $\Lambda_{2g}(\mathbb{F}_{\ell})$-orbits of conjugacy classes in
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$. Now fix an arbitrary
element $\mathcal{C}\in\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$. If
there is an element $b\in\mathbb{F}_{\ell}^{\times}$ such that
$(bI_{2g})\mathcal{C}=\mathcal{C}$, then, by comparing determinants, we have
$b^{2g}=1$. So $b$ may take at most $2g$ elements. By the orbit-stabilizer
theorem in group theory, each $\Lambda_{2g}(\mathbb{F}_{\ell})$-orbit of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$ contains at least
$\frac{\\#\Lambda_{2g}(\mathbb{F}_{\ell})}{2g}$ conjugacy classes. Therefore,

$\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})^{\\#}\leq\frac{\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}}{\frac{\\#\mathbb{F}_{\ell}^{\times}}{2}}\ll_{g}\ell^{g}.$

∎

######  Proposition 11.

The groups $B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$ and
$T_{2g}(\mathbb{F}_{\ell})$ are isomorphic. Consequently, the quotient groups
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$ and
$B_{2g}(\mathbb{F}_{\ell})/U^{\prime}_{2g}(F_{\ell})$ are abelian.

###### Proof.

By comparing the orders of
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$ and
$T_{2g}(\mathbb{F}_{\ell})$, we deduce that the composition
$T_{2g}(\mathbb{F}_{\ell})\hookrightarrow
B_{2g}(\mathbb{F}_{\ell})\twoheadrightarrow
B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$ is a bijection. Then,
recalling that $T_{2g}(\mathbb{F}_{\ell})$ is abelian, we obtain that
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$ is abelian. Furthermore,
by observing that $U_{2g}(\mathbb{F}_{\ell})\leq
U^{\prime}_{2g}(\mathbb{F}_{\ell})\leq B_{2g}(\mathbb{F}_{\ell})$, we deduce
that $B_{2g}(\mathbb{F}_{\ell})/U^{\prime}_{2g}(\mathbb{F}_{\ell})$ is also
abelian. ∎

##  5\. Distinguished conjugacy classes in the general symplectic groups over
a finite prime field

As in the previous section, we fix an arbitrary integer $g\geq 1$ and an
arbitrary rational prime $\ell$. This time, we turn our attention to
particular unions of conjugacy classes in
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

First, we recall that a matrix $M\in M_{2g}(\mathbb{F}_{\ell})$ is called
semisimple if its minimal polynomial over $\mathbb{F}_{\ell}$ has distinct
roots in ${\overline{\mathbb{F}}}_{\ell}$, which is equivalent to the
existence of a matrix
$N\in\operatorname{GL}_{2g}(\overline{\mathbb{F}}_{\ell})$ such that
$NMN^{-1}$ is a diagonal matrix in $M_{2g}(\overline{\mathbb{F}}_{\ell})$.
Next, we recall that the eigenvalues of $M$ are the roots of the
characteristic polynomial
$\operatorname{char}_{M}(X):=\operatorname{det}(XI_{2g}-M)\in\mathbb{F}_{\ell}[X]$
of $M$. Finally, we recall that, when
$M\in\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$, its characteristic
polynomial $\operatorname{char}_{M}(X)$ has the form

$\operatorname{char}_{M}(X)=X^{2g}+b_{1}X^{2g-1}+\ldots+b_{g}X^{g}+\mu
b_{g-1}X^{g-1}+\ldots+\mu^{g-1}b_{1}X+\mu^{g}\in\mathbb{F}_{\ell}[X],$

where $\mu$ is the multiplicator of $M$. Therefore, upon factoring
$\operatorname{char}_{M}(X)$ over $\overline{\mathbb{F}}_{\ell}$, we obtain
that there exist
$\lambda_{1}(M),\ldots,\lambda_{g}(M)\in\overline{\mathbb{F}}_{\ell}$ such
that

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

With notation as above, we introduce the following subsets associated to
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$, an integer $t$, and a positive
real number $z$:

$\displaystyle\mathcal{C}_{0}(\ell,t)$ $\displaystyle:=$
$\displaystyle\left\\{M\in\operatorname{GSp}_{2g}(\mathbb{F}_{\ell}):\
\operatorname{tr}M=-t(\operatorname{mod}\ell)\right\\};$
$\displaystyle\mathcal{C}(\ell,t)$ $\displaystyle:=$
$\displaystyle\left\\{M\in\operatorname{GSp}_{2g}(\mathbb{F}_{\ell}):\lambda_{i}(M)\in\mathbb{F}_{\ell}^{\times}\
1\leq i\leq g,\ \operatorname{tr}M=-t(\operatorname{mod}\ell)\right\\};$
$\displaystyle\mathcal{C}^{ss}(\ell,t)$ $\displaystyle:=$
$\displaystyle\left\\{M\in\mathcal{C}(\ell,t):M\ \text{semisimple}\right\\};$
$\displaystyle\mathcal{C}_{B}(\ell,t)$ $\displaystyle:=$
$\displaystyle{\mathcal{C}}(\ell,t)\cap B_{2g}(\mathbb{F}_{\ell});$
$\displaystyle\widehat{\mathcal{C}_{0}(\ell,0)}$ $\displaystyle:=$
$\displaystyle\;\text{the image of $\mathcal{C}_{0}(\ell,0)$ in
$\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})$};$
$\displaystyle\widehat{\mathcal{C}_{B}(\ell,t)}$ $\displaystyle:=$
$\displaystyle\;\text{the image of $\mathcal{C}_{B}(\ell,t)$ in
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$};$
$\displaystyle\widehat{\mathcal{C}^{\prime}_{B}(\ell,t)}$ $\displaystyle:=$
$\displaystyle\;\text{the image of $\mathcal{C}_{B}(\ell,t)$ in
$B_{2g}(\mathbb{F}_{\ell})/U^{\prime}_{2g}(\mathbb{F}_{\ell})$};$
$\displaystyle\mathcal{C}^{ss}(\ell,|t|\leq z)$ $\displaystyle:=$
$\displaystyle\;\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\mathcal{C}^{ss}(\ell,t);$
$\displaystyle\mathcal{C}(\ell,|t|\leq z)$ $\displaystyle:=$
$\displaystyle\;\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}\mathcal{C}(\ell,t);$
$\displaystyle\mathcal{C}_{B}(\ell,|t|\leq z)$ $\displaystyle:=$
$\displaystyle\;\mathcal{C}(\ell,|t|\leq z)\cap B_{2g}(\mathbb{F}_{\ell});$
$\displaystyle\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\mathcal{C}_{B}(\ell,|t|\leq
z)}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\mathcal{C}_{B}(\ell,|t|\leq
z)}{\tmpbox}$ $\displaystyle:=$ $\displaystyle\;\text{the image of
$\mathcal{C}_{B}(\ell,|t|\leq z)$ in
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$}.$

For the rest of the section, we focus on properties of these subsets that will
be used in the proofs of Theorems 1 \- 3.

######  Proposition 12.

Assume that the prime $\ell$ satisfies $\ell\nmid 2g$. Given any integer $t$
and any positive real number $z$, the following properties hold.

  1. (i)

The sets $\mathcal{C}^{ss}(\ell,t)$ and $\mathcal{C}^{ss}(\ell,|t|\leq z)$ are
non-empty and stable under conjugation by elements of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

  2. (ii)

The sets $\mathcal{C}_{B}(\ell,t)$ and $\mathcal{C}_{B}(\ell,|t|\leq z)$ are
non-empty and stable under conjugation by elements of
$B_{2g}(\mathbb{F}_{\ell})$.

  3. (iii)

Every element in $\mathcal{C}^{ss}(\ell,t)\cup\mathcal{C}^{ss}(\ell,|t|\leq
z)$ is conjugate over $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ to some
element of $B_{2g}(\mathbb{F}_{\ell})$.

  4. (iv)

We have the set inclusions

$U_{2g}(\mathbb{F}_{\ell})\
{\mathcal{C}}_{B}(\ell,t)\subseteq\mathcal{C}_{B}(\ell,t);$
$U^{\prime}_{2g}(\mathbb{F}_{\ell})\
{\mathcal{C}}_{B}(\ell,0)\subseteq\mathcal{C}_{B}(\ell,0);$
$U_{2g}(\mathbb{F}_{\ell})\ {\mathcal{C}}_{B}(\ell,|t|\leq
z)\subseteq\mathcal{C}_{B}(\ell,|t|\leq z).$

###### Proof.

(i) Since $\ell\nmid 2g$, the inverses $2^{-1}(\operatorname{mod}\ell)$ and
$g^{-1}(\operatorname{mod}\ell)$ exist. We consider the cases $\ell\nmid(t+g)$
and $\ell\mid(t+g)$ separately and show that, in either case,
${\mathcal{C}}^{ss}(\ell,t)\neq\emptyset$.

If $\ell\nmid(t+g)$, then the inverse $(t+g)^{-1}(\operatorname{mod}\ell)$
exists. We define

$M:=\operatorname{diag}\left(1(\operatorname{mod}\ell),\ldots,1(\operatorname{mod}\ell),-(t+g)^{-1}g^{-1}(\operatorname{mod}\ell),-(t+g)^{-1}g^{-1}(\operatorname{mod}\ell)\right)\in
T_{2g}(\mathbb{F}_{\ell}).$

Since $M$ is semisimple and $\operatorname{tr}M=-t(\operatorname{mod}\ell)$,
we obtain that $M$ belongs to ${\mathcal{C}}^{ss}(\ell,t)$.

If $\ell\mid(t+g)$, we define

$M:=\operatorname{diag}\left(2^{-1}(\operatorname{mod}\ell),\ldots,2^{-1}(\operatorname{mod}\ell),(1-2^{-1})(\operatorname{mod}\ell),(1-2^{-1})(\operatorname{mod}\ell)\right)\in
T_{2g}(\mathbb{F}_{\ell}).$

Since $(1-2)^{-1}(\operatorname{mod}\ell)$ is non-zero and
$g\equiv-t(\operatorname{mod}\ell)$, we obtain that $M$ is semisimple and
$\operatorname{tr}M=-t(\operatorname{mod}\ell)$, that is, that $M$ belongs to
${\mathcal{C}}^{ss}(\ell,t)$.

We will show now that the set $\mathcal{C}^{ss}(\ell,t)$ is stable under
conjugation by elements of $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.
First, observe that, for any $M\in\mathcal{C}^{ss}(\ell,t)$ and any
$N\in\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$, we have

$\operatorname{tr}(NMN^{-1})=\operatorname{tr}(M)=-t(\operatorname{mod}\ell).$

It remains to check that $NMN^{-1}$ is semisimple. Since $M$ is semisimple,
there exists $A\in\operatorname{GL}_{2g}(\overline{\mathbb{F}}_{\ell})$, such
that $AMA^{-1}$ is a diagonal matrix in
$M_{2g}(\overline{\mathbb{F}}_{\ell})$. Taking
$A^{\prime}:=AN^{-1}\in\operatorname{GL}_{2g}(\overline{\mathbb{F}}_{\ell})$,
we obtain that

$A^{\prime}(NMN^{-1})A^{\prime-1}=AMA^{-1}$

is a diagonal matrix in $M_{2g}(\overline{\mathbb{F}}_{\ell})$. As a
consequence, $\mathcal{C}^{ss}(\ell,t)$ is stable under conjugation by
elements of $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

(ii) Since $\mathcal{C}^{ss}(\ell,t)\subseteq{\mathcal{C}}_{B}(\ell,t)$, we
deduce from (i) that $\mathcal{C}_{B}(\ell,t)\neq\emptyset$. The set
$\mathcal{C}_{B}(\ell,t)$ is stable under conjugation by elements of
$B_{2g}(\mathbb{F}_{\ell})$ because both $\mathcal{C}^{ss}(\ell,t)$ and
$B_{2g}(\mathbb{F}_{\ell})$ are stable under conjugation by elements of
$B_{2g}(\mathbb{F}_{\ell})$. The similar result about
$\mathcal{C}_{B}(\ell,|t|\leq z)$ follows by noting that
$\mathcal{C}_{B}(\ell,|t|\leq z)=\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\
|t|\leq z\end{subarray}}\mathcal{C}_{B}(\ell,t)$.

(iii) Let $M\in{\mathcal{C}}^{ss}(\ell,t)$, with multiplicator $\mu$. Then

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

Define

$M^{\prime}:=\operatorname{diag}\left(\lambda_{1}(M),\ldots,\lambda_{g}(M),\mu\lambda_{1}^{-1}(M),\ldots,\mu\lambda_{g}^{-1}(M)\right)\in
B_{2g}(\mathbb{F}_{\ell}).$

Since $M^{\prime},M$ are semisimple and
$\operatorname{char}_{M^{\prime}}(X)=\operatorname{char}_{M}(X)$, by [Ch97,
Lemma 3.4] we deduce that $M$ is conjugate over
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ to some element of
$B_{2g}(\mathbb{F}_{\ell})$. Since each element $M$ of
$\mathcal{C}^{ss}(\ell,|t|\leq z)$ is an element of $\mathcal{C}^{ss}(\ell,t)$
for some integer $t$ with $|t|\leq z$, by the previous argument we deduce that
$M$ is also conjugate over $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ to
some element of $B_{2g}(\mathbb{F}_{\ell})$.

(iv) Let $M_{1}\in U_{2g}(\mathbb{F}_{\ell})$ and
$M_{2}\in{\mathcal{C}}_{B}(\ell,t)$. Then $M_{1}M_{2}\in
B_{2g}(\mathbb{F}_{\ell})$ and $M_{1}M_{2}$ has the same diagonal entries as
$M_{2}$. Since $\operatorname{tr}M_{2}=-t(\operatorname{mod}\ell)$, we obtain
that $\operatorname{tr}(M_{1}M_{2})=-t(\operatorname{mod}\ell)$. As such,
$U_{2g}(\mathbb{F}_{\ell}){\mathcal{C}}_{B}(\ell,t)\subseteq{\mathcal{C}}_{B}(\ell,t)$.

Now let $M_{1}\in U^{\prime}_{2g}(\mathbb{F}_{\ell})$ and
$M_{2}\in{\mathcal{C}}_{B}(\ell,0)$. Denote the diagonal elements of $M_{1}$
by $a$. Then $M_{1}M_{2}\in B_{2g}(\mathbb{F}_{\ell})$ and
$\operatorname{tr}(M_{1}M_{2})=a\operatorname{tr}M_{2}=0$. As such,
$U^{\prime}_{2g}(\mathbb{F}_{\ell})\
{\mathcal{C}}_{B}(\ell,0)\subseteq\mathcal{C}_{B}(\ell,0)$.

The result about ${\mathcal{C}}_{B}(\ell,|t|\leq z)$ follows from the
observation that

$U_{2g}(\mathbb{F}_{\ell})\ {\mathcal{C}}_{B}(\ell,|t|\leq
z)=\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\ |t|\leq
z\end{subarray}}U_{2g}(\mathbb{F}_{\ell})\mathcal{C}_{B}(\ell,t)\subseteq\bigcup_{\begin{subarray}{c}t\in\mathbb{Z}\\\
|t|\leq z\end{subarray}}\mathcal{C}_{B}(\ell,t).$

∎

######  Proposition 13.

Assume that the prime $\ell$ is odd. Given any integer $t$ and any positive
real number $z$, the following upper bounds hold:

$\\#\mathcal{C}_{0}(\ell,t)\ll\ell^{2g^{2}+g};$
$\\#\widehat{\mathcal{C}_{0}(\ell,0)}\ll\ell^{2g^{2}+g-1};$
$\\#\widehat{\mathcal{C}_{B}(\ell,t)}\leq(\ell-1)^{g};$
$\\#\widehat{\mathcal{C}^{\prime}_{B}(\ell,0)}\leq(\ell-1)^{g-1};$
$\\#\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\mathcal{C}_{B}(\ell,|t|\leq
z)}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\mathcal{C}_{B}(\ell,|t|\leq
z)}{\tmpbox}\leq(2z+1)(\ell-1)^{g}.$

###### Proof.

We recall from the calculation in [CoDaSiSt17, (9), p. 3569] that

$\frac{\\#\mathcal{C}_{0}(\ell,t)}{\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})}=\frac{1}{\ell}+\operatorname{O}\left(\frac{1}{\ell^{3}}\right).$

Then the upper bound for $\\#\mathcal{C}_{0}(\ell,t)$ follows from the
asymptotic above and the upper bound
$\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})\ll\ell^{2g^{2}+g+1}$ derived
from Proposition 9.

Noting that

$\Lambda(\mathbb{F}_{\ell})\mathcal{C}_{0}(\ell,0)=\mathcal{C}_{0}(\ell,0),$

we obtain that the inverse image of $\widehat{\mathcal{C}_{0}(\ell,0)}$ in
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ is $\mathcal{C}_{0}(\ell,0)$.
Therefore,

$\\#\widehat{\mathcal{C}_{0}(\ell,0)}=\frac{\\#\mathcal{C}_{0}(\ell,t)}{\\#\mathbb{F}_{\ell}^{\times}}\ll\ell^{2g^{2}+g-1}.$

Recalling from Proposition 11 that
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})\simeq
T_{2g}(\mathbb{F}_{\ell})$, we deduce that, under this bijection,
$\widehat{\mathcal{C}_{B}(\ell,t)}$ may be identified with the set of matrices

$\\{M\in
T_{2g}(\mathbb{F}_{\ell}):\operatorname{tr}M=-t(\operatorname{mod}\ell)\\}.$

Then, using the definition of $T_{2g}(\mathbb{F}_{\ell})$, we obtain that

$\displaystyle\\#\widehat{\mathcal{C}_{B}(\ell,t)}$ $\displaystyle\leq$
$\displaystyle\\#\\{(a_{1},\ldots,a_{g},\mu)\in(\mathbb{F}_{\ell}^{\times})^{g+1}:\sum_{1\leq
i\leq g}a_{i}+\mu\sum_{1\leq i\leq g}a_{i}^{-1}=-t(\operatorname{mod}\ell)\\}$
$\displaystyle=$
$\displaystyle\sum_{(a_{1},\ldots,a_{g})\in(\mathbb{F}_{\ell}^{\times})^{g}}\\#\\{\mu\in\mathbb{F}_{\ell}^{\times}:\sum_{1\leq
i\leq g}a_{i}+\mu\sum_{1\leq i\leq g}a_{i}^{-1}=-t(\operatorname{mod}\ell)\\}$
$\displaystyle\leq$
$\displaystyle\sum_{(a_{1},\ldots,a_{g})\in(\mathbb{F}_{\ell}^{\times})^{g}}1$
$\displaystyle=$ $\displaystyle(\ell-1)^{g}.$

To estimate $\\#\widehat{\mathcal{C}^{\prime}_{B}(\ell,0)}$, we observe first
that the set $\widehat{\mathcal{C}_{B}(\ell,0)}$ surjects onto
$\widehat{\mathcal{C}^{\prime}_{B}(\ell,0)}$ under the reduction map
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})\twoheadrightarrow
B_{2g}(\mathbb{F}_{\ell})/U^{\prime}_{2g}(\mathbb{F}_{\ell})$. Next, we
observe that the inverse image of $\widehat{\mathcal{C}^{\prime}_{B}(\ell,0)}$
in $B_{2g}(\mathbb{F}_{\ell})$ is of the form
$U^{\prime}_{2g}(\mathbb{F}_{\ell})\mathcal{C}_{B}(\ell,0)$, which is equal to
$\mathcal{C}_{B}(\ell,0)$ by applying part (iv) of Proposition 12 and by
noting that the inclusion is actually an equality. Thus the inverse image of
$\widehat{\mathcal{C}^{\prime}_{B}(\ell,0)}$ in
$B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell})$ is
$\mathcal{C}_{B}(\ell,0)$. Consequently,

$\\#\widehat{\mathcal{C}^{\prime}_{B}(\ell,0)}=\frac{\\#\widehat{\mathcal{C}_{B}(\ell,0)}}{\\#(U^{\prime}_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell}))}\leq(\ell-1)^{g-1}.$

Lastly, to estimate the size of
$\\#\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\mathcal{C}_{B}(\ell,|t|\leq
z)}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\mathcal{C}_{B}(\ell,|t|\leq
z)}{\tmpbox}$, we simply observe that

$\\#\savestack{\tmpbox}{\stretchto{\scaleto{\scalerel*[\widthof{\mathcal{C}_{B}(\ell,|t|\leq
z)}]{\kern-0.6pt\bigwedge\kern-0.6pt}{\rule[-505.89pt]{4.30554pt}{505.89pt}}}{}}{0.5ex}}\stackon[1pt]{\mathcal{C}_{B}(\ell,|t|\leq
z)}{\tmpbox}=\sum_{|t|\leq
z}\\#\widehat{\mathcal{C}_{B}(\ell,t)}\leq(2z+1)(\ell-1)^{g}.$

∎

##  6\. Proofs of Theorems 1 \- 3

In this section, we prove Theorems 1 \- 3 using the preliminary results
presented in Sections 3 - 5.

###  6.1. Background on abelian varieties

We start with a summary of properties of abelian varieties. We refer the
reader to the papers [Fa83], [Ho68], [Oo08], [Se85], [Se86], [Ta66], [Wa69],
[Za18] and to the books [CoSi86], [La83], [Mu70] for the theory of abelian
varieties.

We start by considering an abelian variety $A$ defined over an arbitrary field
$F$ and of dimension $g$. For an integer $n\geq 1$, we write $A[n]$ for the
group of $n$-torsion elements of $A(\overline{F})$ and we write $F(A[n])$ for
the extension of $F$ generated by $A[n]$. For a rational prime
$\ell\neq\operatorname{char}F$, we write $T_{\ell}(A)$ for the $\ell$-adic
Tate module of $A$, defined as the inverse limit
$\displaystyle\lim_{\leftarrow\atop{k}}A[\ell^{k}]$. We recall that, if
$\operatorname{gcd}(n,\operatorname{char}F)=1$, then $A[n]$ is a free
$\mathbb{Z}/n\mathbb{Z}$-module of rank $2g$, endowed with an action of the
absolute Galois group $\operatorname{Gal}(F^{\text{sep}}/F)$, and if
$\ell\neq\operatorname{char}F$, then $T_{\ell}(A)$ is a free
$\mathbb{Z}_{\ell}$-module of rank $2g$, endowed with a continuous action of
$\operatorname{Gal}(F^{\text{sep}}/F)$. These Galois actions give rise to
group homomorphisms

(10)
$\operatorname{Gal}(F^{\text{sep}}/F)\longrightarrow\operatorname{Aut}_{\mathbb{Z}/n\mathbb{Z}}(A[n])$

and

(11)
$\operatorname{Gal}(F^{\text{sep}}/F)\longrightarrow\operatorname{Aut}_{\mathbb{Z}_{\ell}}(T_{\ell}(A)),$

which we call the residual modulo $n$ Galois representation of $A$ and the
$\ell$-adic Galois representation of $A$, respectively. The latter form a
compatible system.

Next, we focus our attention on an abelian variety $A$ defined over
$\mathbb{Q}$ and of dimension $g$. We denote by $N_{A}$ the conductor of $A$.
For a prime $p\nmid N_{A}$, we denote by $A_{p}$ the reduction of $A$ modulo
$p$, which is an abelian variety defined over $\mathbb{F}_{p}$ and of
dimension $g$.

Associated to $A$ and an integer $n\geq 1$, we have the residual modulo $n$
Galois representation

(12)
$\overline{\rho}_{A,n}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\operatorname{Aut}_{\mathbb{Z}/n\mathbb{Z}}(A[n]),$

about which it is known that

(13)
$\mathbb{Q}(A[n])=\overline{\mathbb{Q}}^{\operatorname{Ker}\overline{\rho}_{A,n}}$

and that

(14) $\overline{\rho}_{A,n}$ is unramified outside the primes dividing
$nN_{A}$.

Associated to $A$ and a prime $\ell$, we have the $\ell$-adic Galois
representation

(15)
$\rho_{A,\ell}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow\operatorname{Aut}_{\mathbb{Z}_{\ell}}(T_{\ell}(A)).$

Associated to $A$, a prime $p\nmid N_{A}$, and an integer $n\geq 1$ such that
$p\nmid n$, we have the residual modulo $n$ Galois representation

(16)
$\overline{\rho}_{A_{p},n}:\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\longrightarrow\operatorname{Aut}_{\mathbb{Z}/n\mathbb{Z}}(A_{p}[n]).$

Associated to $A$, a prime $p\nmid N_{A}$, and another prime $\ell\neq p$, we
have the $\ell$-adic Galois representation

(17)
$\rho_{A_{p},\ell}:\operatorname{Gal}(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})\longrightarrow\operatorname{Aut}_{\mathbb{Z}_{\ell}}(T_{\ell}(A_{p})).$

Now let us fix primes $p$ and $\ell$ such that $p\nmid\ell N_{A}$. The $p$-th
power Frobenius
$\overline{\mathbb{F}}_{p}\longrightarrow\overline{\mathbb{F}}_{p}$,
$\alpha\mapsto\alpha^{p}$, gives rise to an $\mathbb{F}_{p}$-endomorphism
$\pi_{p}(A)\in\operatorname{End}_{\mathbb{F}_{p}}(A_{p})$ of $A_{p}$, which,
in turn, gives rise to a $\mathbb{Z}_{\ell}$-linear operator
$\left.\pi_{p}(A)\right|_{T_{\ell}(A_{p})}$ on $T_{\ell}(A_{p})$ and an
$\mathbb{F}_{\ell}$-linear operator $\left.\pi_{p}(A)\right|_{A_{p}[\ell]}$ on
$A_{p}[\ell]$, both of which we shall also call $\pi_{p}(A)$. We denote by
$P_{\pi_{p}(A)}(X)\in\mathbb{Z}_{\ell}[X]$ the characteristic polynomial of
$\pi_{p}(A)$ acting on $T_{\ell}(A_{p})$ and recall the following important
properties.

######  Proposition 14.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $p$ and $\ell$ be primes such that $p\nmid\ell
N_{A}$. Then $P_{\pi_{p}(A)}(X)$ is a $p$-Weil polynomial of degree $2g$,
whose coefficients are integers that are independent of the choice of $\ell$.

###### Proof.

See [Mi86, Theorem 19.1, p. 143]. ∎

We write

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

Over $\overline{\mathbb{Q}}$, we write

(19) $P_{\pi_{p}(A)}(X)=\displaystyle\prod_{1\leq i\leq
2g}(X-\alpha_{i})\in{\overline{\mathbb{Q}}}[X],$

where $\alpha_{1},\ldots,\alpha_{g}\in\overline{\mathbb{Q}}$ have the property
that, for any $\sigma\in\operatorname{Aut}(\mathbb{C})$,

(20) $|\sigma(\alpha_{i})|=\sqrt{p}\quad\forall 1\leq i\leq 2g.$

Note that the discriminant $\operatorname{disc}{P_{\pi_{p}(A)}(X)}$ of the
polynomial $P_{\pi_{p}(A)}(X)$ is an integer, defined by the product

(21) $\operatorname{disc}{P_{\pi_{p}(A)}(X)}=\displaystyle\prod_{1\leq i<j\leq
2g}(\alpha_{i}-\alpha_{j})^{2}.$

Thanks to (20), the absolute value of this discriminant satisfies the upper
bound

(22) $|\operatorname{disc}{P_{\pi_{p}(A)}(X)}|\leq(4p)^{(2g-1)g}.$

In light of Proposition 14, $\pi_{p}(A)$ may be regarded as a $p$-Weil number,
that is, as an algebraic integer for which, for any
$\sigma\in\operatorname{Aut}(\mathbb{C})$, we have that
$|\sigma(\pi_{p}(A))|=\sqrt{p}$. Associated to this algebraic integer, we have
its minimal polynomial $Q_{\pi_{p}(A)}(X)\in\mathbb{Z}[X]$ over $\mathbb{Q}$,
the number field $\mathbb{Q}(\pi_{p}(A))$ that it generates, and the Galois
closure $K_{A,p}$ of $\mathbb{Q}(\pi_{p}(A))$ in $\overline{\mathbb{Q}}$. In
what follows, we record properties of the extensions
$\mathbb{Q}(\pi_{p}(A))/\mathbb{Q}$ and $K_{A,p}/\mathbb{Q}$.

######  Proposition 15.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $p$ be a prime such that $p\nmid N_{A}$.

  1. (i)

The degrees of $\mathbb{Q}(\pi_{p}(A))/\mathbb{Q}$ and $K_{A,p}/\mathbb{Q}$
satisfy the upper bounds

$[\mathbb{Q}(\pi_{p}(A)):\mathbb{Q}]\leq 2g\ \ \text{and}\ \
[K_{A,p}:\mathbb{Q}]\leq g!\ 2^{g}.$

  2. (ii)

The absolute values of the discriminants of
$\mathbb{Q}(\pi_{p}(A))/\mathbb{Q}$ and $K_{A,p}/\mathbb{Q}$ satisfy the upper
bounds

$|d_{K_{A,p}}|\leq|d_{\mathbb{Q}(\pi_{p}(A))}|^{g!\ 2^{g-1}}\leq(4p)^{g!\
2^{g-1}(2g-1)g}.$

###### Proof.

(i) For the degree of $\mathbb{Q}(\pi_{p}(A))/\mathbb{Q}$, we observe that

$[\mathbb{Q}(\pi_{p}(A)):\mathbb{Q}]=\operatorname{deg}Q_{\pi_{p}(A)}(X)\leq\operatorname{deg}P_{\pi_{p}(A)}(X)=2g.$

For the degree of $K_{A,p}/\mathbb{Q}$, we sketch the following well-known
argument. Recall from Proposition 14 that $P_{\pi_{p}(A)}(X)$ is a $p$-Weil
polynomial, having $\pi_{p}(A)$ as one of its roots in
$\overline{\mathbb{Q}}$. The complex roots $\alpha_{1},\ldots,\alpha_{2g}$ of
$P_{\pi_{p}(A)}(X)$, introduced in (19), come in pairs
$\left(\alpha_{1},\frac{\alpha_{1}}{p}\right)$, $\ldots$,
$\left(\alpha_{g},\frac{\alpha_{g}}{p}\right)$. As such, the Galois group
$\operatorname{Gal}(K_{A,p}/\mathbb{Q})$ of the Galois closure of
$\mathbb{Q}(\pi_{p}(A))$ in $\overline{\mathbb{Q}}$ embeds into the subgroup
$W_{2g}$ of the permutation group ${\mathcal{S}}_{2g}$ on $2g$ elements which
induces a permutation on the set of pairs $(1,2)$, $\ldots$, $(2g-1,2g)$. The
group structure of $W_{2g}$ is obtained by noting that the action of $W_{2g}$
on the above pairs of integers gives rise to a short exact sequence of groups
$1\rightarrow H\rightarrow W_{2g}\rightarrow{\mathcal{S}}_{g}\rightarrow 1$,
where $H$ is the group generated by the transpositions $(1,2)$, $\ldots$,
$(2g-1,2g)$ and where ${\mathcal{S}}_{g}$ is the permutation group on $g$
elements. Thus $H\simeq(\mathbb{Z}/2\mathbb{Z})^{g}$ and
$W_{2g}\simeq(\mathbb{Z}/2\mathbb{Z})^{g}\rtimes{\mathcal{S}}_{g}$.
Consequently, $[K_{A,p}:\mathbb{Q}]\leq\\#W_{2g}=g!\ 2^{g}$. For more details,
see, for example, the discussion in [Ch97, pp. 168–169].

(ii) On one hand, we have the inclusion of orders
$\mathbb{Z}[\pi_{p}(A)]\subseteq{\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}$ in
$\mathbb{Q}(\pi_{p}(A))$. By comparing their discriminants, we obtain the
relation

(23)
$\operatorname{disc}Q_{\pi_{p}(A)}(X)=c_{A,p}^{2}d_{\mathbb{Q}(\pi_{p}(A))},$

where

(24)
$c_{A,p}:=\\#\left({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)]\right)\in\mathbb{Z}.$

In particular, we obtain the divisibility

$d_{\mathbb{Q}(\pi_{p}(A))}\mid\operatorname{disc}Q_{\pi_{p}(A)}(X)\;\;\text{in}\;\;\mathbb{Z}.$

Since we also have the divisibility

$Q_{\pi_{p}(A)}(X)\mid P_{\pi_{p}(A)}(X)\;\;\text{in}\;\;\mathbb{Z}[X],$

we deduce the divisibility relations

(25)
$d_{\mathbb{Q}(\pi_{p}(A))}\mid\operatorname{disc}Q_{\pi_{p}(A)}(X)\mid\operatorname{disc}P_{\pi_{p}(A)}(X)\;\;\text{in}\;\;\mathbb{Z}.$

On the other hand, we have the inclusion of number fields
$\mathbb{Q}(\pi_{p}(A))\subseteq K_{A,p}$. Since $K_{A,p}$ is the smallest
normal extension of $\mathbb{Q}(\pi_{p}(A))$ in $\overline{\mathbb{Q}}$, by
comparing discriminants and recalling [La94, p. 327], we obtain the inequality

(26) $|d_{K_{A,p}}|\leq|d_{\mathbb{Q}(\pi_{p}(A))}|^{g!\ 2^{g-1}}.$

Putting together (22), (25), and (26), we deduce the inequalities claimed in
(ii). ∎

Fixing a Frobenius element
$\operatorname{Frob}_{p}\in\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
at $p$, we obtain the $\mathbb{Z}_{\ell}$-linear operator
$\rho_{A,\ell}(\operatorname{Frob}_{p})$ on $T_{\ell}(A)$ and the
$\mathbb{F}_{\ell}$-linear operator
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$ on $A[\ell]$. We denote by
$P_{A,p}(X)\in\mathbb{Z}_{\ell}[X]$ the characteristic polynomial of
$\rho_{A,\ell}(\operatorname{Frob}_{p})$ acting on $T_{\ell}(A)$ and by
${\mathcal{P}}_{A,p}(X)\in\mathbb{F}_{\ell}[X]$ the characteristic polynomial
of $\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$ acting on $A[\ell]$. We
denote by ${\mathcal{Q}}_{A,p}(X)\in\mathbb{F}_{\ell}[X]$ the minimal
polynomial of $\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$ acting on
$A[\ell]$. The latter two polynomials satisfy the following two divisibility
relations:

(27)
${\mathcal{Q}}_{A,p}(X)\mid{\mathcal{P}}_{A,p}(X)\;\;\text{in}\;\;\mathbb{F}_{\ell}[X]$

and

(28)
${\mathcal{P}}_{A,p}(X)\mid{\mathcal{Q}}_{A,p}(X)^{\infty}\;\;\text{in}\;\;\mathbb{F}_{\ell}[X]$

(see, for example, [Al09, Corollary 7.10, Proposition 7.9, p. 376]). In the
next two propositions, we relate the polynomials $P_{A,p}(X)$ and
${\mathcal{P}}_{A,p}(X)$ to $P_{\pi_{p}(A)}(X)$, and the polynomial
${\mathcal{Q}}_{A,p}(X)$ to $Q_{\pi_{p}(A)}(X)$.

######  Proposition 16.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $p$ and $\ell$ be primes such that $p\nmid\ell
N_{A}$. Then

  1. (i)

$P_{A,p}(X)=P_{\pi_{p}(A)}(X)\in\mathbb{Z}[X]$;

  2. (ii)

${\mathcal{P}}_{A,p}(X)=P_{\pi_{p}(A)}(X)(\operatorname{mod}\ell)\in\mathbb{F}_{\ell}[X]$.

###### Proof.

The $\mathbb{Z}_{\ell}$-module isomorphisms

$T_{\ell}(A_{p})\simeq_{\mathbb{Z}_{\ell}}\mathbb{Z}_{\ell}^{2g}\simeq_{\mathbb{Z}_{\ell}}T_{\ell}(A)$

and the $\mathbb{F}_{\ell}$-vector space isomorphisms

$A_{p}[\ell]\simeq_{\mathbb{F}_{\ell}}\mathbb{F}_{\ell}^{2g}\simeq_{\mathbb{F}_{\ell}}A[\ell]$

give rise to the following commutative diagram of ring isomorphisms, in which
the top horizontal row is also an isomorphism of $\mathbb{Z}_{\ell}$-algebras
and the bottom horizontal row is also an isomorphism of
$\mathbb{F}_{\ell}$-algebras:

${\operatorname{End}_{\mathbb{Z}_{\ell}}(T_{\ell}(A))}$${\operatorname{End}(T_{\ell}(A_{p}))}$${\operatorname{End}_{\mathbb{F}_{\ell}}(A[\ell])}$${\operatorname{End}_{\mathbb{F}_{\ell}}(A_{p}[\ell]).}$$\scriptstyle{\simeq_{\mathbb{Z}_{\ell}\text{-alg}}}$$\scriptstyle{\simeq_{\mathbb{F}_{\ell}\text{-alg}}}$

Under this diagram, by moving to the right and down, the operator
$\rho_{A,\ell}(\operatorname{Frob}_{p})\in\operatorname{End}_{\mathbb{Z}_{\ell}}(T_{\ell}(A))$
is mapped to
$\left.\pi_{p}(A)\right|_{{T_{\ell}}(A_{p})}\in\operatorname{End}_{\mathbb{Z}_{\ell}}(T_{\ell}(A_{p}))$
and then to
$\left.\pi_{p}(A)\right|_{A_{p}[\ell]}\in\operatorname{End}_{\mathbb{F}_{\ell}}(A_{p}[\ell])$;
by moving down and to the left, it is mapped to
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})\in\operatorname{End}_{\mathbb{F}_{\ell}}(A[\ell])$
and then again to
$\left.\pi_{p}(A)\right|_{A_{p}[\ell]}\in\operatorname{End}_{\mathbb{F}_{\ell}}(A_{p}[\ell])$.
Hence the polynomial relations (i) and (ii) between the various associated
characteristic polynomials hold. ∎

######  Proposition 17.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $p$ and $\ell$ be primes such that $p\nmid\ell
N_{A}$. Then

${\mathcal{Q}}_{A,p}(X)\mid
Q_{\pi_{p}(A)}(X)(\operatorname{mod}\ell)\;\;\text{in}\;\;\mathbb{F}_{\ell}[X].$

###### Proof.

Viewing $Q_{\pi_{p}(A)}(X)(\operatorname{mod}\ell)\in\mathbb{F}_{\ell}[X]$ as
a polynomial over the ring of $\mathbb{F}_{\ell}$-linear operators on
$A_{p}[\ell]$, we see that $\left.\pi_{p}(A)\right|_{A_{p}[\ell]}$ is one of
its roots. Recall that, under the isomorphism of $\mathbb{F}_{\ell}$-algebras
$\operatorname{End}_{\mathbb{F}_{\ell}}(A_{p}[\ell])\longrightarrow\operatorname{End}_{\mathbb{F}_{\ell}}(A[\ell])$,
the operator $\left.\pi_{p}(A)\right|_{A_{p}[\ell]}$ is mapped to
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$. Then, since
${\mathcal{Q}}_{A,p}(X)$ is the minimal polynomial of
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$, we deduce that
${\mathcal{Q}}_{A,p}(X)\mid Q_{\pi_{p}(A)}(X)(\operatorname{mod}\ell)$ as
polynomials in $\mathbb{F}_{\ell}[X]$. ∎

Finally, we record the following result which will play an important role in
the proofs of our main theorems.

######  Proposition 18.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $p$ and $\ell$ be primes such that $p\nmid\ell
N_{A}$. Assume that $\ell\nmid c_{A,p}$ (defined in (24)) and that $\ell$
splits completely in $\mathbb{Q}(\pi_{p}(A))$. Then the
$\mathbb{F}_{\ell}$-linear operator
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})\in\operatorname{End}_{\mathbb{F}_{\ell}}(A[\ell])$
is semisimple (that is, the minimal polynomial ${\mathcal{Q}}_{A,p}(X)$ has
distinct roots in ${\overline{\mathbb{F}}}_{\ell}$) and has all its
eigenvalues in $\mathbb{F}_{\ell}^{\times}$ (that is, all the roots of the
characteristic polynomial ${\mathcal{P}}_{A,p}(X)$ are in
$\mathbb{F}_{\ell}^{\times}$).

###### Proof.

Since $\ell\nmid c_{A,p}$ and $\ell$ splits completely in
$\mathbb{Q}(\pi_{p}(A))$, by a classical theorem of Dedekind, the reduction
modulo $\ell$ of the minimal polynomial $Q_{\pi_{p}(A)}(X)$ of the algebraic
integer $\pi_{p}(A)$ factors into distinct linear factors in
$\mathbb{F}_{\ell}[X]$. Then, using the divisibility relation
${\mathcal{Q}}_{A,p}(X)\mid Q_{\pi_{p}(A)}(X)(\operatorname{mod}\ell)$ as
polynomials in $\mathbb{F}_{\ell}[X]$, proven in Proposition 17, we deduce
that ${\mathcal{Q}}_{A,p}(X)$ factors into distinct linear factors in
$\mathbb{F}_{\ell}[X]$. Since ${\mathcal{Q}}_{A,p}(X)$ is the minimal
polynomial of the $\mathbb{F}_{\ell}$-linear operator
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$ acting on $A[\ell]$, we
obtain that $\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$ is semisimple.
Furthermore, using (28), we obtain that the characteristic polynomial
${\mathcal{P}}_{A,p}(X)$ has all its roots in $\mathbb{F}_{\ell}$. This means
that all eigenvalues of $\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$
are in $\mathbb{F}_{\ell}$. Since
$\overline{\rho}_{A,\ell}(\operatorname{Frob}_{p})$ is invertible, the
eigenvalues are in fact in $\mathbb{F}_{\ell}^{\times}$.

∎

###  6.2. A variation on counting primes with a fixed Frobenius trace

The first step in the proofs of Theorem 1 and part (i) of Theorem 3 is the
following result established in analogy with [CoWa22, Lemma 18].

######  Proposition 19.

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. For each $x>2$, let $y=y(x)>2$ and $u=u(x)>2$ be such
that

(29) $u\leq y.$

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

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

and

(31)
$\displaystyle\lim_{x\rightarrow\infty}\frac{\operatorname{log}x}{(\operatorname{log}y)^{1+\varepsilon}}=0.$

Assume RH, as well as GRH for the number fields $K_{A,p}$ defined in Section
6.1., where $p\nmid N_{A}$ is an arbitrary prime. Then, for any
$\varepsilon>0$, there exists a constant $c(\varepsilon)>0$ such that:

  1. (i)

for any $t\in\mathbb{Z}$ and any sufficiently large $x$, we have

$\displaystyle\\#\left\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\right\\}$
$\displaystyle\leq$ $\displaystyle
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq y+u}\\#\left\\{p\leq
x:p\nmid\ell N_{A},a_{1,p}(A)=t,\right.$ $\displaystyle\hskip
99.58464pt\left.\ell\nmid
d_{\mathbb{Q}(\pi_{p}(A))}\\#({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)]),\
\ell\ \text{splits completely in}\ \mathbb{Q}(\pi_{p}(A))\right\\}$

  2. (ii)

for any $z>0$ and any sufficiently large $x>0$, we have

$\displaystyle\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|<z}}\\#\left\\{p\leq
x:p\nmid N_{A},a_{1,p}(A)=t\right\\}$ $\displaystyle\leq$ $\displaystyle
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq z}}\\#\left\\{p\leq
x:p\nmid\ell N_{A},a_{1,p}(A)=t,\right.$ $\displaystyle\hskip
122.34692pt\left.\ell\nmid
d_{\mathbb{Q}(\pi_{p}(A))}\\#({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)]),\
\ell\ \text{splits completely in}\ \mathbb{Q}(\pi_{p}(A))\right\\}.$

###### Proof.

The proof follows the main steps of that of [CoWa22, Lemma 9], as we now
explain. To simplify the exposition, for $x>0$, $t\in\mathbb{Z}$, and $\ell$ a
rational prime, we use the notation

(34) $\pi_{A}(x,\ell,t):=\\#\left\\{p\leq x:p\nmid\ell
N_{A},a_{1,p}(A)=t,\ell\nmid d_{\mathbb{Q}(\pi_{p}(A))}c_{A,p},\ \ell\
\text{splits completely in}\ \mathbb{Q}(\pi_{p}(A))\right\\}$

and

(35) $\pi^{\prime}_{A}(x,\ell,t):=\\#\left\\{p\leq x:p\nmid\ell
N_{A},a_{1,p}(A)=t,\ell\nmid d_{K_{A,p}}c_{A,p},\ \ell\ \text{splits
completely in}\ K_{A,p}\right\\},$

where
$c_{A,p}=\\#({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)])$,
as defined in Section 6.1. Since we have the inclusion of number fields
$\mathbb{Q}(\pi_{p}(A))\subseteq K_{A,p}$, we deduce that
$d_{\mathbb{Q}(\pi_{p}(A))}\mid d_{K_{A,p}}$, which implies that

(36) $\pi_{A}^{\prime}(x,\ell,t)\leq\pi_{A}(x,\ell,t).$

Therefore, it is enough to prove that

(37) $\\#\left\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\right\\}\leq
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq y+u}\pi_{A}^{\prime}(x,\ell,t)$

and

(38) $\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|<z}}\\#\left\\{p\leq x:p\nmid
N_{A},a_{1,p}(A)=t\right\\}\leq c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq
z}}\pi_{A}^{\prime}(x,\ell,t).$

First, observe that

(39) $\displaystyle\sum_{y\leq\ell\leq
y+u}\pi^{\prime}_{A}(x,\ell,t)\leq(\pi(y+u)-\pi(y))\displaystyle\max_{y\leq\ell\leq
y+u}\pi^{\prime}_{A}(x,\ell,t).$

Next, observe that

$\displaystyle\displaystyle\sum_{y\leq\ell\leq y+u}\pi^{\prime}_{A}(x,\ell,t)$
$\displaystyle=$ $\displaystyle\displaystyle\sum_{p\leq x\atop{p\nmid
N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\
d_{K_{A,p}}c_{A,p},\ \ell\ \text{splits completely in}\ K_{A,p}\right\\}$
$\displaystyle=$ $\displaystyle\displaystyle\sum_{p\leq x\atop{p\nmid
N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\
d_{K_{A,p}},\ \ell\ \text{splits completely in}\ K_{A,p}\right\\}$
$\displaystyle-$ $\displaystyle\displaystyle\sum_{p\leq x\atop{p\nmid
N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\
d_{K_{A,p}},\ell\mid c_{A,p},\ \ell\ \text{splits completely in}\
K_{A,p}\right\\}$ $\displaystyle\geq$ $\displaystyle\displaystyle\sum_{p\leq
x\atop{p\nmid N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid
p\ d_{K_{A,p}},\ \ell\ \text{splits completely in}\ K_{A,p}\right\\}$
$\displaystyle-$ $\displaystyle\displaystyle\sum_{p\leq x\atop{p\nmid
N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\mid
c_{A,p}\right\\}$ $\displaystyle\geq$ $\displaystyle\displaystyle\sum_{p\leq
x\atop{p\nmid N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid
p\ d_{K_{A,p}},\ \ell\ \text{splits completely in}\ K_{A,p}\right\\}$
$\displaystyle-$ $\displaystyle\displaystyle\sum_{p\leq x\atop{p\nmid
N_{A}\atop{a_{1,p}(A)=t}}}\nu(c_{A,p}),$

where, for an integer $m\geq 1$, we write $\nu(m)$ for the number of its
distinct prime factors.

Using the bound $\nu(m)\leq\frac{\operatorname{log}m}{\operatorname{log}2}$,
we obtain that

$\nu(c_{A,p})\leq\frac{\operatorname{log}c_{A,p}}{\operatorname{log}2}.$

Using (22), (23) and (25), we obtain that

$c_{A,p}^{2}\ll(4p)^{g(2g-1)}.$

Therefore,

$\nu(c_{A,p})\ll_{g}\operatorname{log}p.$

Using this estimate in the last line of the sequence of inequalities (6.2), we
obtain that

$\displaystyle\displaystyle\sum_{y\leq\ell\leq y+u}\pi^{\prime}_{A}(x,\ell,t)$
$\displaystyle\geq$ $\displaystyle\displaystyle\sum_{p\leq x\atop{p\nmid
N_{A}\atop{a_{1,p}(A)=t}}}\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\
d_{K_{A,p}},\ \ell\ \text{splits completely in}\ K_{A,p}\right\\}$
$\displaystyle+$
$\displaystyle\operatorname{O}_{g}\left(\pi_{A}(x,t)\operatorname{log}x\right).$

Now we focus on the summands $\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\
d_{K_{A,p}},\ell\ \text{splits completely in}\ K_{A,p}\right\\}$ appearing in
(6.2). Recalling that we are assuming that GRH holds for each of the fields
$K_{A,p}$, from the effective version of the Chebotarev Density Theorem stated
in part (i) of Theorem (i) we know that

$\displaystyle\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\ d_{K_{A,p}},\ell\
\text{splits completely in}\ K_{A,p}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{[K_{A,p}:\mathbb{Q}]}(\pi(y+u)-\pi(y))$
$\displaystyle+$ $\displaystyle E_{1}(y,u,K_{A,p})+E_{2}(y,u,K_{A,p}),$

where the terms $E_{1}(y,u,K_{A,p})$, $E_{2}(y,u,K_{A,p})$ are real valued
functions of $y,u,$ and $p$, that satisfy the upper bounds

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

with $c_{1}$, $c_{2}$ some absolute positive constants.

To obtain an upper bound for the quotient
$\frac{\operatorname{log}|d_{K_{A,p}}|}{[K_{A,p}:\mathbb{Q}]}$, we invoke
inequality (8):

$\frac{\operatorname{log}|d_{K_{A,p}}|}{[K_{A,p}:\mathbb{Q}]}\leq\displaystyle\sum_{\ell\mid
d_{K_{A,p}}}\operatorname{log}\ell+\operatorname{log}[K_{A,p}:\mathbb{Q}]\leq\operatorname{log}|d_{K_{A,p}}|+\operatorname{log}[K_{A,p}:\mathbb{Q}].$

We recall from Proposition 15 that $[K_{A,p}:\mathbb{Q}]\leq g!\ 2^{g}$ and
$|d_{K_{A,p}}|\leq(4p)^{g!\ 2^{g-1}(2g-1)g}$. Therefore

$\frac{\operatorname{log}|d_{K_{A,p}}|}{[K_{A,p}:\mathbb{Q}]}\leq
c_{3}(g)\operatorname{log}p$

for some positive constant $c_{3}(g)$ that depends on $g$, but not on $p$.
Using this bound in (43) - (44) and then in (6.2), we obtain that

$\displaystyle\\#\left\\{y\leq\ell\leq y+u:\ell\nmid p\ d_{K_{A,p}},\ \ell\
\text{splits completely in}\ K_{A,p}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{[K_{A,p}:\mathbb{Q}]}(\pi(y+u)-\pi(y))+\operatorname{O}_{g}\left((y+u)^{\frac{1}{2}}\operatorname{log}(xy)\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2^{g}g!}(\pi(y+u)-\pi(y))+\operatorname{O}_{g}\left((y+u)^{\frac{1}{2}}\operatorname{log}(xy)\right).$

To understand the growth of the difference $\pi(y+u)-\pi(y)$, we recall that
we are assuming RH and that, as a function of $y$, the parameter $u$ satisfies
(30). Then, for any $\varepsilon>0$ (which we choose arbitrarily and keep
fixed), we know that

$\pi(y+u)-\pi(y)\geq c_{4}(\varepsilon)\frac{u}{\operatorname{log}u}>0$

for some positive constant that depends on $\varepsilon$ (but not on $u$ and
$y$).

Recalling (6.2), we deduce that

$\displaystyle\displaystyle\max_{y\leq\ell\leq y+u}\pi^{\prime}_{A}(x,\ell,t)$
$\displaystyle\geq$ $\displaystyle
c_{5}(g,\varepsilon)\pi_{A}(x,t)+\operatorname{O}_{g,\varepsilon}\left(\frac{(y+u)^{\frac{1}{2}}\operatorname{log}(xy)\operatorname{log}u}{u}\pi_{A}(x,t)\right)$

for some positive constant $c_{5}(g,\varepsilon)$ that depends on $g$ and
$\varepsilon$. Invoking (29) and (31), we see that, as $x\rightarrow\infty$,
the $\operatorname{O}_{g,\varepsilon}$-term above is
$\operatorname{o}(\pi_{A}(x,t))$. This completes the proof of ((i)).

To prove ((ii)), we use the same argument as for ((i)), except for replacing
$\\#\left\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\right\\}$ with $\\#\left\\{p\leq
x:p\nmid N_{A},|a_{1,p}(A)|\leq z\right\\}$. ∎

###  6.3. Proof of Theorem 1 for arbitrary $t\in\mathbb{Z}$

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $t\in\mathbb{Z}$. Let $x>2$ be a real number that
goes to infinity. Our goal in this subsection is to prove the upper bound for
$\pi_{A}(x,t)=\\#\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\\}$ claimed in Theorem
1, under the main assumptions (46) and (50) stated below.

It is known that we may always choose a polarization $e$ on $A$, which we now
do, and that, for any sufficiently large prime $\ell$, the polarization $e$
gives rise to a non-degenerate alternating bilinear form $e_{\ell}$ on
$T_{\ell}(A)$, with respect to which the group
$\operatorname{GSp}(T_{\ell}(A),e_{\ell})$ of symplectic similitudes contains
the image $\operatorname{Im}\rho_{A,\ell}$ of $\rho_{A,\ell}$ (see [Se85, p.
34]). Upon choosing an isomorphism
$\operatorname{GSp}(T_{\ell}(A),e_{\ell})\simeq\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell})$,
we may thus assume that
$\operatorname{Im}\rho_{A,\ell}\leq\operatorname{GSp}_{2g}(\mathbb{Z}_{\ell})$,
and hence that
$\operatorname{Im}\overline{\rho}_{A,\ell}\leq\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$.

We assume that, for any sufficiently large prime $\ell$, the residual modulo
$\ell$ Galois representation $\overline{\rho}_{A,\ell}$ has image isomorphic
to $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$, which implies that

(46)
$\operatorname{Gal}(\mathbb{Q}(A[\ell])/\mathbb{Q})\simeq\operatorname{GSp}_{2g}(\mathbb{F}_{\ell}).$

Considering the subextensions of $\mathbb{Q}(A[\ell])$ fixed by the subgroups
$U_{2g}(\mathbb{F}_{\ell})$ and $B_{2g}(\mathbb{F}_{\ell})$ of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$,

$\mathbb{Q}\subseteq\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\subseteq\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}\subseteq\mathbb{Q}(A[\ell]),$

and recalling that, by part (ii) of Proposition 8, $U_{2g}(\mathbb{F}_{\ell})$
is a normal subgroup of $B_{2g}(\mathbb{F}_{\ell})$, we obtain the following
Galois group structures:

(47)
$\operatorname{Gal}(\mathbb{Q}(A[\ell])/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})})\simeq
B_{2g}(\mathbb{F}_{\ell}),$ (48)
$\operatorname{Gal}(\mathbb{Q}(A[\ell])/\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})})\simeq
U_{2g}(\mathbb{F}_{\ell}),$ (49)
$\operatorname{Gal}(\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})})\simeq
B_{2g}(\mathbb{F}_{\ell})/U_{2g}(\mathbb{F}_{\ell}).$

We assume that

(50) $\text{GRH holds}.$

More precisely, we assume the validity of RH and of GRH for the Dedekind zeta
functions of the number fields
$\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}$ for all sufficiently large
primes $\ell$, as well as of the number fields $K_{A,p}$ for all primes
$p\nmid N_{A}$.

By ((i)) of Proposition 19, for any real numbers $y=y(x)>2$ and $u=u(x)>2$
that depend on and grow with $x$ and that satisfy (29), (30), and (31), we
know that, for any $\varepsilon>0$, there exists $c(\varepsilon)>0$ such that

(51) $\displaystyle\pi_{A}(x,t)$ $\displaystyle\leq$ $\displaystyle
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq y+u}\\#\left\\{p\leq
x:p\nmid\ell N_{A},a_{1,p}(A)=t,\right.$ $\displaystyle\hskip
99.58464pt\left.\ell\nmid
d_{\mathbb{Q}(\pi_{p}(A))}\\#({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)]),\
\ell\ \text{splits completely in}\ \mathbb{Q}(\pi_{p}(A))\right\\}.$

We will use this inequality with $x$ sufficiently large so that any prime
$\ell$ satisfying $y\leq\ell\leq y+u$ is itself sufficiently large and (46)
holds.

By observation (13) and Proposition 18, any prime $p$ for which $\ell\nmid
d_{\mathbb{Q}(\pi_{p}(A))}\\#({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)])$
and $\ell$ splits completely in $\mathbb{Q}(\pi_{p}(A))$ has the property
that, for any prime ideal $\mathfrak{p}$ of $\mathbb{Q}(A[\ell])$ that lies
over $p$, we have that the matrix
$\overline{\rho}_{A,\ell}\left(\left(\frac{\mathbb{Q}(A[\ell])/\mathbb{Q}}{\mathfrak{p}}\right)\right)\in\operatorname{GL}_{2g}(\mathbb{F}_{\ell})$
is semisimple and has all its eigenvalues in $\mathbb{F}_{\ell}$. By part (ii)
of Proposition 16, any prime $p\nmid\ell N_{A}$ for which $a_{1,p}(A)=t$ has
the property that, for any prime ideal $\mathfrak{p}$ of $\mathbb{Q}(A[\ell])$
that lies over $p$, we have
$\operatorname{tr}\overline{\rho}_{A,\ell}\left(\left(\frac{\mathbb{Q}(A[\ell])/\mathbb{Q}}{\mathfrak{p}}\right)\right)=-t(\operatorname{mod}\ell)$.
Thus, every prime $p$ counted on the right hand side of inequality (51)
satisfies that, for any prime ideal $\mathfrak{p}$ of $\mathbb{Q}(A[\ell])$
that lies over $p$,

$\overline{\rho}_{A,\ell}\left(\left(\frac{\mathbb{Q}(A[\ell])/\mathbb{Q}}{\mathfrak{p}}\right)\right)\in{\mathcal{C}}^{ss}(\ell,t).$

Combining these observations with (51), we obtain the inequality

(52) $\displaystyle\pi_{A}(x,t)$ $\displaystyle\leq$ $\displaystyle
c(\varepsilon)\max_{y\leq\ell\leq
y+u}\pi_{{\mathcal{C}}_{0}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q}).$

We estimate
$\pi_{{\mathcal{C}}^{ss}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$ by
invoking Theorem 6 with $K=\mathbb{Q}$, $L=\mathbb{Q}(A[\ell])$,
$H=B_{2g}(\mathbb{F}_{\ell})$, $N=U_{2g}(\mathbb{F}_{\ell})$,
${\mathcal{C}}_{1}={\mathcal{C}}^{ss}(\ell,t)$, and
${\mathcal{C}}_{2}={\mathcal{C}}(\ell,t)$. In this case, the hypotheses of
Theorem 6 hold thanks to parts (iii) and (iv) of Proposition 12, to
Proposition 11, and to our GRH assumption. We obtain that

$\displaystyle\pi_{\mathcal{C}^{ss}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{\\#\widehat{\mathcal{C}_{B}(\ell,t)}\cdot\\#U_{2g}(\mathbb{F}_{\ell})}{\\#B_{2g}(\mathbb{F}_{\ell})}\pi(x)$
$\displaystyle+$
$\displaystyle\left(\\#\widehat{{\mathcal{C}_{B}(\ell,t)}}\right)^{\frac{1}{2}}\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\
\left[\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}:\mathbb{Q}\right]\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\right)$
$\displaystyle+$
$\displaystyle\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q}).$

Using the Chebyshev bound $\pi(x)\ll\frac{x}{\operatorname{log}x}$, the
formulae for $\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$,
$\\#B_{2g}(\mathbb{F}_{\ell})$, $\\#U_{2g}(\mathbb{F}_{\ell})$ recorded in
Proposition 9, and the estimate for $\\#\widehat{{\mathcal{C}}_{B}(\ell,t)}$
given in Proposition 13, we deduce that, under GRH,

$\displaystyle\pi_{\mathcal{C}^{ss}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{1}{\ell}\cdot\frac{x}{\operatorname{log}x}+\ell^{g^{2}+\frac{g}{2}}\cdot\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\cdot\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\right)$
$\displaystyle+$
$\displaystyle\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q}).$

To estimate $\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q})$, we see
that, by definition,

(54)
$\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q})=\operatorname{log}2+\operatorname{log}[\mathbb{Q}(A[\ell]):\mathbb{Q}]+\frac{1}{[\mathbb{Q}(A[\ell]):\mathbb{Q}]}\operatorname{log}\left|d_{\mathbb{Q}(A[\ell])}\right|+\displaystyle\sum_{p\mid
d_{\mathbb{Q}(A[\ell])}}\operatorname{log}p.$

Using inequality (8), we deduce that

(55)
$\frac{\operatorname{log}\left|d_{\mathbb{Q}(A[\ell])}\right|}{[\mathbb{Q}(A[\ell]):\mathbb{Q}]}\leq\displaystyle\sum_{p\mid
d_{\mathbb{Q}(A[\ell])}}\operatorname{log}p+\operatorname{log}[\mathbb{Q}(A[\ell]):\mathbb{Q}].$

Combining (14), (54) and (55) and using the first formula in Proposition 9, we
derive that

(56)
$\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q})\ll_{g}\operatorname{log}(\ell
N_{A}).$

To estimate
$\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\right)$,
we proceed similarly and derive that

(57)
$\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\right)\ll_{g}\operatorname{log}(\ell
N_{A}).$

Putting together (6.3), (56), and (57), we obtain the upper bound

(58)
$\displaystyle\pi_{\mathcal{C}^{ss}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll_{g}$
$\displaystyle\frac{1}{\ell}\cdot\frac{x}{\operatorname{log}x}+\ell^{g^{2}+\frac{g}{2}}\cdot\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\cdot\operatorname{log}(\ell
N_{A}),$

which is valid for any sufficiently large prime $\ell$. Then, recalling (52),
we obtain that

(59) $\displaystyle\pi_{A}(x,t)$ $\displaystyle\leq$ $\displaystyle
c(A,\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\left(\frac{1}{\ell}\cdot\frac{x}{\operatorname{log}x}+\ell^{g^{2}+\frac{g}{2}}\cdot\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\cdot\operatorname{log}\ell\right)$

for some positive constant $c(A,\varepsilon)$ that depends on $A$ and
$\varepsilon$.

Choosing

$y(x):=\frac{x^{\frac{1}{2g^{2}+g+2}}}{(\operatorname{log}x)^{\frac{2}{2g^{2}+g+2}}},$

recalling from (29) that $u\leq y$, and fixing an arbitrary $\varepsilon$, we
deduce that

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

This completes the proof of Theorem 1 for arbitrary $t$.

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

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $x>2$ be a real number that goes to infinity. Our
goal in this subsection is to prove the upper bound for
$\pi_{A}(x,0)=\\#\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=0\\}$ claimed in Theorem
1, under the same two main assumptions (46) and (50) as in Subsection 6.3,
that is, under the assumption that for any sufficiently large prime $\ell$,
the residual modulo $\ell$ Galois representation $\overline{\rho}_{A,\ell}$
has image isomorphic to $\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ and under
the assumption of the validity of RH and of GRH for the Dedekind zeta
functions of the number fields
$\mathbb{Q}(A[\ell])^{U_{2g}(\mathbb{F}_{\ell})}$ for all sufficiently large
primes $\ell$, as well as of the number fields $K_{A,p}$ for all primes
$p\nmid N_{A}$.

Considering the subextensions of $\mathbb{Q}(A[\ell])$ fixed by the subgroups
$U^{\prime}_{2g}(\mathbb{F}_{\ell})$ and $B_{2g}(\mathbb{F}_{\ell})$ of
$\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$,

$\mathbb{Q}\subseteq\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\subseteq\mathbb{Q}(A[\ell])^{U^{\prime}_{2g}(\mathbb{F}_{\ell})}\subseteq\mathbb{Q}(A[\ell]),$

and recalling that, by part (ii) of Proposition 8,
$U^{\prime}_{2g}(\mathbb{F}_{\ell})$ is a normal subgroup of
$B_{2g}(\mathbb{F}_{\ell})$, in addition to (46) and (47), we obtain the
following Galois group structures

(60)
$\operatorname{Gal}(\mathbb{Q}(A[\ell])/\mathbb{Q}(A[\ell])^{U^{\prime}_{2g}(\mathbb{F}_{\ell})})\simeq
U^{\prime}_{2g}(\mathbb{F}_{\ell}),$ (61)
$\operatorname{Gal}(\mathbb{Q}(A[\ell])^{U^{\prime}_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})})\simeq
B_{2g}(\mathbb{F}_{\ell})/U^{\prime}_{2g}(\mathbb{F}_{\ell}),$

where, by Proposition 11, the quotient
$B_{2g}(\mathbb{F}_{\ell})/U^{\prime}_{2g}(\mathbb{F}_{\ell})$ is abelian.

Proceeding identically to the proof given in Subsection 6.3, we obtain that
(51) leads to the inequality

(62) $\displaystyle\pi_{A}(x,0)$ $\displaystyle\leq$ $\displaystyle
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\pi_{{\mathcal{C}}^{ss}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q}),$

where $\varepsilon>0$ is arbitrary and where $y=y(x)>2$ and $u=u(x)>2$ are
arbitrary real numbers that depend on and grow with $x$ and that satisfy (29),
(30), and (31). Then we estimate
$\pi_{{\mathcal{C}}^{ss}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$ by
invoking Theorem 6 with $K=\mathbb{Q}$, $L=\mathbb{Q}(A[\ell])$,
$H=B_{2g}(\mathbb{F}_{\ell})$, $N=U^{\prime}_{2g}(\mathbb{F}_{\ell})$,
${\mathcal{C}}_{1}={\mathcal{C}}^{ss}(\ell,0)$, and
${\mathcal{C}}_{2}={\mathcal{C}}(\ell,0)$. Similarly to the case considered in
Subsection 6.3, the hypotheses of Theorem 6 hold thanks to parts (iii) and
(iv) of Proposition 12, to Proposition 11, and to our GRH assumption. We
obtain that

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

Using the Chebyshev bound $\pi(x)\ll\frac{x}{\operatorname{log}x}$, the
formulae for $\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$,
$\\#B_{2g}(\mathbb{F}_{\ell})$, $\\#U^{\prime}_{2g}(\mathbb{F}_{\ell})$
recorded in Proposition 9, and the estimate for
$\\#\widehat{{\mathcal{C^{\prime}}}_{B}(\ell,0)}$ given in Proposition 13, we
deduce that

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

Using (56) and the estimate
$\operatorname{log}M\left(\mathbb{Q}(A[\ell])^{U^{\prime}_{2g}(\mathbb{F}_{\ell})}/\mathbb{Q}(A[\ell])^{B_{2g}(\mathbb{F}_{\ell})}\right)\ll_{g}\operatorname{log}(\ell
N_{A})$, obtained mutatis mutandis, we infer that

(64)
$\displaystyle\pi_{\mathcal{C}^{ss}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll_{g}$
$\displaystyle\frac{1}{\ell}\cdot\frac{x}{\operatorname{log}x}+\ell^{g^{2}+\frac{g-1}{2}}\cdot\frac{x^{\frac{1}{2}}}{\operatorname{log}x}\cdot\operatorname{log}(\ell
N_{A}).$

Recalling (62), we obtain that

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

for some positive constant $c(A,\varepsilon)$ that depends on $A$ and
$\varepsilon$. Choosing

$y(x):=\frac{x^{\frac{1}{2g^{2}+g+1}}}{(\operatorname{log}x)^{\frac{2}{2g^{2}+g+1}}},$

recalling from (29) that $u\leq y$, and fixing an arbitrary $\varepsilon$, we
deduce that

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

This completes the proof of Theorem 1 for $t=0$.

###  6.5. Proof of Theorem 2 for arbitrary $t\in\mathbb{Z}$

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $x>2$ be a real number that goes to infinity. Our
goal in this subsection is to prove the upper bound for
$\pi_{A}(x,t)=\\#\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=t\\}$ claimed in Theorem
2, under assumptions (46) and (50), as in Subsection 6.3, together with the
assumptions that AHC and PCC hold for the extension
$\mathbb{Q}(A[\ell])/\mathbb{Q}$, where $\ell$ is a sufficiently large
arbitrary prime.

As in Section 6.3, we recall that, by part (ii) of Proposition 16, any prime
$p\nmid\ell N_{A}$ for which $a_{1,p}(A)=t$ has the property that, for any
prime ideal $\mathfrak{p}$ of $\mathbb{Q}(A[\ell])$ that lies over $p$,
$\operatorname{tr}\overline{\rho}_{A,\ell}\left(\left(\frac{\mathbb{Q}(A[\ell])/\mathbb{Q}}{\mathfrak{p}}\right)\right)=-t(\operatorname{mod}\ell)$.
Thus, every prime $p$ counted in $\pi_{A}(x,t)$ satisfies that, for any prime
ideal $\mathfrak{p}$ of $\mathbb{Q}(A[\ell])$ that lies over $p$,

$\overline{\rho}_{A,\ell}\left(\left(\frac{\mathbb{Q}(A[\ell])/\mathbb{Q}}{\mathfrak{p}}\right)\right)\in{\mathcal{C}}_{0}(\ell,t).$

As such, we have the inequality

(66)
$\pi_{A}(x,t)\leq\pi_{{\mathcal{C}}_{0}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q}).$

By invoking Theorem 7 with $K=\mathbb{Q}$, $L=\mathbb{Q}(A[\ell])$,
$H=\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$, $N=\\{I_{2g}\\}$, and
${\mathcal{C}}={\mathcal{C}}_{0}(\ell,t)$, we obtain that, under GRH, AHC, and
PCC,

$\displaystyle\pi_{\mathcal{C}_{0}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{\\#\mathcal{C}_{0}(\ell,t)}{\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})}\pi(x)$
$\displaystyle+$ $\displaystyle\\#\mathcal{C}_{0}(\ell,t)^{\frac{1}{2}}\
\left(\frac{\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}}{\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})}\right)^{\frac{1}{2}}\
x^{\frac{1}{2}}\
\operatorname{log}\left(M(\mathbb{Q}(A[\ell])/\mathbb{Q})x\right)$
$\displaystyle+$
$\displaystyle\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q}).$

Then, by applying the upper bounds for
$\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$ from Proposition 9, for
$\\#{\mathcal{C}}_{0}(\ell,t)$ from Proposition 13, for
$\\#\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$ from Proposition 10, as
well as estimate (56), we deduce that

(67)
$\displaystyle\pi_{\mathcal{C}_{0}(\ell,t)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll_{g}$
$\displaystyle\frac{1}{\ell}\cdot\frac{x}{\operatorname{log}x}+\ell^{\frac{g}{2}}\cdot
x^{\frac{1}{2}}\cdot\operatorname{log}(\ell N_{A}x).$

Finally, by choosing

$y(x):=\frac{x^{\frac{1}{g+2}}}{(\operatorname{log}x)^{\frac{4}{g+2}}},$

we derive that

$\pi_{A}(x,t)\ll_{A}\frac{x^{1-\frac{1}{g+2}}}{(\operatorname{log}x)^{1-\frac{4}{g+2}}}.$

This completes the proof of Theorem 2 for arbitrary $t$.

###  6.6. Proof of Theorem 2 for $t=0$

Let $A$ be an abelian variety defined over $\mathbb{Q}$, of conductor $N_{A}$,
and of dimension $g$. Let $x>2$ be a real number that goes to infinity. Our
goal in this subsection is to prove the upper bound for
$\pi_{A}(x,0)=\\#\\{p\leq x:p\nmid N_{A},a_{1,p}(A)=0\\}$ claimed in Theorem
2, under assumptions (46) and (50), together with the assumptions that AHC and
PCC hold for the extension
$\mathbb{Q}(A[\ell])^{\Lambda(\mathbb{F}_{\ell})}/\mathbb{Q}$, where $\ell$ is
a sufficiently large arbitrary prime.

As in the proof of Theorem 2 for arbitrary $t$, our starting point is
inequality (66) for $t=0$, that is,

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

We estimate the right hand side by invoking Theorem 7 with $K=\mathbb{Q}$,
$L=\mathbb{Q}(A[\ell])$, $H=\operatorname{GSp}_{2g}(\mathbb{F}_{\ell})$,
$N=\Lambda(\mathbb{F}_{\ell})$, and ${\mathcal{C}}={\mathcal{C}}_{0}(\ell,0)$.
We obtain that, under GRH, AHC, and PCC,

$\displaystyle\pi_{\mathcal{C}_{0}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll$
$\displaystyle\frac{\\#\widehat{\mathcal{C}_{0}(\ell,0)}}{\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})}\pi(x)$
$\displaystyle+$
$\displaystyle\\#\widehat{\mathcal{C}_{0}(\ell,0)}^{\frac{1}{2}}\
\left(\frac{\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})^{\\#}}{\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})}\right)^{\frac{1}{2}}\
x^{\frac{1}{2}}\
\operatorname{log}\left(M(\mathbb{Q}(A[\ell])^{\Lambda(\mathbb{F}_{\ell})}/\mathbb{Q})x\right)$
$\displaystyle+$
$\displaystyle\frac{x^{\frac{1}{2}}}{\operatorname{log}x}+\operatorname{log}M(\mathbb{Q}(A[\ell])^{\Lambda(\mathbb{F}_{\ell})}/\mathbb{Q}).$

Then, by applying the upper bounds for
$\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})$ from Proposition 9, for
$\\#\widehat{{\mathcal{C}}_{0}(\ell,0)}$ from Proposition 13, and for
$\\#\operatorname{PGSp}_{2g}(\mathbb{F}_{\ell})^{\\#}$ from Proposition 10,
and by observing that

$\operatorname{log}M(\mathbb{Q}(A[\ell])^{\Lambda(\mathbb{F}_{\ell})}/\mathbb{Q})\leq\operatorname{log}M(\mathbb{Q}(A[\ell])/\mathbb{Q}),$

we deduce that

(68)
$\displaystyle\pi_{\mathcal{C}_{0}(\ell,0)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$
$\displaystyle\ll_{g}$
$\displaystyle\frac{1}{\ell}\cdot\frac{x}{\operatorname{log}x}+\ell^{\frac{g-1}{2}}\cdot
x^{\frac{1}{2}}\cdot\operatorname{log}(\ell N_{A}x).$

Finally, by choosing

$y(x):=\frac{x^{\frac{1}{g+1}}}{(\operatorname{log}x)^{\frac{4}{g+1}}},$

we derive that

$\pi_{A}(x,0)\ll_{A}\frac{x^{1-\frac{1}{g+1}}}{(\operatorname{log}x)^{1-\frac{4}{g+1}}}.$

This completes the proof of Theorem 2 for $t=0$.

###  6.7. Proof of part (i) of Theorem 3

Once again, let $A$ be an abelian variety defined over $\mathbb{Q}$, of
conductor $N_{A}$, and of dimension $g$. We keep the two main assumptions (46)
and (50). as in Subsections 6.3 and 6.4. Our goal in this subsection is to
prove that, for any $\varepsilon>0$, the lower bound $|a_{1,p}(A)|\geq
p^{\frac{1}{2g^{2}+g+1}-\varepsilon}$ holds for a set of primes $p$ of density
1. The proof proceeds similarly to that of [CoWa22, Theorem 3], as follows.

Let $x>2$ be a real number that goes to infinity. Let $z=z(x)>0$, $y=y(x)>2$,
and $u=u(x)>2$ be real numbers that depend on and grow with $x$ and satisfy
(29), (30), and (31). By ((ii)) of Proposition 19, we know that, for any
$\varepsilon>0$, there exists $c(\varepsilon)>0$ such that

$\displaystyle\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq z}}\pi_{A}(x,t)$
$\displaystyle\leq$ $\displaystyle
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq z}}\\#\left\\{p\leq
x:p\nmid\ell N_{A},a_{1,p}(A)=t,\right.$ $\displaystyle\hskip
122.34692pt\left.\ell\nmid
d_{\mathbb{Q}(\pi_{p}(A))}\\#({\mathcal{O}}_{\mathbb{Q}(\pi_{p}(A))}/\mathbb{Z}[\pi_{p}(A)]),\
\ell\ \text{splits completely in}\ \mathbb{Q}(\pi_{p}(A))\right\\}.$

Proceeding similarly to the proofs of (52) and (58), we obtain that

$\displaystyle\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq z}}\pi_{A}(x,t)$
$\displaystyle\leq$ $\displaystyle
c(\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\pi_{{\mathcal{C}}^{ss}(\ell,|t|\leq
z)}(x,\mathbb{Q}(A[\ell])/\mathbb{Q})$ $\displaystyle\leq$ $\displaystyle
c(A,\varepsilon)\displaystyle\max_{y\leq\ell\leq
y+u}\left(\frac{1}{\ell}\cdot\frac{xz}{\operatorname{log}x}+\ell^{g^{2}+\frac{g}{2}}\cdot\frac{x^{\frac{1}{2}}z^{\frac{1}{2}}}{\operatorname{log}x}\cdot\operatorname{log}\ell\right)$

for some positive constant $c(A,\varepsilon)$ that depends on $A$ and
$\varepsilon$. Choosing

$y(x):=z(x)(\operatorname{log}x)^{\eta}$

and

$z(x):=\frac{x^{\frac{1}{2g^{2}+g+1}}}{(\operatorname{log}x)^{\eta+\varepsilon}}$

for some fixed arbitrary $\eta>0$ and $\varepsilon>0$, we derive that

$\displaystyle\sum_{t\in\mathbb{Z}\atop{|t|\leq
z(x)}}\pi_{A}(x,t)\ll_{A,\varepsilon,\eta}\frac{x}{(\operatorname{log}x)^{\eta+1}}=\operatorname{o}(\pi(x)).$

Using this estimate, we deduce that

$\displaystyle\pi(x)$ $\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>\frac{p^{\frac{1}{2g^{2}+g+1}}}{(\operatorname{log}p)^{\eta+\varepsilon}}\right\\}$
$\displaystyle+$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|\leq\frac{p^{\frac{1}{2g^{2}+g+1}}}{(\operatorname{log}p)^{\eta+\varepsilon}}\right\\}+\\#\\{p\leq
x:p\mid N_{A}\\}$ $\displaystyle=$ $\displaystyle\\#\left\\{p\leq x:p\nmid
N_{A},|a_{1,p}(A)|>\frac{p^{\frac{1}{2g^{2}+g+1}}}{(\operatorname{log}p)^{\eta+\varepsilon}}\right\\}+\operatorname{o}(\pi(x)).$

This completes the proof of part (i) of Theorem 3.

###  6.8. Proof of part (ii) of Theorem 3

Yet again, let $A$ be an abelian variety defined over $\mathbb{Q}$, of
conductor $N_{A}$, and of dimension $g$. We keep the four main assumptions
(46), (50), AHC, and PCC, as in Subsections 6.5 and 6.6. Our goal in this
subsection is to prove that, for any $\varepsilon>0$, the lower bound
$|a_{1,p}(A)|\geq p^{\frac{1}{g+2}-\varepsilon}$ holds for a set of primes $p$
of density 1. The proof proceeds similarly to that of part (i), as follows.

Let $x>2$ be a real number that goes to infinity and let $\varepsilon>0$.
Using Theorem 2 , we deduce that

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

This completes the proof of part (ii) of Theorem 3.

##  7\. Final remarks

We conclude with brief remarks about the current conditional approaches
towards obtaining upper bounds for $\pi_{A}(x,t)$ for a given generic abelian
variety $A$ defined over $\mathbb{Q}$ and of dimension $g$, where by generic
we mean that $\operatorname{Im}\rho_{A}$ is open in
$\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$.

The main goal of the present paper is to prove the currently best upper bound
for $\pi_{A}(x,t)$ under GRH. Theorem 1 does provide such bounds,
substantially improving upon [CoDaSiSt17, Theorem 1, pp. 3560-3561] for any
value of $g$ and any value of $t$. The secondary goal of the paper is to prove
the currently best upper bound for $\pi_{A}(x,t)$ under other sensible
unproven hypotheses. Theorem 2 does provide such bounds, substantially
improving upon [Be16, Corollaire 17, p. 51]. However, while in the first
comparison, both results assume some version of GRH, in the second comparison,
the two results assume different hypotheses. Indeed, Theorem 2 assumes GRH,
AHC, and PCC, in contrast with [Be16, Corollaire 17, p. 51], which assumes GRH
and AHC. It is plausible that the methods of these two results may be combined
to produce an improvement to the upper bounds for $\pi_{A}(x,t)$ under GRH,
AHC, and PCC. We relegate such work to a future project.

In the case $g=1$, a different conditional approach for proving upper bounds
for $\pi_{A}(x,t)$ was initiated in [Mu85]. Therein, Murty assumed an
effective version of the Sato-Tate Conjecture for $A$ and used it to deduce
the upper bound
$\pi_{A}(x,t)\ll_{A,t}x^{1-\frac{1}{4}}(\operatorname{log}x)^{\frac{1}{2}}$.
He also proved that the effective version of the Sato-Tate Conjecture for $A$
invoked in the above bound follows from the assumptions that each of the
symmetric power L-functions of $A$ has an analytic continuation to
$\mathbb{C}$, satisfies an appropriate functional equation, and satisfies an
analogue of the Riemann Hypothesis. Thanks to recent work of Newton and
Thorner [NeTh21], the assumption about the analytic continuation is now known
to hold. Murty’s approach was exploited further by Rouse and Thorner [RoTh16],
who proved, under similar hypotheses as those of Murty’s, together with the
additional assumption that the conductor $N_{A}$ of the elliptic curve $A$ is
squarefree, that
$\pi_{A}(x,t)\ll_{A,t}\frac{x^{1-\frac{1}{4}}}{(\operatorname{log}x)^{\frac{1}{2}}}$,
a bound which improves on that of [Mu85], but not on that of [MuMuWo18]. In
the case $g\geq 2$, Bucur, Fité and Kedlaya [BuFiKe20] proved an analogue of
Murty’s result concerning the validity of an effective version of the Sato-
Tate Conjecture for $A$, under assumptions similar to the ones of [Mu85],
together with the additional assumption that the Mumford-Tate Conjecture holds
for the abelian variety $A$. Following the strategy of [Mu85], the result of
[BuFiKe20] seems to give
$\pi_{A}(x,t)\ll_{A,t}x^{1-\frac{1}{2g^{2}+2g}}(\operatorname{log}x)^{c}$ for
some (possibly non-negative) constant $c$, a bound which does not improve upon
Theorem 1. We plan to explore this approach in depth in future work.

Finally, let us note that similar methods may be used to obtain upper bounds
for $\pi_{A}(x,t)$ for other types of abelian varieties $A$. For example, in
the case of an abelian variety isogenous to the product
$E_{1}\times\ldots\times E_{g}$ of elliptic curves $E_{1},\ldots,E_{g}$
defined over $\mathbb{Q}$, pairwise non-isogenous over
$\overline{\mathbb{Q}}$, and each with a trivial
$\overline{\mathbb{Q}}$-endomorphism ring, the refined mod $\ell$ method of
[MuMuSa88], under GRH, was used in [CoWa22], while the direct mod $\ell$
method of [MuMuWo18], under GRH, AHC, and PCC, as well as the Sato-Tate method
of [Mu85], under assumptions similar to the case $g=1$, will be used in an
upcoming paper of the authors.

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