# On the Frobenius fields of abelian varieties over number fields

Ashay A. Burungale  Department of Mathematics
UT Austin
Austin, TX 78712<EMAIL_ADDRESS>,  Haruzo Hida  Department of
Mathematics
UCLA
Los Angeles, CA 90095-1555, USA<EMAIL_ADDRESS>and  Shilin Lai  Department
of Mathematics
UT Austin
Austin, TX 78712<EMAIL_ADDRESS>
###### Abstract.

Let $A$ be a non-CM simple abelian variety over a number field $K$. For a
place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote
the Frobenius field generated by the corresponding Frobenius eigenvalues.
Assuming $A$ has connected monodromy groups, we show that the set of places
$v$ such that $F(A,v)$ is isomorphic to a fixed number field has upper
Dirichlet density zero. Assuming the GRH, we give a power saving upper bound
for the number of such places.

###### Contents

  1. 1 Introduction
  2. 2 Backdrop
  3. 3 Results on finite reductive groups
  4. 4 Proof of Main Theorem

##  1\. Introduction

For a simple abelian variety over a number field and a place of good reduction
of the number field, a basic invariant is the Frobenius field generated by the
corresponding Frobenius eigenvalues. The arithmetic of the abelian variety
seems to be reflected through the Frobenius fields.

Based on CM theory of Shimura–Taniyama–Weil, the Frobenius fields are readily
seen to be contained in a fixed number field for a simple CM abelian variety
and equal to the fixed field for a set of places of positive Dirichlet
density. For a non-CM simple abelian variety, it is known that the set of
places such that the corresponding Frobenius fields equal a fixed number field
cannot be the complement of a finite set of places. A natural question is to
explore the upper Dirichlet density of the set of places at which the
Frobenius field coincides with a given one up to an isomorphism. We show that
the density is zero under a connected-ness hypothesis (see Theorem 1.1). Thus,
if a set of places with equal Frobenius fields has positive upper Dirichlet
density, then the abelian variety is CM.

### Main result

Let $K$ be a number field and ${\varSigma}_{K}$ the set of its finite places.
For $\overline{K}$ an algebraic closure, let $G_{K}=\mathop{\rm
Gal}\nolimits(\overline{K}/K)$ be the corresponding absolute Galois group. For
a place $v\in{\varSigma}_{K}$, let ${\rm Frob}_{v}$ be a geometric Frobenius
element at $v$.

Let $A$ be an abelian variety over $K$ of dimension $d$ and conductor
$\mathfrak{N}$. For a prime $p$, let $T_{p}A$ be the $p$-adic Tate module of
$A$ and

$\rho_{A,p}:G_{K}\rightarrow\mathop{\rm Aut}\nolimits_{\mathbb{Z}_{p}}T_{p}A$

the associated $p$-adic Galois representation. For $v\nmid p\mathfrak{N}$, it
is unramified at $v$. Let $F(A,v)$ be the splitting field of the
characteristic polynomial of $\rho_{A,p}({\rm Frob}_{v})$. As
$(\rho_{A,p})_{p}$ is a compatible system of Galois representations, $F(A,v)$
is a number field independent of $p$, referred to as the Frobenius field
associated to the pair $(A,v)$.

Let $M$ be a number field and $S_{A,M}$ a subset of finite places
${\varSigma}_{K}$ of $K$ given by

$S_{A,M}=\\{v\in{\varSigma}_{K}|v\nmid\mathfrak{N},F(A,v)\cong M\\}.$

When $A$ is a simple CM abelian variety, the Frobenius fields are a
subfield111More precisely, the Frobenius fields are a subfield of the CM
endomorphism field $K$ given by $\alpha^{\Phi^{\prime}}\in K$ for some
$\alpha\in K^{\prime}$ for the reflex CM type $(K^{\prime},\Phi^{\prime})$. of
the corresponding CM field (cf. [31]). For a set of places of Dirichlet
density one, the Frobenius fields equal a fixed subfield of the CM field up to
isomorphisms.

In this article we consider dependence of the Frobenius fields $F(A,v)$ on the
place $v$ for non-CM simple abelian varieties. To state the results, we recall
the following notion. For $S\subset{\varSigma}_{K}$, the upper Dirichlet
density ${\rm ud}(S)$ is given by

${\rm
ud}(S)=\limsup_{X\rightarrow\infty}\frac{\\#\\{v\in{\varSigma}_{K}:\mathbf{N}_{K/\mathbb{Q}}v\leq
X,X\in S\\}}{\\#\\{v\in{\varSigma}_{K}:\mathbf{N}_{K/\mathbb{Q}}v\leq X\\}}$

for $\mathbf{N}_{K/\mathbb{Q}}$ the norm of the extension $K/\mathbb{Q}$.

The qualitative version of our main result is the following.

######  Theorem 1.1.

Let $A$ be an absolutely simple non-CM abelian variety defined over a number
field $K$. Suppose that the monodromy groups associated to $A$ over $K$ are
connected (cf. Hypothesis 2.2). Then for a number field $M$, we have

${\rm ud}(S_{A,M})=0.$

The connectivity hypothesis 2.2 is satisfied by any $A$ over a finite
extension of its field of definition (cf. Remark 2.3). Moreover, it holds for
a class of abelian varieties over their field of definition. For instance, it
holds for most abelian varieties in a typical family (e.g. hyperelliptic
Jacobians).

######  Remark 1.2.

Theorem 1.1 gives a criterion for characterisation of CM/non-CM abelian
varieties. For other criteria in the case of $\mathop{\rm
GL}\nolimits_{2}$-type abelian varieties over $\mathbb{Q}$, the reader may
refer to [12, §3.1.1].

We now state the quantitative version. Let $\mathbf{G}$ be the Mumford–Tate
group associated to $A$. Let $\mathbf{G}^{\mathrm{ss}}$ be its semisimple
quotient, with dimension $d$. Let $r=\mathop{\rm
rank}\nolimits\mathbf{G}^{\mathrm{ss}}$ denote its absolute rank, namely the
dimension of a maximal torus of $\mathbf{G}^{\mathrm{ss}}$.

######  Theorem 1.3.

Let $A$ be an absolutely simple non-CM abelian variety defined over a number
field $K$. Assume that the monodromy groups associated to $A$ over $K$ are
connected (cf. Hypothesis 2.2), and the generalized Riemann hypothesis (GRH)
for all Hecke $L$-functions holds.

Fix a number field $M$. Then for all $\varepsilon>0$, there exists a constant
$c$ depending on $(A,K,\varepsilon)$ such that for all sufficiently large $X$
(depending on $M$), we have

$|\\{v\in S_{A,M}\,|\,\mathbf{N}_{K/\mathbb{Q}}v\leq X\\}|\leq
cX^{1-\frac{1}{3d+r+6}+\varepsilon}$

######  Corollary 1.4.

Suppose further that $A$ is generic222That is, the associated adelic Galois
representation has open image in ${\rm GSp}_{2g}(\widehat{\mathbb{Z}})$.. Then
for all $\varepsilon>0$, there exists a constant $c$ depending on
$(A,K,\varepsilon)$ such that

$|\\{v\in{\varSigma}_{K}\,|\,\mathbf{N}_{K/\mathbb{Q}}v\leq X,v\text{ has
Frobenius field M}\\}|\leq cX^{1-\frac{1}{6g^{2}+2g+6}+\varepsilon}$

for all sufficiently large $X$ depending on $M$.

### About the proof

Our approach is primarily Galois and group theoretic. We formulate the problem
in terms of (a variant of) Frobenius tori in the algebraic monodromy group of
the abelian variety. A volume computation of conjugacy classes arising from
the tori and large Galois image results are the keys of the proof. We now
describe the strategy in more detail.

For primes $p$ which split completely in $M$, we show that if $v\in S_{A,M}$,
then the image of $\mathrm{Frob}_{v}$ in the mod-$p$ monodromy group lies in a
$\mathbb{F}_{p}$-rational Borel subgroup. We also give a volume upper bound
for this bounding subset (cf. Corollary 3.4), which may be of independent
interest. This step uses standard results on reductive groups, several of
which require the group to be connected.

Combined with results about large Galois image for non-CM abelian varieties
due to Wintenberger [32] and Hui–Larsen [13] (cf. Theorem 2.5), we obtain
upper bound to the density of $S_{A,M}$ by the Chebotarev density theorem. In
the other approaches this upper bound decreases as $p$ increases. However, our
upper bound does not decay in $p$, and so we need to consider multiple primes
at the same time. This is allowed by Serre’s result on potential
$\ell$-independence (cf. Theorem 2.4(2)).

To derive the power-saving upper bound, a naïve aplication of the effective
Chebotarev density theorem of Lagarias–Odlyzko appears to be insufficient.
Instead, we reformulate the problem as a sieving problem. Again using
potential $\ell$-independence, we write the adèlic Galois image as a union of
cosets of a subgroup which is a direct product. On each coset, the coefficient
of the main term is multiplicative, so the Selberg sieve can be directly
applied. We further improve the exponent by observing that the Borel subgroup
of a finite reductive group satisfies the Artin holomorphy conjecture (AHC),
which allows us to use the better error term of Murty–Murty–Saradha (cf.
Theorem 4.8).

### The prior work and prospects

These investigations are partly inspired by the Lang–Trotter conjecture [19]
and a consideration of Serre for Hecke eigenvalues of an elliptic newform [27,
§7]. For elliptic curves, the result was first stated333albeit without proof
or explicit exponent by Serre [27, §8.2]. Its various explicit forms have
appeared in the work of Cojocaru–Fouvry–Murty [7], Cojocaru–David [5], Zywina
[33], and Kulkarni–Patankar–Rajan [16]. Cojocaru–David [6] also consider the
analogous question for certain Drinfeld modules.

For generic abelian varieties, the result was established by Bloom [1]. In
general, the only known result seems to be that of Khare [15]: for a non-CM
simple abelian variety, the set of places such that the corresponding
Frobenius fields equal a fixed number field cannot be the complement of a
finite set of places.

Our result applies to any abelian variety up to a base change. In the generic
case, no base change is needed, and it gives a better exponent than the one
obtained by Bloom [1] under the same assumptions, though with the caveat that
the “sufficiently large $X$” constant depends on $M$. The qualitative part of
the argument also applies to function fields, and we expect a suitable
modification can be used to derive the quantitative version. Moreover, our
method seems to apply to any compatible system of Galois representations for
which Frobenius semisimplicity holds (cf. Remark 2.6).

Our approach differs from the previous approaches. It is akin to a group-
theoretic reinterpretation of the square sieve method. One sign is that
directly applying the effective Chebotarev density theorem gives the same
exponents as Bloom’s result. In Remark 4.19 we sketch a distilled argument
which removes the dependency on $M$ and still improves the exponent. A refined
analysis of conjugacy classes in finite reductive groups likely yields sharper
results.

Finally, to remove Hypothesis 2.2, we would need to study the structure of
semisimple elements in a disconnected finite reductive group. We plan to
return to this question in the near future.

### Plan

In section 2 we describe preliminaries regarding Galois representations,
monodromy groups, and large image theorems. In section 3 we present
preliminaries regarding finite redcutive groups and define the bounding set
(cf. Definition 3.3). In section 4 we prove the main theorems.

######  Acknowledgments.

We thank Brian Conrad, Chun-Yin Hui, Chandrashekhar Khare, Vijay Patankar,
Will Sawin, and Victor Wang for helpful comments.

##  2\. Backdrop

We describe some preliminaries regarding $p$-adic Galois representation
associated to an abelian variety over a number field. The reader may refer to
[13, 24, 32] for some details.

Let the setting be as in section 1. In particular, $A$ is an abelian variety
over a number field $K$ of dimension $g$. Put
$\Delta_{A}=\mathbf{N}_{K/\mathbb{Q}}\mathfrak{N}$, so $A$ has good reduction
away from places of $K$ which lie above a prime dividing $\Delta_{A}$.

###  2.1. Galois representations

For a prime $p$, let $T_{p}A$ be the $p$-adic Tate module of $A$. Let

$\rho_{p}:G_{K}\rightarrow\mathop{\rm Aut}\nolimits_{\mathbb{Z}_{p}}T_{p}A$

be the associated $p$-adic Galois representation.

######  Theorem 2.1.

Let $v$ be a finite place of $K$ such that $v\nmid p\Delta_{A}$.

  1. (1)

(Serre–Tate) The Galois representation $\rho_{p}$ is unramified at $v$.

  2. (2)

(Weil) The characteristic polynomial of $\rho_{p}({\rm Frob}_{v})$ has
integral coefficients and is independent of $p$.

For a set $T$ of primes, put

$\rho^{T}=\prod_{p\notin T}\rho_{p},\quad\rho_{T}=\prod_{p\in T}\rho_{p}.$

For $N$ a positive integer, put $\rho^{N}=\rho^{T(N)},\ \rho_{N}=\rho_{T(N)}$,
where $T(N)$ is the set of primes dividing $N$.

####  2.1.1. Monodromy groups

Let

$\Gamma_{p}=\rho_{p}(G_{K})\subseteq\mathop{\rm
Aut}\nolimits_{\mathbb{Z}_{p}}T_{p}A,$

and $\mathbf{G}_{p}$ denote its Zariski closure. This is a linear algebraic
group over $\mathbb{Z}_{p}$. The generic fibre of $\mathbf{G}_{p}$ is the
usual $p$-adic monodromy group. Throughout this paper, we make the following
assumption.

######  Hypothesis 2.2.

The group $\mathbf{G}_{p}$ is connected for a prime $p$.

######  Remark 2.3.

A result of Serre [28, Corollaire p. 15] shows that the component group
$\mathbf{G}_{p}/\mathbf{G}_{p}^{\circ}$ is independent of the prime $p$. So
the hypothesis holds for all primes $p$ as soon as it holds for one prime. It
is also proven in _op. cit._ that there exists a finite extension
$K^{\mathrm{conn}}/K$ such that this hypothesis holds for the base change
$A_{/K^{\mathrm{conn}}}$.

We will use the following key results on the monodromy group.

######  Theorem 2.4.

  1. (1)

If $p$ is sufficiently large, then $\mathbf{G}_{p}$ is reductive.

  2. (2)

There exists a finite Galois extension $L/K$ and an integer $N$ such that

$\rho^{N}(G_{L})=\prod_{p\nmid N}\rho_{p}(G_{L})$

Part (1) is due to Larsen–Pink (cf. the proof of [21, Theorem 3.2]). Part (2)
is another result of Serre [28, Théorème p. 56] (see also [14, 30]). The
reference does not include the Galois requirement, but we can take the Galois
closure, which preserves independence away from a finite set of primes.

####  2.1.2. Galois image

This subsection will recall some large image results for non-CM abelian
varieties due to Hui–Larsen [13, Theorem 1.3]. An essentially equivalent
result was obtained earlier by Wintenberger [32, Théorème 2], but this
formulation is more convenient for us.

For a prime $p$ such that $\mathbf{G}_{p}$ is reductive, let
$\mathbf{G}_{p}^{\mathrm{ss}}$ denote the quotient of $\mathbf{G}_{p}$ by its
radical, and $\Gamma_{p}^{\rm ss}$ the image of $\Gamma_{p}$ in
$\mathbf{G}_{p}^{\mathrm{ss}}$. Let
$\pi:\mathbf{G}_{p}^{\mathrm{sc}}\to\mathbf{G}_{p}^{\mathrm{ss}}$ be the
algebraic universal cover. This is a finite morphism.

######  Theorem 2.5.

For all sufficiently large $p$, we have

$\mathop{\rm
Im}\nolimits(\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{Z}_{p})\to\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{Z}_{p}))\subseteq\Gamma_{p}^{\mathrm{ss}}.$

###### Proof.

The cited result of Hui–Larsen is that for $p\gg 0$, the group
$\mathbf{G}_{p}(\mathbb{Q}_{p})$ is unramified, and
$\pi^{-1}(\Gamma_{p}^{\mathrm{ss}})$ is a hyperspecial maximal compact
subgroup of $\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{Q}_{p})$. It is not easy to
extract the integral model directly from the proof, so we conclude in an
indirect way. We have the inclusions

$\pi^{-1}(\Gamma_{p}^{\mathrm{ss}})\subseteq\pi^{-1}(\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{Z}_{p}))\subseteq\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{Z}_{p})$

The first is by definition. The second holds since $\pi$ is finite, hence
proper. The final group is also a hyperspecial maximal compact subgroup of
$\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{Q}_{p})$, and so all three groups above
are equal. ∎

######  Remark 2.6.

If we have a compatible system of semisimple Galois representations coming
from geometry, then Theorem 2.4 still holds. Larsen showed that Theorem 2.5
holds for a density one set of primes $p$ [20, Theorem 3.17]. The exceptional
set in that proof is again cut out using Chebotarev sets. Therefore, it’s
likely that our approach works in such generality, though we have not checked
the details carefully.

Define the reduced residual representation

$\overline{\rho}^{\mathrm{ss}}_{p}:G_{K}\to\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p}),$

Its image is also the image of $\Gamma_{p}^{\mathrm{ss}}$ under the reduction
map
$\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{Z}_{p})\to\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})$.
Denote it by $\overline{\Gamma}_{p}^{\mathrm{ss}}$. A simple consequence of
Theorem 2.5 is the following.

######  Corollary 2.7.

Let $I_{p}=\mathop{\rm
Im}\nolimits(\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{F}_{p})\to\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p}))$,
then for all sufficiently large $p$, $\overline{\Gamma}_{p}^{\mathrm{ss}}$
contains $I_{p}$.

###  2.2. A characterisation of CM abelian varieties

######  Proposition 2.8.

Let $A$ be a simple abelian variety over a number field $K$. If the associated
$p$-adic monodromy group $\mathbf{G}_{p/\mathbb{Q}_{p}}$ is abelian for one
prime $p$, then it is so for all primes, and $A$ has CM.

###### Proof.

Suppose that ${\bf G}_{p/\mathbb{Q}_{p}}$ is abelian. The $p$-adic Galois
representation $\rho_{p}$ is semisimple by Faltings, and rational by
construction. Hence, this Galois representation arises from an arithmetic
Hecke character over $K$ by a result of Henniart [11] (cf. [26, Theorem 2 in
III-13]). Considering infinity type of the Hecke character, it follows that
$\rho_{p}$ corresponds to a CM abelian variety $A^{\prime}$. Note that $A$ and
$A^{\prime}$ are isogenous by Falting’s isogeny theorem. Hence $A$ has CM, and
the assertion follows. ∎

##  3\. Results on finite reductive groups

Let $\mathbf{G}$ be a connected reductive group defined over $\mathbb{F}_{p}$.
In this section we will give a volume upper bound for the union of certain
conjugacy classes.

###  3.1. Tori and Weyl groups

We recall some basic results about maximal tori of $\mathbf{G}$. A detailed
exposition can be found in [4, Section 3.3].

Let $W_{G}$ be the absolute Weyl group of $\mathbf{G}$. It carries an action
of the Galois group $\mathop{\rm
Gal}\nolimits(\overline{\mathbb{F}}_{p}/\mathbb{F}_{p})$. Let $\gamma$ be the
action corresponding to the Frobenius element. The general theory of forms
gives a bijection

$\\{\text{Conjugacy class of maximal tori defined over
}\mathbb{F}_{p}\\}\longleftrightarrow\\{\gamma\text{-conjugacy classes in
}W_{G}\\}.$

Let $\mathbf{T}$ be a maximal torus in $\mathbf{G}$. Define its Weyl group by

$W(\mathbf{G},\mathbf{T})=N_{\mathbf{G}(\mathbb{F}_{p})}(\mathbf{T})/\mathbf{T}(\mathbb{F}_{p}).$

It can also be described as the stabilizer of the $\gamma$-conjugacy class
corresponding to $\mathbf{T}$ under the above bijection. In particular, the
$\gamma$-twisted class equation gives

(3.1) $\sum_{\mathbf{T}}\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}=1$

where the sum is over a set of representatives for the conjugacy classes of
maximal tori in $\mathbf{G}$.

######  Lemma 3.1.

If $\mathbf{G}$ is not abelian, then $W(\mathbf{G},\mathbf{T})$ is non-trivial
for any maximal torus $\mathbf{T}$.

###### Proof.

By the class equation (3.1), if the lemma holds for one choice of
$\mathbf{T}$, then it holds for all choices of $\mathbf{T}$. Let $\mathbf{B}$
be a Borel subgroup of $\mathbf{G}$ defined over $\mathbb{F}_{p}$. Its
existence is guaranteed by Lang’s theorem [18]. Let $\mathbf{S}$ be a maximal
$\mathbb{F}_{p}$-split torus contained in $\mathbf{B}$, and let $\mathbf{T}$
be its centralizer. This is a Levi subgroup of $\mathbf{B}$, and hence a
maximal torus. In addition, $N_{\mathbf{G}}(\mathbf{S})\subseteq
N_{\mathbf{G}}(\mathbf{T})$, so the Weyl group $W(\mathbf{G},\mathbf{T})$
contains the relative Weyl group $W(\mathbf{G},\mathbf{S})$.444We in fact have
equality here, but this will not be needed.

Since $\mathbf{G}$ is not abelian, its derived subgroup $\mathbf{G}^{\rm der}$
is non-trivial. By Lang’s theorem, $\mathbf{G}^{\rm der}$ has a Borel
subgroup, which implies that $\mathbf{G}^{\rm der}$ has positive rank [3,
Corollaire 4.17]. The second paragraph of the proof of [3, Théorème 5.3] then
shows that the relative Weyl group $W(\mathbf{G},\mathbf{S})$ is non-trivial,
so $W(\mathbf{G},\mathbf{T})$ is also non-trivial for this choice of
$\mathbf{T}$. ∎

###  3.2. Conjugacy classes

Given a maximal torus $\mathbf{T}$, let $C_{\mathbf{T}}$ denote the set of
elements in $\mathbf{G}(\mathbb{F}_{p})$ which are conjugate to an element of
$\mathbf{T}(\mathbb{F}_{p})$. We will also use the superscript “reg” to denote
the subset of regular semisimple elements. The volume is the counting measure
normalized so that $\mathop{\rm vol}\nolimits(\mathbf{G}(\mathbb{F}_{p}))=1$.

######  Proposition 3.2.

There exists a constant $C$ depending only on the (absolute) rank of the group
$\mathbf{G}$ such that the following holds for all $p$

  1. (1)

$\mathop{\rm
vol}\nolimits(\mathbf{G}(\mathbb{F}_{p})^{\mathrm{reg}})>1-Cp^{-1}$,

  2. (2)

If $\mathbf{T}$ is a maximal torus, then

$\left|\mathop{\rm
vol}\nolimits(C_{\mathbf{T}}^{\mathrm{reg}})-\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\right|<Cp^{-1}.$

###### Proof.

Let $\mathbf{T}$ be a maximal torus. The map
$\mathbf{G}/\mathbf{T}\times\mathbf{T}\to\mathbf{G},\ (g,t)\mapsto gtg^{-1}$
is finite of degree $\left|W(\mathbf{G},\mathbf{T})\right|$ above the regular
elements in its image. Indeed, if
$gtg^{-1}=g^{\prime}t^{\prime}(g^{\prime})^{-1}$, then $g^{-1}g^{\prime}$
conjugates $t^{\prime}$ to $t$, so it conjugates $Z_{\mathbf{G}}(t^{\prime})$
to $Z_{\mathbf{G}}(t)$. If $t$ is regular, then the centralizer is just
$\mathbf{T}$, so $g^{-1}g^{\prime}\in N_{\mathbf{G}}(\mathbf{T})$. On
$\mathbb{F}_{p}$-points, the image of this map is exactly $C_{\mathbf{T}}$, so

(3.2) $\mathop{\rm
vol}\nolimits(C_{\mathbf{T}}^{\mathrm{reg}})=\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\cdot\frac{\left|\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\right|}.$

Summing over a set of representatives of conjugacy classes of maximal tori, we
get

$\mathop{\rm
vol}\nolimits(\mathbf{G}(\mathbb{F}_{p})^{\mathrm{reg}})=\sum_{\mathbf{T}}\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\cdot\frac{\left|\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\right|}.$

We now give a coarse estimate for the right hand side.

Let $r=\mathop{\rm rank}\nolimits\mathbf{T}$. The set of non-regular elements
in $\mathbf{T}(\mathbb{F}_{p})$ is the $\mathbb{F}_{p}$-points of a variety
$\mathbf{T}^{\mathrm{nr}}$ of dimension $r-1$. When base changed to
$\overline{\mathbb{F}}_{p}$, it can be described as the subset of
$\mathbf{G}_{m,\overline{\mathbb{F}}_{p}}^{r}$ where at least two of the
entries are equal. Let $E_{/\mathbb{Z}}$ be the subscheme of
$\mathbf{G}_{m,\mathbb{Z}}^{r}$ defined this way, then we see that
$\mathbf{T}^{\mathrm{nr}}$ is a form of $E\times_{\mathbb{Z}}\mathbb{F}_{p}$.
It follows from the Weil bound555One may also proceed by an elementary
argument: counting $\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}$ using
inclusion-exclusion, but the details are messy. that there exist a constant
$C$ (depending only on $r$) such that

(3.3)
$\left|\frac{\left|\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\right|}-1\right|\leq
Cp^{-1}$

for all $p$ (cf. the proof of Lemma 4.15).

This estimate together with equation (3.2) immediately gives part (2) of the
proposition. Using equation (3.1), we see that

$\mathop{\rm
vol}\nolimits(\mathbf{G}(\mathbb{F}_{p})^{\mathrm{reg}})\geq\sum_{\mathbf{T}}\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\cdot(1-Cp^{-1})=1-Cp^{-1}.$

This proves (1). ∎

######  Definition 3.3.

Let $\mathbf{G}$ be a reductive group over $\mathbb{F}_{p}$. Its _bounding
set_ is the set of elements in $\mathbf{G}(\mathbb{F}_{p})$ which lie in a
Borel subgroup defined over $\mathbb{F}_{p}$.

A consequence of Proposition 3.2 is the following.

######  Corollary 3.4.

Let $\mathcal{B}$ be the bounding set of $\mathbf{G}$. There exists an integer
$N$ depending only on the rank of $\mathbf{G}$ such that if $\mathbf{G}$ is
non-abelian and $p>N$, then

$\mathop{\rm vol}\nolimits(\mathcal{B})<\frac{3}{4}.$

###### Proof.

Let $\mathbf{T}$ be a maximal torus in a Borel subgroup $\mathbf{B}$. Let
$x\in\mathcal{B}$ be regular semisimple. We will show that $x$ is conjugate to
an element of $\mathbf{T}$. Indeed, all Borel subgroups are conjugate [3,
Théorème 4.13(b)], so we may assume $x\in\mathbf{B}$. Any maximal torus of
$\mathbf{B}$ containing $x$ is a Levi component, so it is conjugate to
$\mathbf{T}$ [3, Proposition 4.7]. It follows that $\mathcal{B}$ is contained
in the union of non-regular semisimple elements and $C_{\mathbf{T}}$.
Therefore,

$\mathop{\rm vol}\nolimits(\mathcal{B})\leq\mathop{\rm
vol}\nolimits(\mathbf{G}(\mathbb{F}_{p})-\mathbf{G}(\mathbb{F}_{p})^{\mathrm{reg}})+\mathop{\rm
vol}\nolimits(C_{\mathbf{T}}^{\mathrm{reg}})<2Cp^{-1}+\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}$

by Proposition 3.2. Lemma 3.1 shows that $W(\mathbf{G},\mathbf{T})$ is non-
trivial, so its order is at least 2. By taking $p$ sufficiently large, the
right hand side can be made smaller than any real number greater than
$\frac{1}{2}$. ∎

###  3.3. Central isogeny

This subsection refines Proposition 3.2 and Corollary 3.4. This refinement is
the key ingredient which allows us to apply the large image results of Theorem
2.5 and Corollary 2.7.

Suppose $\mathbf{G}$ is semisimple, then it has an algebraic universal cover
$\pi:\mathbf{G}^{\mathrm{sc}}\to\mathbf{G}$. Its kernel, denoted by $Z$, is a
finite group scheme over $\mathbb{F}_{p}$ and contained in the centre of
$\mathbf{G}^{\mathrm{sc}}$. Let $\mathbf{T}$ be a maximal torus of
$\mathbf{G}$, then $\mathbf{T}^{\mathrm{sc}}:=\pi^{-1}(\mathbf{T})$ is a
maximal torus in $\mathbf{G}^{\mathrm{sc}}$ containing $Z$.

We have the following exact sequence of groups

${1}$${Z(\mathbb{F}_{p})}$${\mathbf{T}^{\mathrm{sc}}(\mathbb{F}_{p})}$${\mathbf{T}(\mathbb{F}_{p})}$${\mathrm{H}^{1}(\mathbb{F}_{p},Z)}$${1}$${1}$${Z(\mathbb{F}_{p})}$${\mathbf{G}^{\mathrm{sc}}(\mathbb{F}_{p})}$${\mathbf{G}(\mathbb{F}_{p})}$${\mathrm{H}^{1}(\mathbb{F}_{p},Z)}$${1.}$$\scriptstyle{\delta}$

The final 1 in both rows are by Lang’s theorem. Note that _a priori_ , the
connecting morphisms are between pointed sets, but since $Z$ is central, they
are actually group homomorphisms (cf. [29, §I.5.6]).

######  Lemma 3.5.

We have
$\left|\mathbf{G}^{\mathrm{sc}}(\mathbb{F}_{p})\right|=\left|\mathbf{G}(\mathbb{F}_{p})\right|$.

###### Proof.

Since $Z(\overline{\mathbb{F}}_{p})$ is finite, its Herbrand quotient is 1.
Hence the assertion follows by considering the bottom row of the above
diagram. We remark that this is also a special case of [2, Chapter V,
Proposition 16.8]. ∎

The main result of this section is that the various conjugacy classes we have
considered are approximately equally distributed in the fibres of $\delta$.
More precisely, we have the following.

######  Proposition 3.6.

Let $\mathbf{G}$ be a non-abelian semisimple group. Let $I=\mathop{\rm
Im}\nolimits(\mathbf{G}^{\mathrm{sc}}(\mathbb{F}_{p})\to\mathbf{G}(\mathbb{F}_{p}))$.

  1. (1)

There exists a constant $C$ depending only on the rank of $\mathbf{G}$ such
that for any maximal torus $\mathbf{T}$ and $g\in\mathbf{G}(\mathbb{F}_{p})$,
we have

$\bigg{|}\frac{\left|C_{\mathbf{T}}^{\mathrm{reg}}\cap
gI\right|}{\left|I\right|}-\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\bigg{|}<Cp^{-1}.$

  2. (2)

There exists an integer $N$ depending only on the rank of $\mathbf{G}$ such
that for all $p>N$, we have

$\frac{\left|\mathcal{B}\cap gI\right|}{\left|I\right|}<\frac{3}{4}$

where $\mathcal{B}$ is the bounding set for $\mathbf{G}$.

###### Proof.

Let
$n=[\mathbf{G}(\mathbb{F}_{p}):I]=\left|\mathrm{H}^{1}(\mathbb{F}_{p},Z)\right|=\left|Z(\mathbb{F}_{p})\right|$.
The group $Z$ is a subgroup of the center of a semisimple simply connected
group over $\overline{\mathbb{F}}_{p}$. Since the rank is fixed, there are a
finite number of such groups, so $n$ is bounded from above, independently of
$\mathbf{G}$.

From the above diagram, $I=\mathop{\rm ker}\nolimits\delta$ is a normal
subgroup with an abelian quotient, so if two elements of
$\mathbf{G}(\mathbb{F}_{p})$ are conjugate, then they lie in the same coset of
$I$. Moreover, $\delta$ is still surjective when restricted to
$\mathbf{T}(\mathbb{F}_{p})$, so it identifies
$\mathbf{T}(\mathbb{F}_{p})/(\mathbf{T}(\mathbb{F}_{p})\cap I)$ with
$\mathrm{H}^{1}(\mathbb{F}_{p},Z)$. Fix $g\in\mathbf{G}(\mathbb{F}_{p})$, then
we get

$\left|\mathbf{T}(\mathbb{F}_{p})\cap
gI\right|=\frac{\left|\mathbf{T}(\mathbb{F}_{p})\right|}{\left|\mathrm{H}^{1}(\mathbb{F}_{p},Z)\right|}.$

The conjugation action $\mathbf{G}/\mathbf{T}\times\mathbf{T}\to\mathbf{G},\
(x,t)\mapsto xtx^{-1}$, as in the proof of Proposition 3.2, restricts to an
action on the coset $gI$, so

$\displaystyle\left|C_{\mathbf{T}}^{\mathrm{reg}}\cap gI\right|$
$\displaystyle=\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\cdot\frac{\left|\mathbf{G}(\mathbb{F}_{p})\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\right|}\cdot\left|\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}\cap
gI\right|$
$\displaystyle=\frac{\left|I\right|}{\left|W(\mathbf{G},\mathbf{T})\right|}\cdot\frac{\left|\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}\cap
gI\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\cap gI\right|}.$

Therefore, we have

$\bigg{|}\frac{\left|C_{\mathbf{T}}^{\mathrm{reg}}\cap
gI\right|}{\left|I\right|}-\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}\bigg{|}=\frac{\left|(\mathbf{T}(\mathbb{F}_{p})-\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}})\cap
gI\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\cap
gI\right|}\leq\frac{\left|\mathbf{T}(\mathbb{F}_{p})-\mathbf{T}(\mathbb{F}_{p})^{\mathrm{reg}}\right|}{\left|\mathbf{T}(\mathbb{F}_{p})\right|/n}\leq
nCp^{-1}$

where $C$ is the constant from the equation (3.3).

For the second part, take $\mathbf{T}$ to be the maximal torus in a Borel
subgroup as before. Let $C$ be the constant in Proposition 3.2, then

$\displaystyle\frac{\left|\mathcal{B}\cap gI\right|}{\left|I\right|}$
$\displaystyle\leq\frac{\left|\mathbf{G}(\mathbb{F}_{p})-\mathbf{G}(\mathbb{F}_{p})^{\mathrm{reg}}\right|}{\left|I\right|}+\frac{\left|C_{\mathbf{T}}^{\mathrm{reg}}\cap
gI\right|}{\left|I\right|}$
$\displaystyle<C(n+1)p^{-1}+\frac{1}{\left|W(\mathbf{G},\mathbf{T})\right|}.$

The same argument as in Corollary 3.4 gives the desired conclusion. ∎

##  4\. Proof of Main Theorem

We now prove the main results. The first subsection sets up some notation.
Then the second contains a short proof of the qualitative version (cf. Theorem
1.1). Finally, the third subsection is dedicated to the quantitative version
(cf. Theorem 1.3).

###  4.1. Preliminaries

####  4.1.1. Setting

Let $A$ be a simple abelian variety of dimension $g$ defined over a number
field $K$ such that $A_{/\bar{K}}$ does not have CM. Let $\Delta_{A}$ be the
norm of the conductor of $A$. Let $M$ be a number field and

$S_{A,M}=\\{v\in{\varSigma}_{K}|v\nmid\Delta_{A},F(A,v)\cong M\\}.$

For $X$ a positive real number, define

$S_{A,M}(X)=\\{v\in S_{A,M}\,|\,\mathbf{N}_{K/\mathbb{Q}}v\leq X\\}.$

Our goal is to bound $S_{A,M}(X)$ from above, supposing Hypothesis 2.2.

Let $L$ be a finite Galois extension of $K$ such that the conclusion of
Theorem 2.4(2) holds, and let $N$ be the bound given therein. Increase $N$ so
that for all $p>N$, the following holds:

  1. (1)

The monodromy group $\mathbf{G}_{p}$ is reductive, cf. Theorem 2.4(1).

  2. (2)

The large image result in Corollary 2.7 holds for $A_{/L}$

  3. (3)

The density estimate in Proposition 3.6(2) holds for all groups of rank at
most $2g$.

This $N$ depends only on $A$ and $K$. Let ${\mathcal{P}}$ be the set of all
primes greater than $N$ which split completely in $M$.

####  4.1.2. Preliminary lemmas

######  Lemma 4.1.

Pick $p\in{\mathcal{P}}$ and $v\in S_{A,M}$ which does not lie above $p$. Then
the Frobenius
$\overline{\rho}_{p}^{\mathrm{ss}}(\mathrm{Frob}_{v})\in\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})$
lives in a Borel subgroup defined over $\mathbb{F}_{p}$.

###### Proof.

Let
$F_{v}=\rho_{p}^{\mathrm{ss}}(\mathrm{Frob}_{v})\in\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{Z}_{p})$.
By Faltings’ theorem, this is a semisimple element in
$\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{Q}_{p})$. Moreover, our assumptions on
$p$ and $v$ together imply that $F_{v}$ lies in a split torus, and hence in a
minimal parabolic subgroup of $\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{Q}_{p})$,
which is necessarily a Borel subgroup since $\mathbf{G}_{p}^{\mathrm{ss}}$ is
unramified. The scheme of Borel subgroups over $\mathbb{Z}_{p}$ is proper [10,
XXII, 5.8.3(i)], so we can extend this Borel subgroup to one defined over
$\mathbb{Z}_{p}$. Its fibre over $\mathbb{F}_{p}$ is a Borel subgroup of
$\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})$ containing the reduction of
$F_{v}$. ∎

######  Lemma 4.2.

The image of the Galois representation

$\overline{\rho}^{\mathrm{ss}}_{{\mathcal{P}}}:G_{K}\to\prod_{p\in{\mathcal{P}}}\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})$

is a union of cosets of $\prod_{p\in{\mathcal{P}}}I_{p}$, where $I_{p}$ is
defined as in Corollary 2.7.

###### Proof.

By the choice of $L$, we have

$\overline{\rho}^{\mathrm{ss}}_{{\mathcal{P}}}(G_{L})=\prod_{p\in{\mathcal{P}}}\overline{\rho}_{p}^{\mathrm{ss}}(G_{L})\subseteq\prod_{p\in{\mathcal{P}}}\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p}).$

Item (2) of the choice of ${\mathcal{P}}$ implies that
$\overline{\rho}_{p}^{\mathrm{ss}}(G_{L})$ contains the subgroup $I_{p}$ for
all $p\in{\mathcal{P}}$. Therefore,

$\overline{\rho}_{{\mathcal{P}}}^{\mathrm{ss}}(G_{K})\supseteq\overline{\rho}^{\mathrm{ss}}_{{\mathcal{P}}}(G_{L})\supseteq\prod_{p\in{\mathcal{P}}}I_{p}.$

In other words, $\overline{\rho}_{{\mathcal{P}}}^{\mathrm{ss}}(G_{K})$ is a
subgroup of
$\prod_{p\in{\mathcal{P}}}\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})$ which
contains the subgroup $\prod_{p\in{\mathcal{P}}}I_{p}$. ∎

###  4.2. Proof of Theorem 1.1

Let $d$ be a square-free product of primes in ${\mathcal{P}}$. Let
${\mathcal{F}}_{d}$ be the union of all conjugacy classes
$\overline{\rho}_{d}^{\mathrm{ss}}(\mathrm{Frob}_{v})$ for $v\in S_{A,M}$ not
dividing $d$.

By Lemma 4.1, we have

${\mathcal{F}}_{d}\subseteq\Big{(}\prod_{p|d}\mathcal{B}_{p}\Big{)}\cap\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})$

where $\mathcal{B}_{p}$ is the bounding set for $\mathbf{G}_{p}^{\mathrm{ss}}$
introduced in Definition 3.3. The Chebotarev density theorem implies that

(4.1)
$\mathrm{ud}(S_{A,M})\leq\frac{\left|{\mathcal{F}}_{d}\right|}{\left|\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})\right|}\leq\frac{\big{|}\big{(}\prod_{p|d}\mathcal{B}_{p}\big{)}\cap\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})\big{|}}{\left|\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})\right|}.$

To estimate the right hand side, we utilise the following decomposition

$\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})=\bigsqcup_{i=1}^{s}\Big{(}g_{i}\prod_{p|d}I_{p}\Big{)}=\bigsqcup_{i=1}^{s}\Big{(}\prod_{p|d}g_{i,p}I_{p}\Big{)}$

given by Lemma 4.2. In view of Proposition 2.8, $\mathbf{G}_{p}^{\rm ss}$ is
non-abelian666The generic fibre is abelian if and if the special fibre is,
since $\mathbf{G}_{p}^{\mathrm{ss}}$ is reductive over $\mathbb{Z}_{p}$..
Therefore, Proposition 3.6 can be applied, giving the upper bound

(4.2)
$\bigg{|}\Big{(}\prod_{p|d}\mathcal{B}_{p}\Big{)}\cap\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})\bigg{|}=\sum_{i=1}^{s}\bigg{|}\prod_{p|d}\big{(}\mathcal{B}_{p}\cap
g_{i,p}I_{p}\big{)}\bigg{|}\leq
s\prod_{p|d}\Big{(}\frac{3}{4}\left|I_{p}\right|\Big{)}=\Big{(}\frac{3}{4}\Big{)}^{\omega(d)}\left|\overline{\rho}_{d}^{\mathrm{ss}}(G_{K})\right|$

where $\omega(d)$ is the number of prime divisors of $d$.

Combining equations (4.1) and (4.2), we get

$\mathrm{ud}(S_{A,M})\leq\Big{(}\frac{3}{4}\Big{)}^{\omega(d)}$

for any $d$ which is a square-free product of primes in ${\mathcal{P}}$. Since
${\mathcal{P}}$ is infinite, this implies that

$\mathrm{ud}(S_{A,M})=0.$

###  4.3. Proof of Theorem 1.3

We will now use the effective Chebotarev theorem of Lagarias–Odlyzko and the
Selberg sieve to obtain a power saving upper bound for the number of places
with a given Frobenius field $M$. This is possibly the approach suggested by
Serre in [27, §8.2].777Our optimisation of exponents appearing in the analysis
is based on the readily accessible methods in the literature, thereby amenable
to improvements.

In this section we will write $f(X)=O(g(X))$ or $f(X)\ll g(X)$ to mean the
existence of an absolute constant $c$ such that $\left|f(X)\right|\leq
c\left|g(X)\right|$ for all $X$. If the constant is allowed to depend on some
other object, we will indicate so in the subscripts. All constants are in fact
effectively computable.

######  Remark 4.3.

This section will be conditional on the GRH. Unconditionally, we expect that
the standard methods lead to an upper bound of the form
$O\big{(}\frac{X}{(\log X)^{\alpha}}\big{)}$ for some explicit $\alpha>1$, but
we have not worked out the details.

####  4.3.1. Sieving setup

In this subsection, we recast the problem into a form where the Selberg sieve
can be directly applied. There are notational complications since we are only
assuming potential independence of the Galois images at different primes $p$.
No important idea is lost if one assumes $L=K$, i.e. the family of Galois
representations over $K$ has independent image.

Fix a positive real number $X$. Let ${\varSigma}_{K}(X)$ be the set of all
finite places of $K$ of norm at most $X$ where $A$ has good reduction. Recall
that in §4.1, we have chosen a finite Galois extension $L/K$ and an infinite
set of primes ${\mathcal{P}}$ defined by a splitting condition.

Let ${\mathcal{R}}$ be the set of square-free product of primes in
${\mathcal{P}}$. Let $d\in{\mathcal{R}}\cup\\{\infty\\}$. In the infinite
case, the condition “$p|d$” should be interpreted as “$p\in{\mathcal{P}}$”.
Let $G_{d}=\prod_{p|d}\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})$, so we
have a Galois representation

$\overline{\rho}_{d}^{\mathrm{ss}}:G_{K}\to G_{d}.$

Denote its image by $G_{d}^{\prime}$ and the fixed field of its kernel by
$K_{d}$, so we have an isomorphism

$\mathop{\rm Gal}\nolimits(K_{d}/K)\simeq G_{d}^{\prime},$

and $K_{d}$ is the compositum of all $K_{p}$ for $p|d$. Define $H_{d}$ and
$L_{d}$ similarly using the restriction
$\overline{\rho}_{d}^{\mathrm{ss}}|_{G_{L}}$. By the choice of $L$, we have
$H_{d}=\prod_{p|d}H_{p}$. In these notations, we have

$I_{d}:=\prod_{p|d}I_{p}\subseteq H_{d}\subseteq G_{d}^{\prime}\subseteq
G_{d}$

where the first inclusion is due to Corollary 2.7.

Here is a summary of the above definitions

$K$$K_{d}\cap
L$$L$$K_{d}$$L_{d}$$H_{d}=\prod_{p|d}H_{p}$$G_{d}^{\prime}$$\mathop{\rm
Gal}\nolimits(L/K)$

Let $\mathcal{I}=G_{\infty}^{\prime}/H_{\infty}$, then from the above diagram,
$\mathcal{I}=\mathop{\rm Gal}\nolimits(K_{\infty}\cap L/K)$ is finite. Fix an
integer $\mathtt{d}$ such that $K_{\mathtt{d}}\cap L=K_{\infty}\cap L$, so if
$\mathtt{d}|d$, then $G_{d}^{\prime}/H_{d}\simeq\mathcal{I}$. Write down a
coset decomposition

$G_{\infty}^{\prime}=\bigsqcup_{i\in\mathcal{I}}g_{i}H_{\infty}=\bigsqcup_{i\in\mathcal{I}}\prod_{p\in{\mathcal{P}}}g_{i,p}H_{p}.$

For any $d$ as above, define $g_{i,d}=(g_{i,p})_{p|d}\in G_{d}^{\prime}$, so
we have a decomposition
$G_{d}^{\prime}=\bigcup_{i\in\mathcal{I}}g_{i,d}H_{d}$. By comparing indices,
this is a disjoint union if $\mathtt{d}|d$.

For each $p\in{\mathcal{P}}$, let $\mathcal{B}_{p}\subseteq G_{p}$ be the
bounding set as in the previous section (cf. Definition 3.3). Let
$\mathcal{C}_{p}=G_{p}-\mathcal{B}_{p}$. For a general
$d\in{\mathcal{R}}\cup\\{\infty\\}$, define $\mathcal{C}_{d}$ and
$\mathcal{B}_{d}$ as a product of the corresponding sets for $p$ dividing $d$.
Let $\mathcal{C}_{d}^{\prime}=\mathcal{C}_{d}\cap G_{d}^{\prime}$ and
$\mathcal{B}_{d}^{\prime}=\mathcal{B}_{d}\cap G_{d}^{\prime}$. For an index
$i\in\mathcal{I}$, define

$\mathcal{C}_{d}^{(i)}=\mathcal{C}_{d}\cap\prod_{p|d}g_{i,p}H_{p}.$

So we have $\mathcal{C}_{d}^{(i)}=\prod_{p|d}\mathcal{C}_{p}^{(i)}$. Moreover,
$\mathcal{C}_{d}^{\prime}=\bigcup_{i\in\mathcal{I}}\mathcal{C}_{d}^{(i)}$,
which is a disjoint union if $\mathtt{d}|d$. Define $\mathcal{B}_{d}^{(i)}$
similarly. If $p$ is a prime, then
$g_{i,p}H_{p}=\mathcal{B}_{p}^{(i)}\sqcup\mathcal{C}_{p}^{(i)}$.

Let $C$ be any union of conjugacy classes in $G_{d}^{\prime}$. Define

$E_{d}(C)=\Big{\\{}v\in{\varSigma}_{K}(X)\,\Big{|}\,v\text{ does not lie above
any }p|d,\ \overline{\rho}_{d}^{\mathrm{ss}}(\mathrm{Frob}_{v})\in
C\Big{\\}}.$

The proof of Theorem 1.1 in the previous subsection then implies
$S_{A,M}(X)\subseteq E_{d}(\mathcal{B}_{d}^{\prime})$. This form does not
immediately yield a power saving upper bound. Instead, we re-write it in a
different way.

######  Proposition 4.4.

We have an inclusion

$S_{A,M}(X)\subseteq\bigsqcup_{i\in\mathcal{I}}\bigg{(}E_{\mathtt{d}}(\mathcal{B}_{\mathtt{d}}^{(i)})-\bigcup_{p\in{\mathcal{P}}^{(\mathtt{d})}}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{)}$

where ${\mathcal{P}}^{(\mathtt{d})}$ is any subset of ${\mathcal{P}}$ not
containing the primes dividing $\mathtt{d}$.

###### Proof.

Implicit in the statement is that $\mathcal{B}_{d}^{(i)}$ is a union of
conjugacy classes in $G_{d}^{\prime}$ for all $d$. To see this, observe that
$\mathcal{B}_{d}^{\prime}$ is a union of conjugacy classes, and
$\mathcal{B}_{d}^{(i)}$ is its intersection with a coset of $H_{d}$ in
$G_{d}^{\prime}$. It remains to observe that $H_{d}$ is a normal subgroup of
$G_{d}^{\prime}$, and the quotient is abelian, since it is a subquotient of
the abelian group $G_{d}/I_{d}$ (cf. §3.3). The same argument works for the
second term.

The containment is apparent: let $v\in S_{A,M}(X)$ and
$g=(g_{p})_{p\in{\mathcal{P}}}=\overline{\rho}_{\infty}^{\mathrm{ss}}(\mathrm{Frob}_{v})\in
G_{\infty}^{\prime}$. Then there exists a unique $i\in\mathcal{I}$ such that
$g\in g_{i}H_{\infty}$. Moreover, we know that
$g_{p}\in\mathcal{B}_{p}^{\prime}$ for all $p\in{\mathcal{P}}$, so
$g_{p}\notin\mathcal{C}_{p}^{\prime}$. It follows that
$(g_{p})_{p|\mathtt{d}}\in\mathcal{B}_{\mathtt{d}}^{(i)}$, but it is not in
$\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)}$ for any prime $p$.
∎

Each term in the disjoin union in Proposition 4.4 is a sieving problem. We now
recall a form of the Selberg sieve.

######  Theorem 4.5.

Let ${\mathcal{A}}$ be a finite set. Let ${\mathcal{P}}$ be a set of primes.
For each $p\in{\mathcal{P}}$, let ${\mathcal{A}}_{p}\subseteq{\mathcal{A}}$.
For $d\in{\mathcal{R}}$, define
${\mathcal{A}}_{d}=\bigcap_{p|d}{\mathcal{A}}_{p}$. If $d=1$, set
${\mathcal{A}}_{1}={\mathcal{A}}$. Suppose for all $d$, we can write

$\left|{\mathcal{A}}_{d}\right|=\beta_{d}c\,\mathrm{Li}(X)+R_{d}$

where $R_{d}$ is some real number, the function $d\mapsto\beta_{d}$ is
multiplicative, and $\beta_{d}>\beta$ for some $\beta>0$.

Under these assumptions, for any positive real number $z$, we have

$\bigg{|}{\mathcal{A}}-\bigcup_{p\in{\mathcal{P}}}{\mathcal{A}}_{p}\bigg{|}\leq\frac{1-\beta}{\beta}c\,\mathrm{Li}(X)\cdot\frac{1}{\pi_{\mathcal{P}}(z)}+O\bigg{(}\sum_{\begin{subarray}{c}d_{1},d_{2}\leq
z\\\ d_{1},d_{2}\in{\mathcal{R}}\end{subarray}}R_{[d_{1},d_{2}]}\bigg{)}$

where $\pi_{\mathcal{P}}(z)=\left|\\{p\in{\mathcal{P}}\,|\,p\leq z\\}\right|$.

###### Proof.

This follows from [8, Theorem 7.2.1] by taking $X$ to be $c\mathrm{Li}(X)$ in
their statement. In their notation, $f(d)=\beta_{d}^{-1}$,
$f_{1}(d)=\prod_{p|d}\frac{1-\beta_{p}}{\beta_{p}}$, and

$V(z)=\sum_{\begin{subarray}{c}d\leq z\\\
d|P(z)\end{subarray}}\frac{\mu^{2}(d)}{f_{1}(d)}\geq\sum_{\begin{subarray}{c}p\leq
z\\\
p\in{\mathcal{P}}\end{subarray}}\frac{1}{f_{1}(p)}=\sum_{\begin{subarray}{c}p\leq
z\\\
p\in{\mathcal{P}}\end{subarray}}\frac{\beta_{p}}{1-\beta_{p}}\geq\frac{\beta}{1-\beta}\pi_{{\mathcal{P}}}(z).$

The upper bound follows from the conclusion in _loc. cit._. ∎

We will take ${\mathcal{A}}=E_{\mathtt{d}}(\mathcal{B}_{\mathtt{d}}^{(i)})$,
the set of sieving primes to be ${\mathcal{P}}^{(\mathtt{d})}$, and
${\mathcal{A}}_{p}=E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})$.
For their intersections, we have a simple lemma.

######  Lemma 4.6.

Let $d\in{\mathcal{R}}$ be coprime to $\mathtt{d}$, then

$\bigcap_{p|d}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})=E_{\mathtt{d}}(\mathcal{B}_{\mathtt{d}}^{(i)})-\bigcup_{p|d}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{dp}}^{(i)}).$

###### Proof.

Let $g=(g_{p})_{p\in{\mathcal{P}}}\in G_{\infty}^{\prime}$. Suppose
$g\in\mathcal{B}_{\mathtt{d}}^{(i)}$, then by the choice of $\mathtt{d}$, we
have $g\in g_{i}H_{\infty}$. So for all $p\in{\mathcal{P}}$, $g_{p}\in
g_{i,p}H_{p}$. The lemma is now clear from the relation
$g_{i,p}H_{p}=\mathcal{B}_{p}^{(i)}\sqcup\mathcal{C}_{p}^{(i)}$. ∎

Therefore, we just need to estimate terms of the form
$E_{d}(\mathcal{B}_{d}^{(i)})$ for coprime $d\in{\mathcal{R}}$. This will be
done using versions of the effective Chebotarev density theorem, which we now
recall.

####  4.3.2. Background on effective Chebotarev

For any number field $k$, let $n_{k}=[k:\mathbb{Q}]$ and $\Delta_{k}$ be its
absolute discriminant. Given a finite extension $l/k$, let $\mathcal{D}(l/k)$
denote the set of rational primes that lie below the ramified finite places of
$l/k$. Define

$M(l/k)=\left|G\right|\Delta_{k}^{\frac{1}{n_{k}}}\prod_{p\in\mathcal{D}(l/k)}p.$

Suppose that $l/k$ is Galois with Galois group $G$. Let $C\subseteq G$ be a
union of conjugacy classes. For any positive real number $X$, define

$\pi(X,C,l/k):=|\\{\mathfrak{p}\,|\,\mathfrak{p}\text{ is unramified in }l,\
\mathrm{Frob}_{\mathfrak{p}}\in C,\ \mathbf{N}_{k/\mathbb{Q}}\mathfrak{p}\leq
X\\}|.$

We are interested in the error term

$R(X,C,l/k):=\pi(X,C,l/k)-\frac{\left|C\right|}{\left|G\right|}\mathrm{Li}(X).$

The following result is equation ($20_{\mathrm{R}}$) in [27, §2.4], which is
an improvement to Lagarias–Odlyzko’s original result [17].

######  Theorem 4.7.

Assume the GRH holds for the Artin $L$-functions of irreducible
representations of $l/k$, then

$\left|R(X,C,l/k)\right|\ll\left|C\right|n_{k}X^{\frac{1}{2}}\big{(}\log
X+\log M(l/k)\big{)}.$

Assuming the AHC, Murty–Murty–Saradha obtained the following improvement [22,
Corollary 3.10].

######  Theorem 4.8.

Let $H$ be a subgroup of $G$ such that the GRH and AHC hold for the Artin
$L$-functions of the irreducible characters of $H$. Suppose $H$ meets every
conjugacy class contained in $C$, then

$R(X,C,l/k)\ll\left|C\right|^{\frac{1}{2}}[G:H]^{\frac{1}{2}}n_{k}X^{\frac{1}{2}}\big{(}\log
X+\log M(l/k)\big{)}.$

If $\left|C\right|$ is of roughly the same size as $\left|G\right|$, then this
estimate saves a factor of $\left|H\right|^{\frac{1}{2}}$ compared to Theorem
4.7. We therefore want to take $\left|H\right|$ as large as possible.

######  Remark 4.9.

This theorem is a corollary of a precise result [22, Proposition 3.9]. While
it suffices for our purpose, we expect that a finer analysis of centralizers
in finite reductive groups can give a better exponent.

####  4.3.3. Applications

We will now use the effective Chebotarev density theorem to estimate various
terms appearing in Theorem 4.5. The main result is Proposition 4.13 below.

First fix $i\in\mathcal{I}$. For each $d\in{\mathcal{R}}$, define

$\alpha_{d}=\frac{|\mathcal{B}_{d}^{(i)}|}{\left|H_{d}\right|},\quad\beta_{d}=\frac{|\mathcal{C}_{d}^{(i)}|}{\left|H_{d}\right|}.$

Then we have the relations

$\alpha_{d}=\prod_{p|d}\alpha_{p},\quad\beta_{d}=\prod_{p|d}\beta_{p},\quad\alpha_{p}+\beta_{p}=1$

for $p\in{\mathcal{P}}$ and $d\in{\mathcal{R}}$. These values depend on $i$,
but we suppress this for notational simplicity. In the notation of the
previous subsection, we can write

$\big{|}E_{d}(\mathcal{B}_{d}^{(i)})\big{|}=\pi(X,\mathcal{B}_{d}^{(i)},K_{d}/K)=\frac{1}{[G_{d}^{\prime}:H_{d}]}\alpha_{d}\mathrm{Li}(X)+R(X,\mathcal{B}_{d}^{(i)},K_{d}/K).$

For each prime $p$, fix a Borel subgroup
$\mathbf{B}_{p}\subseteq\mathbf{G}_{p}^{\mathrm{ss}}$, and let

$B_{p}=\mathbf{B}_{p}(\mathbb{F}_{p}).$

Define $B_{d}=\prod_{p|d}B_{p}$ and $B_{d}^{\prime}=B_{d}\cap G_{d}^{\prime}$.
This is a subgroup of $G_{d}^{\prime}$. The next two lemmas verify that it
satisfies the conditions in Theorem 4.8.

######  Lemma 4.10.

All conjugacy classes in $\mathcal{B}_{d}^{(i)}$ intersect $B_{d}^{\prime}$.

###### Proof.

By the Borel conjugacy theorem [3, Théorème 4.13(b)], this result holds in
$G_{d}$. To conclude, we will show that the conjugating element can be chosen
to lie in $I_{d}$. It is enough to do this for $d=p$, a prime. Suppose
$b\in\mathcal{B}_{p}^{\prime}$ and $gbg^{-1}\in B_{p}^{\prime}$, where $g\in
G_{p}$.

Let $\mathbf{T}_{p}\subseteq\mathbf{B}_{p}$ be a maximal torus. Then the
commutative diagram appearing $\S\ref{ss:Isog}$ shows that
$\mathbf{T}_{p}(\mathbb{F}_{p})$ intersects all cosets of $I_{p}$. Choose
$t\in\mathbf{T}_{p}(\mathbb{F}_{p})$ so that $t$ and $g$ are in the same
coset, then $t^{-1}g$ is an element of $I_{p}$ conjugating $b$ to
$B_{p}^{\prime}$. ∎

######  Lemma 4.11.

All irreducible representations of $B_{d}^{\prime}$ are induced from abelian
characters. Consequently, the AHC holds for $K_{d}/K_{d}^{B_{d}^{\prime}}$,
and the GRH holds provided it holds for all Hecke $L$-functions.

###### Proof.

Let $p$ be a prime. By choosing a faithful representation
$\mathbf{G}_{p}^{\mathrm{ss}}\hookrightarrow\mathop{\rm GL}\nolimits_{N}$
defined over $\mathbb{F}_{p}$, we see that $B_{p}$ is isomorphic to a subgroup
of $R_{N}$, the group of upper triangular matrices in $\mathop{\rm
GL}\nolimits_{N}(\mathbb{F}_{p})$. Write the derived series of $R_{N}$ as

$R_{N}\triangleright S_{N}\triangleright
S_{N-1}\triangleright\cdots\triangleright S_{0}=\\{1\\}.$

Then a standard calculation shows that

$S_{k}=\\{g\in R_{N}\,|\,g_{ii}=1,\ g_{ij}=0\text{ if }0<j-i\leq N-k\\}.$

For $v=1,\cdots,k-1$, let

$S_{k,v}=\\{g\in S_{k}\,|\,g_{i,i+N-k}=0\text{ if }i\leq v\\}.$

Then we have a normal series $S_{k}\triangleright
S_{k,1}\triangleright\cdots\triangleright S_{k,k-1}=S_{k-1}$ where each
quotient is isomorphic to $\mathbb{F}_{p}$. Conjugation by a diagonal matrix
preserves $S_{k,v}$, so all of the groups are normal in $R_{N}$. This normal
series shows that $R_{N}$ is a supersolvable group.

Now, $B_{p}$ is a subgroup of $R_{N}$, $B_{d}=\prod_{p|d}B_{p}$, and
$B_{d}^{\prime}$ is a subgroup of $B_{d}$. Each of the operations preserve
supersolvability, so $B_{d}^{\prime}$ is also supersolvable. The first claim
of the lemma now follows from [25, Theorem 12.8.5]. The second part of the
lemma is a standard consequence of class field theory. ∎

######  Proposition 4.12.

For each prime $p$, let

(4.3)
$N_{p}=\left|\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})\right|\cdot\left|\mathbf{B}_{p}(\mathbb{F}_{p})\right|^{-\frac{1}{2}}.$

Assume the GRH for Hecke $L$-functions. Then for all $d\in{\mathcal{R}}$, we
have

$\big{|}E_{d}(\mathcal{B}_{d}^{(i)})\big{|}=\frac{1}{[G_{d}^{\prime}:H_{d}]}\alpha_{d}\mathrm{Li}(X)+O_{A,K}\Big{(}\prod_{p|d}N_{p}\cdot
X^{\frac{1}{2}}(\log X+\log d)\Big{)}.$

###### Proof.

Applying Theorem 4.8 to the extension $l/k=K_{d}/K$ with the subgroup
$H=B_{d}^{\prime}$ gives the error term

$\displaystyle\left|R(X,\mathcal{B}_{d}^{(i)},K_{d}/K)\right|$
$\displaystyle\ll(|\mathcal{B}_{d}^{(i)}|\cdot[G_{d}^{\prime}:B_{d}^{\prime}])^{\frac{1}{2}}n_{K}X^{\frac{1}{2}}\big{(}\log
X+\log M(K_{d}/K)\big{)}$
$\displaystyle\leq(\left|G_{d}\right|\cdot[G_{d}:B_{d}])^{\frac{1}{2}}n_{K}X^{\frac{1}{2}}\big{(}\log
X+\log M(K_{d}/K)\big{)}$ $\displaystyle=\prod_{p|d}N_{p}\cdot
n_{K}X^{\frac{1}{2}}\big{(}\log X+\log M(K_{d}/K)\big{)}.$

For the terms beside $N_{p}$, recall that $K_{d}/K$ is unramified away from
the places of $K$ dividing $d\Delta_{A}$, so

$\displaystyle\log M(K_{d}/K)$
$\displaystyle=\log\left|G_{d}^{\prime}\right|+\frac{\log(\Delta_{K})}{n_{K}}+\sum_{p|d}\log
p+\sum_{p|\Delta_{A}}\log p$
$\displaystyle\leq\sum_{p|d}\log\left|\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})\right|+\log
d+O_{A,K}(1)$ $\displaystyle\ll_{A,K}\log d$

where for the final estimate, we used the trivial bound
$\left|\mathbf{G}_{p}^{\mathrm{ss}}(\mathbb{F}_{p})\right|\leq\left|\mathrm{GSp}_{2g}(\mathbb{F}_{p})\right|$.
∎

We now apply this to calculate the intersection terms in the Selberg sieve.

######  Proposition 4.13.

Assume the GRH for Hecke $L$-functions. If $d\in{\mathcal{R}}$ is coprime to
$\mathtt{d}$, then

$\bigg{|}\bigcap_{p|d}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{|}=\frac{1}{\left|\mathcal{I}\right|}\alpha_{\mathtt{d}}\beta_{d}\mathrm{Li}(X)+O_{A,K}\Big{(}2^{\omega(d)}\prod_{p|\mathtt{d}d}N_{p}\cdot
X^{\frac{1}{2}}\big{(}\log X+\log(\mathtt{d}d)\big{)}\Big{)}.$

###### Proof.

By Lemma 4.6 and the principle of inclusion-exclusion, we have

$\bigg{|}\bigcap_{p|d}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{|}=\sum_{d^{\prime}|d}\mu(d^{\prime})\big{|}E_{\mathtt{d}d^{\prime}}(\mathcal{B}_{\mathtt{d}d^{\prime}}^{(i)})\big{|}.$

Apply the previous proposition to the right hand side. There are
$2^{\omega(d)}$ terms in the above sum, so the error term has the required
form by the triangle inequality. In the main term, the coefficient in front of
$\mathrm{Li}(X)$ is given by

$\sum_{d^{\prime}|d}\mu(d^{\prime})\cdot\frac{1}{\left|\mathcal{I}\right|}\alpha_{\mathtt{d}d^{\prime}}=\frac{1}{\left|\mathcal{I}\right|}\alpha_{\mathtt{d}}\prod_{p|d}(1-\alpha_{p})=\frac{1}{\left|\mathcal{I}\right|}\alpha_{\mathtt{d}}\beta_{d}$

where we used the observation that
$[G_{d}^{\prime}:H_{d}]=[G_{\infty}^{\prime}:H_{\infty}]=\left|\mathcal{I}\right|$
for $\mathtt{d}|d$. ∎

####  4.3.4. Point counting

We will now estimate $N_{p}$, defined in (4.3).

######  Proposition 4.14.

Let $\mathbf{G}$ be the Mumford–Tate group of $A_{/K}$ and
$\mathbf{G}^{\mathrm{ss}}$ its semisimple quotient. Put

$\gamma=\frac{1}{4}(3\dim\mathbf{G}^{\mathrm{ss}}+\mathop{\rm
rank}\nolimits\mathbf{G}^{\mathrm{ss}}).$

There exists a constant $C$ depending only on $g$ such that

$N_{p}\leq Cp^{\gamma}$

for all primes $p$.

The key feature of this proposition is that neither $C$ nor $\gamma$ depend on
the prime $p$, even though the monodromy groups $\mathbf{G}_{p}$ are not
assumed to interpolate into a group over $\mathbb{Q}$. The proposition follows
by combining Lemma 4.15 and Corollary 4.17 below.

######  Lemma 4.15.

There exists a constant $C$ depending only on $g$ such that for all prime $p$,

$N_{p}\leq
Cp^{\dim\mathbf{G}_{p}^{\mathrm{ss}}-\frac{1}{2}\dim\mathbf{B}_{p}}.$

###### Proof.

In view of Lemma 3.5 we may replace $\mathbf{G}_{p}^{\mathrm{ss}}$ with
$\mathbf{G}_{p}^{\mathrm{sc}}$ in the definition of $N_{p}$. The base change
$(\mathbf{G}_{p}^{\mathrm{sc}})_{/\overline{\mathbb{F}}_{p}}$ is a simply
connected semisimple group, so it is a product of simply connected split
simple groups. These are classified by Dynkin diagrams. Since the rank is
bounded (say by $2g$), there are only a finite number of possibilities.
Moreover, each of them is obtained from a Chevalley group over $\mathbb{Z}$ by
base change.

Now, $\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{F}_{p})$ is the set of fixed points
of the Frobenius operator $F$ acting on
$\mathbf{G}_{p}^{\mathrm{sc}}(\overline{\mathbb{F}}_{p})$. The
Grothendieck–Lefschetz trace formula gives

$\left|\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{F}_{p})\right|=\sum_{i=0}^{2\dim\mathbf{G}_{p}^{\mathrm{sc}}}(-1)^{i}\mathrm{Tr}\big{(}F,\mathrm{H}^{i}_{c}\big{(}(\mathbf{G}_{p}^{\mathrm{sc}})_{/\overline{\mathbb{F}}_{p}},\mathrm{\mathbb{Q}}_{\ell}\big{)}\big{)}$

The top degree term is $p^{\dim\mathbf{G}_{p}^{\mathrm{sc}}}$ since
$\mathbf{G}_{p}^{\mathrm{sc}}$ is geometrically connected. Deligne’s Weil
bound gives an upper bound for the Frobenius eigenvalues on each of the
remaining terms. In particular, it implies

$\left|p^{-\dim\mathbf{G}_{p}^{\mathrm{sc}}}\left|\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{F}_{p})\right|-1\right|\leq
bp^{-\frac{1}{2}}$

where $b$ is the sum of the dimensions of all lower degree cohomology groups.
The argument in the previous paragraph gives a finite list of possibilities
for $b$ independently of $p$. Therefore, we get positive constants
$c_{1}(g),c_{2}(g)$ depending only on $g$ such that

$c_{1}p^{\dim\mathbf{G}_{p}^{\mathrm{sc}}}\leq\left|\mathbf{G}_{p}^{\mathrm{sc}}(\mathbb{F}_{p})\right|\leq
c_{2}p^{\dim\mathbf{G}_{p}^{\mathrm{sc}}}$

for all primes $p$.

By applying the exact same argument to the term
$\left|\mathbf{B}_{p}(\mathbb{F}_{p})\right|$, the assertion follows. ∎

Let $\mathbf{G}$ be the Mumford–Tate group of $A$. For our purpose, it
suffices to know that this is a reductive group over $\mathbb{Q}$ which
“contains” all $p$-adic monodromy groups, made precise by the following
theorem of Deligne [9].

######  Theorem 4.16.

For all primes $p$, we have888Recall that we are assuming all monodromy groups
are connected.
$\mathbf{G}_{p/\mathbb{Q}_{p}}\subseteq\mathbf{G}\times_{\mathbb{Q}}\mathbb{Q}_{p}$.

######  Corollary 4.17.

Let $\mathbf{G}^{\mathrm{ss}}$ denote the semisimple quotient of $\mathbf{G}$,
then for all primes $p$,

$\dim\mathbf{G}_{p}^{\mathrm{ss}}-\frac{1}{2}\dim\mathbf{B}_{p}\leq\frac{1}{4}(3\dim\mathbf{G}^{\mathrm{ss}}+\mathop{\rm
rank}\nolimits\mathbf{G}^{\mathrm{ss}})$

###### Proof.

Fix a prime $p$. For simplicity, we base change the pertinent groups to
$\overline{\mathbb{Q}}_{p}$. This does not change any dimension.

Recall that $\mathbf{B}_{p}$ is a Borel subgroup of
$\mathbf{G}_{p}^{\mathrm{ss}}$. It follows that

$\dim\mathbf{G}_{p}^{\mathrm{ss}}=2\dim\mathbf{B}_{p}-\mathop{\rm
rank}\nolimits\mathbf{G}_{p}^{\mathrm{ss}}.$

Rearranging this, we get

$\dim\mathbf{G}_{p}^{\mathrm{ss}}-\frac{1}{2}\dim\mathbf{B}_{p}=\frac{1}{4}(3\dim\mathbf{G}^{\mathrm{ss}}_{p}+\mathop{\rm
rank}\nolimits\mathbf{G}_{p}^{\mathrm{ss}})$

It remains to show that
$\dim\mathbf{G}_{p}^{\mathrm{ss}}\leq\dim\mathbf{G}^{\mathrm{ss}}$ and
$\mathop{\rm rank}\nolimits\mathbf{G}_{p}^{\mathrm{ss}}\leq\mathop{\rm
rank}\nolimits\mathbf{G}^{\mathrm{ss}}$.

The previous theorem gives an embedding $\mathbf{G}_{p}\subseteq\mathbf{G}$ of
reductive groups, and hence the inclusion of the derived subgroups
$\mathbf{G}_{p}^{\rm der}\subseteq\mathbf{G}^{\rm der}$. Moreover, the derived
subgroup is isogenous to the semisimple quotient, so they have the same
dimension and rank. The first inequality follows. For the rank part, let
$\mathbf{T}$ be a maximal torus in $\mathbf{G}_{p}^{\rm der}$, then its image
in $\mathbf{G}^{\rm der}$ is again a torus, so it is contained in a maximal
torus. Hence $\mathop{\rm rank}\nolimits\mathbf{G}_{p}^{\rm
der}\leq\mathop{\rm rank}\nolimits\mathbf{G}^{\rm der}$, as desired. ∎

####  4.3.5. Conclusion of the proof

We now assemble the pieces to conclude the proof.

######  Theorem 4.18.

Assume the GRH for all Hecke $L$-functions. Let $\varepsilon>0$. Then there
exists a constant $C$ depending on $(A,K,\varepsilon)$, and $M$ such that
whenever $X>C$, we have

$\left|S_{A,M}(X)\right|\ll_{A,K,\varepsilon}X^{1-\frac{1}{4\gamma+6}+\varepsilon}$

where $\gamma=\frac{1}{4}(3\dim\mathbf{G}^{\mathrm{ss}}+\mathop{\rm
rank}\nolimits\mathbf{G}^{\mathrm{ss}})$ from Proposition 4.14.

###### Proof.

Fix $\varepsilon>0$. Let $N$ be the lower bound on $p$ chosen in §4.1. Let
$C_{1}$ be the constant denoted by $C$ in Proposition 4.14. Let
$N_{\varepsilon}$ be a positive integer greater than $N$,
$(2C_{1})^{\frac{1}{\varepsilon}}$, and the largest prime divisor of
$\mathtt{d}$. Let ${\mathcal{P}}_{\varepsilon}$ be the subset of
${\mathcal{P}}$ consisting of all primes greater than $N_{\varepsilon}$. We
will use ${\mathcal{P}}_{\varepsilon}$ as a sieving set.

From Proposition 4.4, we have

$\left|S_{A,M}(X)\right|=\sum_{i\in\mathcal{I}}\bigg{|}E_{\mathtt{d}}(\mathcal{B}_{\mathtt{d}}^{(i)})-\bigcup_{p\in{\mathcal{P}}_{\varepsilon}}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{|}$

By combing Proposition 4.13 and Proposition 4.14, we get the estimate

(4.4)
$\bigg{|}\bigcap_{p|d}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{|}=\frac{1}{\left|\mathcal{I}\right|}\alpha_{\mathtt{d}}\beta_{d}\mathrm{Li}(X)+O_{A,K}\big{(}(2C_{1})^{\omega(d)}d^{\gamma}X^{\frac{1}{2}}(\log
X+\log d)\big{)}$

for all $d\in{\mathcal{R}}$. If moreover all prime divisors of $d$ are greater
than $N_{\varepsilon}$, then

$d^{\varepsilon}=\prod_{p|d}p^{\varepsilon}\geq
N_{\varepsilon}^{\varepsilon\omega(d)}\geq(2C_{1})^{\omega(d)}.$

So we may replace $(2C_{1})^{\omega(d)}d^{\gamma}$ by $d^{\gamma+\varepsilon}$
in the error term.

We are now ready to apply Theorem 4.5. Fix an index $i\in\mathcal{I}$. Take
${\mathcal{A}}=E_{\infty}(g_{i}H_{\infty})$, sieving primes
${\mathcal{P}}_{\varepsilon}$, and excluded subsets
${\mathcal{A}}_{p}=E_{\mathtt{d}p}(\mathcal{C}_{\mathtt{d}p}^{(i)})$. The
above discussion gives

$\left|{\mathcal{A}}_{d}\right|=\big{|}E_{\mathtt{d}d}(\mathcal{C}_{\mathtt{d}d}^{(i)})\big{|}=\frac{1}{\left|\mathcal{I}\right|}\alpha_{\mathtt{d}}\beta_{d}\mathrm{Li}(X)+O_{A,K}(d^{\gamma+\varepsilon}X^{\frac{1}{2}}(\log
X+\log d)).$

By definition, the function $d\mapsto\beta_{d}$ is multiplicative. Let
$p\in{\mathcal{P}}$, then

$\beta_{p}=\frac{\left|\mathcal{C}_{p}\cap
g_{i,p}H_{p}\right|}{\left|H_{p}\right|}=1-\frac{\left|\mathcal{B}_{p}\cap
g_{i,p}H_{p}\right|}{\left|H_{p}\right|}>\frac{1}{4}.$

Similar to the proof of Theorem 1.1, the final inequality follows from
Proposition 3.6 since $H_{p}$ is a union of cosets of $I_{p}$

Having verified all of the hypotheses, Theorem 4.5 gives the estimate

$\bigg{|}E_{\mathtt{d}}(\mathcal{B}_{\mathtt{d}}^{(i)})-\bigcup_{p\in{\mathcal{P}}_{\varepsilon}}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{|}\\\
\leq\frac{1}{\left|\mathcal{I}\right|}\cdot
3\alpha_{\mathtt{d}}\cdot\mathrm{Li}(X)\cdot\frac{1}{\pi_{\mathcal{P}}(z)}+O_{A,K}\bigg{(}X^{\frac{1}{2}}\sum_{\begin{subarray}{c}d_{1},d_{2}\leq
z\\\
d_{1},d_{2}\in{\mathcal{R}}_{\varepsilon}\end{subarray}}[d_{1},d_{2}]^{\gamma+\varepsilon}(\log
X+\log[d_{1},d_{2}])\bigg{)}$

where ${\mathcal{R}}_{\varepsilon}$ is the set of square-free products of
primes in ${\mathcal{P}}_{\varepsilon}$. In the sum for the error term, use
the trivial bound $[d_{1},d_{2}]\leq d_{1}d_{2}$ and extend the sum to all
pairs $d_{1},d_{2}\leq z$. In particular, $\log[d_{1},d_{2}]\leq\log(z^{2})$.
Therefore,

$\displaystyle\sum_{\begin{subarray}{c}d_{1},d_{2}\leq z\\\
d_{1},d_{2}\in{\mathcal{R}}_{\varepsilon}\end{subarray}}R_{[d_{1},d_{2}]}$
$\displaystyle\ll_{A,K}X^{\frac{1}{2}}\sum_{d_{1},d_{2}\leq
z}(d_{1}d_{2})^{\gamma+\varepsilon}(\log X+\log z)$
$\displaystyle=X^{\frac{1}{2}}(\log X+\log z)\Big{(}\sum_{d\leq
z}d^{\gamma+\varepsilon}\Big{)}^{2}$
$\displaystyle\ll_{\gamma}z^{2(\gamma+1)+2\varepsilon}X^{\frac{1}{2}}(\log
X+\log z).$

For the main term, we need to have a lower bound for

$\pi_{{\mathcal{P}}_{\varepsilon}}(z)=\pi(z,\\{1\\},M/\mathbb{Q})-\pi(N_{\varepsilon},\\{1\\},M/\mathbb{Q}).$

The second term is a constant for our purpose. The field $M$ is Galois over
$\mathbb{Q}$ with degree999Otherwise it cannot be a Frobenius field for $A$,
and the conclusion holds trivially. at most $(2g)!$. So the qualitative form
of the Chebotarev density theorem gives the weak but unconditional estimate

$\pi(z,\\{1\\},M/\mathbb{Q})\geq\frac{1}{(2g)!}\cdot\frac{z}{\log
z}(1+o_{M}(1)).$

If $z\gg_{\varepsilon,M}1$, then the right hand side of the above equation is
bounded below by $\frac{1}{(2g)!}z^{1-\varepsilon}$. Therefore,

$\pi_{{\mathcal{P}}_{\varepsilon}}(z)\gg_{A,K,\varepsilon}z^{1-\varepsilon}$

whenever $z\gg_{A,K,\varepsilon,M}1$.

In summary, we have obtained a bound

$\bigg{|}E_{\mathtt{d}}(\mathcal{B}_{\mathtt{d}}^{(i)})-\bigcup_{p\in{\mathcal{P}}_{\varepsilon}}E_{\mathtt{d}p}(\mathcal{B}_{\mathtt{d}}^{(i)}\times\mathcal{C}_{p}^{(i)})\bigg{|}\ll_{A,K,\varepsilon}z^{-(1-\varepsilon)}\mathrm{Li}(X)+z^{2(\gamma+1)+2\varepsilon}X^{\frac{1}{2}}(\log
X+\log z)$

whenever $z$ is sufficiently large. Let $z=X^{\beta}$, then the right hand
side is optimized with $\beta=\frac{1}{4\gamma+6}$. Summing over all
$i\in\mathcal{I}$, we obtain the desired result. ∎

######  Remark 4.19.

The reason we have a dependency on $M$ is that the above analysis only
involves primes which split completely. For $p\notin{\mathcal{P}}$, if the
Frobenius image in $\mathbf{G}_{p}(\mathbb{F}_{p})$ is semisimple, then it
must lie in a non-split torus. The results in §3 can be used to give a uniform
upper bound in this case as well. Using the error term in Theorem 4.7, we may
take $\gamma=\dim\mathbf{G}^{\mathrm{ss}}$ in the above discussion, giving a
bound

$\left|S_{A,M}(X)\right|\ll_{A,K,\varepsilon}X^{1-\frac{1}{4\dim\mathbf{G}^{\mathrm{ss}}+6}+\varepsilon}$

for all $X$.

To improve the exponent, note that on the semisimple conjugacy classes,
Theorem 4.8 can still be applied to tori to save at least
$\frac{1}{2}\mathop{\rm rank}\nolimits\mathbf{G}^{\mathrm{ss}}$ from $\gamma$.
However, on the non-semisimple classes, it is not immediately clear to us how
to improve the error term of Lagarias–Odlyzko. The volume of the these classes
is $O(p^{-1})$, so we can take $\gamma=\dim\mathbf{G}^{\mathrm{ss}}-1$, giving
an exponent of $1-\frac{1}{4\dim\mathbf{G}^{\mathrm{ss}}+2}+\varepsilon$.

######  Remark 4.20.

We only needed the error term in the effective Chebotarev theorem on average.
The situation is analogous to the Bombieri–Vinogradov theorem, except that it
deals with the family of abelian extensions
$\\{\mathbb{Q}(\zeta_{q})/\mathbb{Q}\\}$, and we have a family of non-abelian
Lie-extensions $\\{K_{d}/K\\}$. There are some works when the Galois group is
fixed (cf. [23]), but we are not aware of any work in our setting.

## References

  * [1] S. Bloom.  The square sieve and a Lang-Trotter question for generic abelian varieties.  J. Number Theory, 191:119–157, 2018.
  * [2] A. Borel.  Linear algebraic groups, volume 126 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1991.
  * [3] A. Borel and J. Tits.  Groupes réductifs.  Inst. Hautes Études Sci. Publ. Math., 27:55–150, 1965.
  * [4] R. W. Carter.  Finite groups of Lie type: Conjugacy classes and complex characters.  Pure and Applied Mathematics (New York). John Wiley & Sons, Inc., New York, 1985.
  * [5] A. C. Cojocaru and C. David.  Frobenius fields for Drinfeld modules of rank 2.  Compos. Math., 144(4):827–848, 2008.
  * [6] A. C. Cojocaru and C. David.  Frobenius fields for elliptic curves.  Amer. J. Math., 130(6):1535–1560, 2008.
  * [7] A. C. Cojocaru, E. Fouvry, and M. R. Murty.  The square sieve and the Lang-Trotter conjecture.  Canad. J. Math., 57(6):1155–1177, 2005.
  * [8] A. C. Cojocaru and M. R. Murty.  An introduction to sieve methods and their applications, volume 66 of London Mathematical Society Student Texts.  Cambridge University Press, Cambridge, 2006.
  * [9] P. Deligne.  Hodge cycles on abelian varieties.  In Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., pages 9–100. Springer, Berlin, 1982.
  * [10] M. Demazure and A. Grothendieck.  Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, volume 152 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin-New York, 1970.  (SGA 3).
  * [11] G. Henniart.  Représentations $l$-adiques abéliennes.  In Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981), volume 22 of Progr. Math., pages 107–126. Birkhäuser Boston, Boston, MA, 1982.
  * [12] H. Hida.  Elliptic curves and arithmetic invariants.  Springer Monographs in Mathematics. Springer, New York, 2013.
  * [13] C. Y. Hui and M. Larsen.  Maximality of Galois actions for abelian and hyper-Kähler varieties.  Duke Math. J., 169(6):1163–1207, 2020.
  * [14] L. Illusie.  Independence of families of $l$-adic representations and uniform constructibility (after J.-P. Serre).  Available at http://www.math.u-psud.fr/~illusie, 2010.
  * [15] C. Khare.  $F$-split Galois representations are potentially abelian.  Proc. Amer. Math. Soc., 131(10):3021–3023, 2003.
  * [16] M. Kulkarni, V. M. Patankar, and C. S. Rajan.  Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves.  J. Number Theory, 164:87–102, 2016.
  * [17] J. C. Lagarias and A. M. Odlyzko.  Effective versions of the Chebotarev density theorem.  In Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pages 409–464. Academic Press, London-New York, 1977.
  * [18] S. Lang.  Algebraic groups over finite fields.  Amer. J. Math., 78:555–563, 1956.
  * [19] S. Lang and H. Trotter.  Frobenius distributions in ${\rm GL}_{2}$-extensions.  Lecture Notes in Mathematics, Vol. 504. Springer-Verlag, Berlin-New York, 1976.  Distribution of Frobenius automorphisms in ${{\rm{G}}L}_{2}$-extensions of the rational numbers.
  * [20] M. Larsen.  Maximality of Galois actions for compatible systems.  Duke Math. J., 80(3):601–630, 1995.
  * [21] M. Larsen and R. Pink.  Abelian varieties, $l$-adic representations, and $l$-independence.  Math. Ann., 302(3):561–579, 1995.
  * [22] M. R. Murty, V. K. Murty, and N. Saradha.  Modular forms and the Chebotarev density theorem.  Amer. J. Math., 110(2):253–281, 1988.
  * [23] L. B. Pierce, C. L. Turnage-Butterbaugh, and M. M. Wood.  An effective Chebotarev density theorem for families of number fields, with an application to $\ell$-torsion in class groups.  Invent. Math., 219(2):701–778, 2020.
  * [24] R. Pink.  $l$-adic algebraic monodromy groups, cocharacters, and the Mumford-Tate conjecture.  J. Reine Angew. Math., 495:187–237, 1998.
  * [25] W. R. Scott.  Group theory.  Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964.
  * [26] J.-P. Serre.  Abelian $l$-adic representations and elliptic curves.  W. A. Benjamin, Inc., New York-Amsterdam, 1968.  McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute.
  * [27] J.-P. Serre.  Quelques applications du théorème de densité de Chebotarev.  Publications Mathématiques de l’IHÉS, 54:123–201, 1981.
  * [28] J.-P. Serre.  Œuvres. Collected papers. IV.  Springer-Verlag, Berlin, 2000.  1985–1998.
  * [29] J.-P. Serre.  Galois cohomology.  Springer Monographs in Mathematics. Springer-Verlag, Berlin, english edition, 2002.  Translated from the French by Patrick Ion and revised by the author.
  * [30] J.-P. Serre.  Un critère d’indépendance pour une famille de représentations $\ell$-adiques.  Comment. Math. Helv., 88(3):541–554, 2013.
  * [31] G. Shimura.  Abelian varieties with complex multiplication and modular functions, volume 46 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 1998.
  * [32] J.-P. Wintenberger.  Démonstration d’une conjecture de Lang dans des cas particuliers.  J. Reine Angew. Math., 553:1–16, 2002.
  * [33] D. Zywina.  Bounds for the Lang-Trotter conjectures.  In SCHOLAR—a scientific celebration highlighting open lines of arithmetic research, volume 655 of Contemp. Math., pages 235–256. Amer. Math. Soc., Providence, RI, 2015.