# Smallness of Faltings heights of CM abelian varieties

Xunjing Wei

###### Abstract

We prove that assuming the Colmez conjecture and the “no Siegel zeros”
conjecture, the stable Faltings height of a CM abelian variety over a number
field is less than or equal to the logarithm of the root discriminant of the
field of definition of the abelian variety times an effective constant
depending only on the dimension of the abelian variety. In view of the fact
that the Colmez conjecture for abelian CM fields, the averaged Colmez
conjecture, and the “no Siegel zeros” conjecture for CM fields with no complex
quadratic subfields are already proved, we prove unconditional analogues of
the result above. In addition, we also prove that the logarithm of the root
discriminant of the field of everywhere good reduction of CM abelian varieties
can be “small”.

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###### Contents

  1. 1 Introduction
  2. 2 The Faltings height
  3. 3 The Colmez conjecture revisited
  4. 4 The zero of the Artin $L$-function near $1$
    1. 4.1 Relation between the zero near $1$ and the logarithmic derivative at $0$ of the Artin $L$-function
    2. 4.2 Sufficient conditions for the nonexistence of the zero near $1$ of the Artin $L$-function
    3. 4.3 Proofs of section 1 and section 1
  5. 5 The (proved) averaged Colmez conjecture
  6. 6 Field of everywhere good reduction of CM abelian varieties

##  1 Introduction

Let $E$ be a CM-field, and let $\Phi$ be a CM-type of $E$. Let $A$ be an
abelian variety over a number field $K$ such that we have an embedding
$i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K}(A)$ such that $(A,i)$
has CM-type $\Phi$. It is proved by Colmez in [Col93] that the stable Faltings
height $h_{\mathrm{Falt}}^{\mathrm{st}}(A)$ of the abelian variety $A$ depends
only on the CM-field $E$ and the CM-type $\Phi$ and not on the abelian variety
$A$. We denote it as $h_{(E,\Phi)}^{\mathrm{Falt}}$. In [Col93] Colmez
proposed a conjecture relating $h_{(E,\Phi)}^{\mathrm{Falt}}$ to the
logarithmic derivatives at $s=0$ of certain Artin $L$-functions defined by
$(E,\Phi)$. We will refer to this conjecture as the Colmez conjecture. The
precise statement is as follows:

We define a function $A_{(E,\Phi)}^{0}$ from
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to $\mathbb{C}$ by

$A_{(E,\Phi)}^{0}(\sigma)=\frac{1}{[\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}):\mathrm{Stab}(\Phi)]}\sum_{\nu\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})/\mathrm{Stab}(\Phi)}|\nu\Phi\cap\sigma\nu\Phi|,\forall\sigma\in\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}),$

where $\mathrm{Stab}(\Phi)$ is the subgroup of
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ consisting of the stabilizers
of $\Phi$.

This function is locally constant and constant on conjugacy classes.
Therefore, there is a unique decomposition of $A_{(E,\Phi)}^{0}$ into
$\mathbb{C}$-linear combinations of irreducible Artin characters $\chi$ (i.e.
characters $\chi$ of irreducible continuous representations of
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on finite-dimensional
$\mathbb{C}$-vector spaces)

${A_{(E,\Phi)}^{0}}=\sum_{\chi}m_{(E,\Phi)}(\chi)\chi,\ \ \ \ \ \ \
m_{(E,\Phi)}(\chi)\in\mathbb{C}.$

It can be shown that for any irreducible Artin character $\chi$ such that
$m_{(E,\Phi)}(\chi)\neq 0$, the Artin $L$-function $L(s,\chi,\mathbb{Q})$ is
defined and nonzero at $s=0$. We define

$Z_{(E,\Phi)}\coloneqq-\frac{1}{2}g\log(2\pi)+\sum_{\chi}m_{(E,\Phi)}(\chi)\frac{L^{\prime}(0,\chi,\mathbb{Q})}{L(0,\chi,\mathbb{Q})}$

and

$\mu_{(E,\Phi)}\coloneqq\sum_{\chi}m_{(E,\Phi)}(\chi)\log(\mathfrak{f}(\chi,\mathbb{Q})),$

where $\mathfrak{f}(\chi,\mathbb{Q})$ is the Artin conductor of the Artin
character $\chi$ (a positive integer).

The Colmez conjecture says that we have

$h_{(E,\Phi)}^{\mathrm{Falt}}=-Z_{(E,\Phi)}-\frac{1}{2}\mu_{(E,\Phi)}.$

When $E$ is a complex quadratic field, the Colmez conjecture is the same as
the classical Chowla-Selberg formula (see for example Page 91 and 92 of
[Wei76]), and so it is in fact a theorem.

Colmez [Col93] and Obus [Obu13] proved that the Colmez conjecture is true if
the extension $E/\mathbb{Q}$ is Galois with abelian Galois group (subsection
4.3). Yuan–Zhang [YZ18] and Andreatta–Goren–Howard–Madapusi-Pera [AGHMP18]
independently proved that the Colmez conjecture is true when one averages over
all CM-types of a given CM-field (section 5).

Let $-d\in\mathbb{Z}_{\leq-2}$ be a fundamental discriminant, so
$\mathrm{disc}(\mathbb{Q}(\sqrt{-d}))=d$. Let $\chi_{d}$ be the quadratic
character associated to the quadratic field extension
$\mathbb{Q}(\sqrt{-d})/\mathbb{Q}$, so $\chi_{d}(p)=(-d|p)$ for any prime $p$.
Let $L(s,\chi_{d})$ be the Dirichlet $L$-function of the character $\chi_{d}$,
so
$L(s,\chi_{d})=\frac{\zeta_{\mathbb{Q}(\sqrt{-d})}(s)}{\zeta_{\mathbb{Q}}(s)}$.

It is known that there is at most one zero of $L(s,\chi_{d})$ in the region

$1-\frac{1}{4\log(d)}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log(d)},$

and if such a zero exists it is real and simple.

For any $0<c\leq\frac{1}{4}$, we define the $c$-Siegel zero of $L(s,\chi_{d})$
to be the zero of $L(s,\chi_{d})$ in the region

$1-\frac{c}{\log(d)}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log(d)}$

(if it exists). We define the Siegel zero of $L(s,\chi_{d})$ to be the
$\frac{1}{4}$-Siegel zero of $L(s,\chi_{d})$.

The conjecture Introduction is as follows:

######  Conjecture  (No $\frac{1}{O(1)}$-Siegel zero of $L(s,\chi_{d})$).

There exists some effectively computable absolute constant
$C_{\mathrm{zero}}\in\mathbb{R}_{\geq 4}$ such that for any fundamental
discriminant $-d\in\mathbb{Z}_{\leq-2}$, the Dirichlet $L$-function
$L(s,\chi_{d})$ has no zeros in the region
$1-\frac{1}{C_{\mathrm{zero}}\log(d)}\leq\mathrm{Re}(s)<1$,
$|\operatorname{Im}(s)|\leq\frac{1}{4\log(d)}$.

Now let $E$ be an arbitrary CM-field. Let $F$ be the maximal totally real
subfield of $E$, and so $E$ is a totally complex quadratic field extension of
$F$. Let $\chi_{E/F}$ be the quadratic character associated to the quadratic
field extension $E/F$, so for any prime ideal $\mathfrak{p}$ of
$\mathcal{O}_{F}$,

$\chi_{E/F}(\mathfrak{p})=\begin{cases}1&\mbox{if }\mathfrak{p}\mbox{ splits
completely in }\mathcal{O}_{E},\\\ -1&\mbox{if }\mathfrak{p}\mbox{ is
unramified but does not split completely in }\mathcal{O}_{E},\\\ 0&\mbox{if
}\mathfrak{p}\mbox{ is ramified in }\mathcal{O}_{E}.\end{cases}$

Let $L(s,\chi_{E/F})$ be the $L$-function of the character $\chi_{E/F}$, and
so $L(s,\chi_{E/F})=\frac{\zeta_{E}(s)}{\zeta_{F}(s)}$.

Similarly to the case where $E$ is a complex quadratic field, by Lemma 3 of
[Sta74], for any CM-field $E$ with maximal totally real subfield $F$,
$L(s,\chi_{E/F})$ has at most one zero in the region

$1-\frac{1}{4\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|}.$

If such a zero exists, it is real and simple.

For any $0<c\leq\frac{1}{4}$, we define the generalized $c$-Siegel zero of
$L(s,\chi_{E/F})$ to be the zero of $L(s,\chi_{E/F})$ in the region

$1-\frac{c}{\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|}$

(if it exists). We define the generalized Siegel zero of $L(s,\chi_{E/F})$ to
be the generalized $\frac{1}{4}$-Siegel zero of $L(s,\chi_{E/F})$.

The conjecture Introduction is as follows:

######  Conjecture  (No generalized $\frac{1}{O_{g}(1)}$-Siegel zero of
$L(s,\chi_{E/F})$).

For any $g\in\mathbb{Z}_{\geq 1}$, there exists some effectively computable
constant $C_{\mathrm{zero}}(g)\in\mathbb{R}_{\geq 4}$ depending only on $g$
such that for any CM-field $E$ with maximal totally real subfield $F$ such
that $[F:\mathbb{Q}]=g$, the function $L(s,\chi_{E/F})$ has no zeros in the
region

$1-\frac{1}{C_{\mathrm{zero}}(g)\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|}.$

It is proved by Stark (Lemma 9 of [Sta74]) that section 1 implies section 1.
He also proved that section 1 is true whenever the CM-field $E$ contains no
complex quadratic subfields.

We show that assuming the Colmez conjecture, the nonexistence of the
generalized Siegel zero of $L$-functions of quadratic characters associated to
CM extensions is closely related to the stable Faltings height of CM abelian
varieties being bounded by the logarithm of the root discriminant of the field
of definition. More precisely, we prove the following theorem:

######  Theorem .

Suppose that the Colmez conjecture holds. Suppose further that Introduction
holds. Then for any $g\in\mathbb{Z}_{\geq 1}$, there exist effectively
computable constants $C_{1}(g)>0$, $C_{2}(g)\in\mathbb{R}$ depending only on
$g$ such that

$\begin{split}h_{\mathrm{Falt}}^{\mathrm{st}}(A)\leq
C_{1}(g)\frac{1}{[K:\mathbb{Q}]}\log|\mathrm{disc}(K)|+C_{2}(g),\end{split}$

for any dimension-$g$ abelian variety $A$ defined over a number field $K$ with
complex multiplication by $\mathcal{O}_{E}$ for some CM-field $E$.

Since the Colmez conjecture for abelian CM-fields is already proved, and since
section 1 is true when the CM-field $E$ contains no complex quadratic
subfields, we can also prove an unconditional version of the theorem above:

######  Theorem .

For any $g\in\mathbb{Z}_{\geq 1}$, there exists effectively computable
constants $C_{3}(g)>0$, $C_{4}(g)\in\mathbb{R}$ depending only on $g$ such
that

$\begin{split}h_{\mathrm{Falt}}^{\mathrm{st}}(A)\leq
C_{3}(g)\frac{1}{[K:\mathbb{Q}]}\log|\mathrm{disc}(K)|+C_{4}(g),\end{split}$

for any dimension-$g$ abelian variety $A$ over a number field $K$ with complex
multiplication by $\mathcal{O}_{E}$ for some CM-field $E$ such that the
extension $E/\mathbb{Q}$ is Galois with abelian Galois group and $E$ does not
contain any complex quadratic subfields.

######  Remark .

To show that the condition “$E$ does not contain any complex quadratic
subfields” in the hypotheses in section 1 is possible, we give examples of CM
fields $E$ containing no complex quadratic subfields such that the extension
$E/\mathbb{Q}$ is Galois with abelian Galois group.

Let $n$ be an integer greater than or equal to $3$ such that the group
$(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a cyclic group and such that
$\\#(\mathbb{Z}/n\mathbb{Z})^{\times}$ divides $4$. (Equivalently, $n=p^{k}$
or $n=2p^{k}$ for some odd prime $p$ such that $p\equiv 1\mod 4$.)

Let $E$ be the $n$-th cyclotomic field $\mathbb{Q}(\mu_{n})$, where $\mu_{n}$
denotes a primitive $n$-th root of unity. Then $E$ is a CM-field with maximal
totally real subfield $F=\mathbb{Q}(\mu_{n}+\mu_{n}^{-1})$. The extension
$E/\mathbb{Q}$ is Galois and $\mathrm{Gal}(E/\mathbb{Q})$ is isomorphic to
$(\mathbb{Z}/n\mathbb{Z})^{\times}$. Since $\mathrm{Gal}(E/\mathbb{Q})$ is
cyclic and of even order, there is a unique subgroup $H$ of
$\mathrm{Gal}(E/\mathbb{Q})$ of index $2$, and so there is a unique quadratic
subfield $K$ of $E$. Let $\iota$ be the nontrivial element of
$\mathrm{Gal}(E/F)\subset\mathrm{Gal}(E/\mathbb{Q})$. Then $\iota$ is the
unique element in $\mathrm{Gal}(E/\mathbb{Q})$ of order $2$. Since
$\\#\mathrm{Gal}(E/\mathbb{Q})=\\#(\mathbb{Z}/n\mathbb{Z})^{\times}$ divides
$4$, we have $\iota\in H$. Thus, $K$ is fixed by the element $\iota$.
Therefore, $K$ is a real quadratic field and so $E$ contains no complex
quadratic subfields.

More generally, let $E$ be any totally imaginary number field such that the
extension $E/\mathbb{Q}$ is Galois with abelian Galois group. Then $E$ is a
CM-field (any abelian extension over $\mathbb{Q}$ is either totally real or a
CM field). We know that

$\mathrm{Gal}(E/\mathbb{Q})\cong\mathbb{Z}/q_{1}\mathbb{Z}\times\mathbb{Z}/q_{2}\mathbb{Z}\times\cdots\times\mathbb{Z}/q_{k}\mathbb{Z},$

where $q_{1},q_{2},\cdots q_{m}$ are powers of prime numbers
($q_{1},q_{2},\cdots q_{m}$ are not necessarily distinct). Suppose further
that the number “$2$” does not appear in $q_{1},q_{2},\cdots q_{m}$, i.e. each
$q_{i}$ is either $2^{k_{i}}$ for $k_{i}\geq 2$ or a power of an odd prime.
Then each component $\mathbb{Z}/q_{i}\mathbb{Z}$ such that $q_{i}$ is
$2^{k_{i}}$ ($k_{i}\geq 2$) contains a unique subgroup $H_{i}$ of index $2$
and a unique element $\sigma_{i}$ of order $2$, and $\sigma_{i}\in H_{i}$. Let
$\iota$ be the nontrivial element of
$\mathrm{Gal}(E/F)\subset\mathrm{Gal}(E/\mathbb{Q})$, where $F$ is the maximal
real subfield of $E$. Then $\iota\in H$ for any subgroup $H$ of
$\mathrm{Gal}(E/\mathbb{Q})$ of index $2$. Therefore, $E$ contains no complex
quadratic subfields.

Since the averaged Colmez conjecture is already proved, we can also prove
averaged analogues of the theorems above.

######  Theorem .

For any $g\in\mathbb{Z}_{\geq 1}$, there exists effectively computable
constants $C_{5}(g)>0$, $C_{6}(g)\in\mathbb{R}$ depending only on $g$ such
that

$\begin{split}&\frac{1}{2}\biggl{(}h_{\mathrm{Falt}}^{\mathrm{st}}(A_{1})+h_{\mathrm{Falt}}^{\mathrm{st}}(A_{2})\biggr{)}\\\
\leq&C_{5}(g)\cdot\frac{1}{2}\biggl{(}\frac{1}{[K_{1}:\mathbb{Q}]}\log|\mathrm{disc}(K_{1})|+\frac{1}{[K_{1}:\mathbb{Q}]}\log|\mathrm{disc}(K_{2})|\biggr{)}+C_{6}(g),\end{split}$

for any pair $A_{1},A_{2}$ of dimension-$g$ abelian varieties defined over
number fields $K_{1},K_{2}$ respectively, such that the following holds:

  * •

There exists a CM-field $E$ of degree $[E:\mathbb{Q}]=2g$ and embeddings
$i_{1}\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K_{1}}(A_{1})$,
$i_{2}\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K_{2}}(A_{2})$ such
that $E$ does not contain any complex quadratic subfields and the CM-type
$\Phi_{1}$ of $(A_{1},i_{1})$ and the CM-type $\Phi_{2}$ of $(A_{2},i_{2})$
satisfy:

$|\Phi_{1}\cap\Phi_{2}|=g-1.$

######  Theorem .

Let $g$ be a positive integer. Suppose that there exists some effectively
computable constant $C_{\mathrm{zero}}(g)\in\mathbb{R}_{\geq 4}$ depending
only on $g$ such that for any CM-field $E$ with maximal totally real subfield
$F$ such that $[F:\mathbb{Q}]=g$, the function $L(s,\chi_{E/F})$ has no zeros
in the region

$1-\frac{1}{C_{\mathrm{zero}}(g)\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|},$

i.e. the conjecture Introduction holds for $g$.

Then there exist effectively computable constants $C_{7}(g)>0$,
$C_{8}(g)\in\mathbb{R}$ depending only on $g$ such that

$\begin{split}&\frac{1}{2}\biggl{(}h_{\mathrm{Falt}}^{\mathrm{st}}(A_{1})+h_{\mathrm{Falt}}^{\mathrm{st}}(A_{2})\biggr{)}\\\
\leq&C_{7}(g)\cdot\frac{1}{2}\biggl{(}\frac{1}{[K_{1}:\mathbb{Q}]}\log|\mathrm{disc}(K_{1})|+\frac{1}{[K_{1}:\mathbb{Q}]}\log|\mathrm{disc}(K_{2})|\biggr{)}+C_{8}(g),\end{split}$

for any pair $A_{1},A_{2}$ of dimension-$g$ abelian varieties defined over
number fields $K_{1},K_{2}$ respectively, such that the following holds:

  * •

There exists a CM-field $E$ of degree $[E:\mathbb{Q}]=2g$ and embeddings
$i_{1}\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K_{1}}(A_{1})$,
$i_{2}\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K_{2}}(A_{2})$ such
that the CM-type $\Phi_{1}$ of $(A_{1},i_{1})$ and the CM-type $\Phi_{2}$ of
$(A_{2},i_{2})$ satisfy:

$|\Phi_{1}\cap\Phi_{2}|=g-1.$

It might be interesting to know that if we only make use of the (proved)
averaged Colmez conjecture, then we cannot obtain results stronger than
section 1 and section 1 (i.e. the “average” condition in these theorems cannot
be dropped), even if we further assume that the abelian variety over the
number field has everywhere good reduction. In particular, we prove the
following theorems, which show that the logarithm of the root discriminant of
the field of everywhere good reduction of CM abelian varieties can be “small”.

######  Theorem .

Assume the Generalized Riemann Hypothesis. For any $g\in\mathbb{Z}_{\geq 1}$,
there exist effectively computable constants $C_{13}(g)>0$,
$C_{14}(g)\in\mathbb{R}$, such that for any CM-field $E$ with
$[E:\mathbb{Q}]=2g$, for any CM-type $\Phi$ of $E$, there exists a number
field $K^{\prime}$ and a CM abelian variety
$(A,i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K^{\prime}}(A))$ over
$K^{\prime}$ of CM-type $\Phi$ such that the abelian variety $A$ over
$K^{\prime}$ has everywhere good reduction and

$\begin{split}&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log|\mathrm{disc}(K^{\prime})|\\\
\leq&C_{13}(g)\log\log|\mathrm{disc}(E)|+\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|+C_{14}(g).\end{split}$
(1)

######  Theorem .

For any $g\in\mathbb{Z}_{\geq 1}$, there exist effectively computable
constants $C_{15}(g)>0$, $C_{16}(g)\in\mathbb{R}$, such that for any CM-field
$E$ with $[E:\mathbb{Q}]=2g$, for any CM-type $\Phi$ of $E$, there exists a
number field $K^{\prime}$ and a CM abelian variety
$(A,i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K^{\prime}}(A))$ over
$K^{\prime}$ of CM-type $\Phi$ such that the abelian variety $A$ over
$K^{\prime}$ has everywhere good reduction and

$\begin{split}\frac{1}{[K^{\prime}:\mathbb{Q}]}\log|\mathrm{disc}(K^{\prime})|\leq
C_{15}(g)\log|\mathrm{disc}(E)|+C_{16}(g).\end{split}$ (2)

This will be discussed in detail in section 6.

In Theorem 6(ii) of [Col98], Colmez has proved that there exist effectively
computable absolute constants $C_{\mathrm{Col},1}>0$,
$C_{\mathrm{Col},2}\in\mathbb{R}$ such that for any CM-field $E$ of degree
$[E:\mathbb{Q}]=2g$ and any CM-type $\Phi$ of $E$ such that the following
hold:

  1. 1.

$(E,\Phi)$ satisfies the Colmez conjecture,

  2. 2.

For any irreducible Artin character $\chi$ such that $m_{(E,\Phi)}(\chi)\neq
0$, the Artin conjecture for $\chi$ holds (i.e. the Artin $L$-function
$L(s,\chi,\mathbb{Q})$ is holomorphic everywhere except possibly for a simple
pole at $s=1$),

  3. 3.

For any irreducible Artin character $\chi$ such that $m_{(E,\Phi)}(\chi)\neq
0$, the Artin $L$-function $L(s,\chi,\mathbb{Q})$ has no zeros on the ball of
radius $\frac{1}{4}$ centered at $0$,

we have

$\begin{split}h_{(E,\Phi)}^{\mathrm{Falt}}\leq
C_{\mathrm{Col},1}\cdot\mu_{(E,\Phi)}+gC_{\mathrm{Col},2}.\end{split}$

The proofs of section 1 and section 1 show that that we can actually remove
the second hypothesis that the Artin $L$-functions involved satisfy the Artin
conjecture. Moreover, in the remark after Theorem 6 of [Col98], Colmez asked
whether it is possible to remove the third hypothesis that the Artin
$L$-functions involved have no zeros on the ball of radius $\frac{1}{4}$
centered at $0$ and making use of “no Siegel zeros” instead. The proofs of
section 1 and section 1 is more or less a positive answer to this question.

### Acknowledgements

The author is deeply grateful to Professor Wei Zhang for suggesting this
problem to the author, supervising the author on this project, and teaching,
mentoring and guiding the author all along. For many times the author would
have been in distress had it not been for the kind and generous help of
Professor Zhang. The author also thanks the Undergraduate Research
Opportunities Program of MIT for providing this opportunity for the author to
do undergraduate research.

##  2 The Faltings height

Let $A$ be a dimension-$g$ abelian variety defined over a number field $K$.
Let $\pi\colon\mathcal{A}\to\mathrm{Spec}(\mathcal{O}_{K})$ be the Néron model
of $A$, and take $\omega$ to be any global section of
$\mathcal{L}\coloneqq\pi_{*}\Omega^{g}_{\mathcal{A}/\mathrm{Spec}\mathcal{O}_{K}}$.
We define the unstable Faltings height of $A$ as follows:

$\begin{split}h_{\mathrm{Falt}}^{\mathrm{unst}}(A/K)\coloneqq\frac{1}{[K:\mathbb{Q}]}\bBigg@{4}(&\log\\#\biggl{(}H^{0}(\mathrm{Spec}\mathcal{O}_{K},\mathcal{L})/(\mathcal{O}_{K}\cdot\omega)\biggr{)}\\\
&-\frac{1}{2}\sum_{\sigma\colon
K\to\mathbb{C}}\log\Biggl{(}\frac{1}{(2\pi)^{g}}\biggl{|}\int_{A(\mathbb{C})}\sigma(\omega\wedge\overline{\omega})\biggr{|}\Biggr{)}\bBigg@{4}).\end{split}$

This definition is independent of the choice of $\omega\in
H^{0}(\mathrm{Spec}\mathcal{O}_{K},\mathcal{L})$.

We define the stable Faltings height of $A$ to be

$h_{\mathrm{Falt}}^{\mathrm{st}}(A)\coloneqq
h_{\mathrm{Falt}}^{\mathrm{unst}}(A_{K^{\prime}}/K^{\prime}),$

where $K^{\prime}$ is a finite extension of $K$ such that
$A_{K^{\prime}}/K^{\prime}$ has everywhere semistable reduction. This
definition does not depend on the choice of the finite extension
$K^{\prime}/K$. Unlike the unstable Faltings height, the stable Faltings
height does not depend on the field of definition of the abelian variety.

The following is a theorem of Bost ([Bos96]).

######  Theorem .

There exists an effectively computable absolute constant
$C_{\mathrm{lower}}>0$ such that for any dimension-$g$ abelian variety $A$
over a number field, we have

$h_{\mathrm{Falt}}^{\mathrm{st}}(A)\geq-gC_{\mathrm{lower}}.$

As we have mentioned in section 1, it is proved by Colmez in [Col93] that for
any CM-field $E$ and any CM-type $\Phi$ of $E$, if
$(A_{1},i_{1}\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K_{1}}(A_{1}))$
and
$(A_{2},i_{2}\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K_{1}}(A_{2}))$
are CM abelian varieties over number fields $K_{1}$ and $K_{2}$, both with CM-
type $\Phi$, then

$h_{\mathrm{Falt}}^{\mathrm{st}}(A_{1})=h_{\mathrm{Falt}}^{\mathrm{st}}(A_{2}).$

We denote this stable Faltings height as $h_{(E,\Phi)}^{\mathrm{Falt}}$.

##  3 The Colmez conjecture revisited

Throughout this section $g$ is an arbitrary positive integer, $E$ is an
arbitrary CM-field of degree $[E:\mathbb{Q}]=2g$ and $\Phi$ is an arbitrary
CM-type of $E$. We denote as $E^{*}_{\Phi}$ the reflex field of $(E,\Phi)$.

Let $A_{(E,\Phi)}^{0}$ be the function from
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ to $\mathbb{C}$ defined as in
section 1. Then since
$\mathrm{Stab}(\Phi)\subset\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is
equal to $\mathrm{Gal}(\overline{\mathbb{Q}}/E^{*}_{\Phi})$, we have:
$A_{(E,\Phi)}^{0}$ factors as

$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\twoheadrightarrow\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})\to\mathbb{C},$

where we denote as $\widetilde{E^{*}_{\Phi}}$ the Galois closure of the
extension $E^{*}_{\Phi}/\mathbb{Q}$. For the following we view the function
$A_{(E,\Phi)}^{0}$ as a (class) function from
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ to $\mathbb{C}$. For any
irreducible Artin character $\chi$ such that $m_{(E,\Phi)}(\chi)\neq 0$,
$\chi$ also factors as

$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\twoheadrightarrow\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})\to\mathbb{C},$

and so we also view $\chi$ as a character of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$.

We fix an embedding $\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and let
$\iota$ be the element of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$
induced by complex conjugation. Let $\chi$ be an irreducible Artin character.
We say that $\chi$ is odd if $\chi(\iota)=-\chi(1)$. Some computations show
that for the trivial character $\chi=\mathbf{1}$, we have
$m_{(E,\Phi)}(\mathbf{1})=\frac{1}{2}g$; and for any nontrivial irreducible
Artin character $\chi$, $m_{(E,\Phi)}(\chi)=0$ unless $\chi$ is odd.

Therefore, we have

$A_{(E,\Phi)}^{0}=\frac{1}{2}g\cdot\mathbf{1}+\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
\rm{odd}}\end{subarray}}m_{(E,\Phi)}(\chi)\chi.$ (3)

This implies that for any irreducible Artin character $\chi$ such that
$m_{(E,\Phi)}(\chi)\neq 0$, the Artin $L$-function $L(s,\chi,\mathbb{Q})$ is
defined and nonzero at $s=0$.

Let $Z_{(E.\Phi)}$ and $\mu_{(E,\Phi)}$ be as in section 1. Since

$\frac{\zeta^{\prime}_{\mathbb{Q}}(0)}{\zeta_{\mathbb{Q}}(0)}=\log(2\pi)$

and

$\mathfrak{f}(\mathbf{1},\mathbb{Q})=0,$

we can deduce that

$Z_{(E,\Phi)}=\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
\rm{odd}}\end{subarray}}m_{(E,\Phi)}(\chi)\frac{L^{\prime}(0,\chi,\mathbb{Q})}{L(0,\chi,\mathbb{Q})},$
(4)

and

$\mu_{(E,\Phi)}=\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
\rm{odd}}\end{subarray}}m_{(E,\Phi)}(\chi)\log\mathfrak{f}(\chi,\mathbb{Q}).$
(5)

##  4 The zero of the Artin $L$-function near $1$

###  4.1 Relation between the zero near $1$ and the logarithmic derivative at
$0$ of the Artin $L$-function

Throughout this subsection $g$ is an arbitrary positive integer, $E$ is an
arbitrary CM-field of degree $[E:\mathbb{Q}]=2g$ and $\Phi$ is an arbitrary
CM-type of $E$. We denote as $E^{*}_{\Phi}$ the reflex field of $(E,\Phi)$. We
denote as $\widetilde{E^{*}_{\Phi}}$ the Galois closure of the extension
$E^{*}_{\Phi}/\mathbb{Q}$.

By Chapter 2, Section 5 of [MM97], for any nontrivial irreducible character
$\chi$ of $\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$, the functions
$\frac{\zeta_{\widetilde{E^{*}_{\Phi}}}(s)}{L(s,\chi,\mathbb{Q})}$ and
$\zeta_{\widetilde{E^{*}_{\Phi}}}(s)L(s,\chi,\mathbb{Q})$ are both holomorphic
except for a simple pole at $s=1$. By Lemma 3 of [Sta74], for any number field
$K$ such that $K\neq\mathbb{Q}$, the function $\zeta_{K}(s)$ has at most one
zero in the region

$1-\frac{1}{4\log|\mathrm{disc}(K)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(K)|}.$

If such a zero exists, it is real and simple.

Therefore, the function $L(s,\chi,\mathbb{Q})$ has at most one zero in the
region

$1-\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}.$

If such a zero exists, it is real and simple.

######  Proposition .

Let $\chi$ be a nontrivial odd irreducible character of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$. Denote as $\beta_{0}$ the
(necessarily real and simple) zero of $L(s,\chi,\mathbb{Q})$ in the region

$1-\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}$

(if it exists).

Let $\delta_{\chi}$ be $1$ if $\beta_{0}$ exists, and let $\delta_{\chi}$ be
$0$ otherwise. We have

$\begin{split}&-\biggl{(}\frac{L^{\prime}(0,\chi,\mathbb{Q})}{L(0,\chi,\mathbb{Q})}+\frac{L^{\prime}(0,\overline{\chi},\mathbb{Q})}{L(0,\overline{\chi},\mathbb{Q})}+\frac{2\delta_{\chi}}{1-\beta_{0}}\biggr{)}\\\
>&-75\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|+\Biggl{(}\log(\frac{\mathfrak{f}(\chi,\mathbb{Q})}{\pi^{\chi(1)}})+\chi(1)\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}\Biggr{)},\end{split}$

and

$\begin{split}&-\biggl{(}\frac{L^{\prime}(0,\chi,\mathbb{Q})}{L(0,\chi,\mathbb{Q})}+\frac{L^{\prime}(0,\overline{\chi},\mathbb{Q})}{L(0,\overline{\chi},\mathbb{Q})}+\frac{2\delta_{\chi}}{1-\beta_{0}}\biggr{)}\\\
<&75\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|+\Biggl{(}\log(\frac{\mathfrak{f}(\chi,\mathbb{Q})}{\pi^{\chi(1)}})+\chi(1)\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}\Biggr{)}.\end{split}$
(6)

###### Proof.

We define the function $\Lambda(s,\chi,\mathbb{Q})$ to be

$\Lambda(s,\chi,\mathbb{Q})\coloneqq(\mathfrak{f}(\chi,\mathbb{Q}))^{s/2}\Biggl{(}\pi^{-(s+1)/2}\Gamma\bigl{(}\frac{s+1}{2}\bigr{)}\Biggr{)}^{\chi(1)}L(s,\chi,\mathbb{Q}).$

We have the functional equation

$\Lambda(s,\chi,\mathbb{Q})=W(\chi)\Lambda(1-s,\overline{\chi},\mathbb{Q})$

for some $W(\chi)\in\mathbb{C}$ with absolute value $1$.

We define the function $\xi_{\widetilde{E^{*}_{\Phi}}}$ to be

$\xi_{\widetilde{E^{*}_{\Phi}}}(s)\coloneqq
s(s-1)|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|^{s/2}\Biggl{(}2(2\pi)^{-s}\Gamma(s)\Biggr{)}^{[\widetilde{E^{*}_{\Phi}}:\mathbb{Q}]/2}\zeta_{\widetilde{E^{*}_{\Phi}}}(s).$

We have the functional equation

$\xi_{\widetilde{E^{*}_{\Phi}}}(s)=\xi_{\widetilde{E^{*}_{\Phi}}}(1-s).$

First consider the function
$f_{1}(s)\coloneqq(\xi_{\widetilde{E^{*}_{\Phi}}}(s))^{2}\Lambda(s,\chi,\mathbb{Q})\Lambda(s,\overline{\chi},\mathbb{Q})$.
It is entire and satisfies the functional equation

$f_{1}(s)=f_{1}(1-s).$

Since $f_{1}(s)$ is real for $s$ real, for any $\rho\in\mathbb{C}$ the order
of the zero of $f_{1}(s)$ at $s=\rho$ is equal to that at $s=\overline{\rho}$.
Moreover, all zeros of $f_{1}(s)$ lie in the critical strip
$0<\mathrm{Re}(s)<1$. Therefore, by logarithmically differentiating the
Hadamard product formula for $f_{1}(s)$ at $s=1$ we get

$\sum_{\rho\colon
f_{1}(\rho)=0}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}=\frac{f^{\prime}_{1}(1)}{f_{1}(1)},$

i.e.

$\begin{split}&\sum_{\rho\colon
f_{1}(\rho)=0}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}\\\
=&2\frac{\xi^{\prime}_{\widetilde{E^{*}_{\Phi}}}(1)}{\xi_{\widetilde{E^{*}_{\Phi}}}(1)}+\frac{\Lambda^{\prime}(1,\chi,\mathbb{Q})}{\Lambda(1,\chi,\mathbb{Q})}+\frac{\Lambda^{\prime}(1,\overline{\chi},\mathbb{Q})}{\Lambda(1,\overline{\chi},\mathbb{Q})}.\end{split}$
(7)

Let $\delta_{\widetilde{E^{*}_{\Phi}}}$ be $1$ if $\beta_{0}$ is a zero of
$\zeta_{\widetilde{E^{*}_{\Phi}}}(s)$, and let
$\delta_{\widetilde{E^{*}_{\Phi}}}$ be $0$ otherwise. Then
$\delta_{\widetilde{E^{*}_{\Phi}}}-\delta_{\chi}$ is equal to $0$ or $1$ and
the order of the zero of $f_{1}(s)$ at $s=\beta_{0}$ is equal to
$2\delta_{\widetilde{E^{*}_{\Phi}}}+2\delta_{\chi}$. Thus, we have

$\begin{split}&\sum_{\begin{subarray}{c}\rho\colon f_{1}(\rho)=0\\\
\rho\neq\beta_{0}\end{subarray}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}\\\
=&\Biggl{(}2\frac{\xi^{\prime}_{\widetilde{E^{*}_{\Phi}}}(1)}{\xi_{\widetilde{E^{*}_{\Phi}}}(1)}-\frac{2\delta_{\widetilde{E^{*}_{\Phi}}}}{1-\beta_{0}}\Biggr{)}+\Biggl{(}\frac{\Lambda^{\prime}(1,\chi,\mathbb{Q})}{\Lambda(1,\chi,\mathbb{Q})}+\frac{\Lambda^{\prime}(1,\overline{\chi},\mathbb{Q})}{\Lambda(1,\overline{\chi},\mathbb{Q})}-\frac{2\delta_{\chi}}{1-\beta_{0}}\Biggr{)}.\end{split}$
(8)

Since the function
$\frac{(\zeta_{\widetilde{E^{*}_{\Phi}}}(s))^{2}}{L(s,\chi,\mathbb{Q})L(s,\overline{\chi},\mathbb{Q})}$
is holomorphic on $0<\mathrm{Re}(s)<1$, for any $\rho\in\mathbb{C}$ such that
$0<\mathrm{Re}(\rho)<1$, the order of the zero at $s=\rho$ of the function
$f_{1}(s)$ is less than or equal to $4$ times the order of the zero at
$s=\rho$ of the function $\zeta_{\widetilde{E^{*}_{\Phi}}}(s)$. In view of the
fact that all zeros of $f_{1}(s)$ lie in the critical strip
$0<\mathrm{Re}(s)<1$, we have

$0\leq\sum_{\begin{subarray}{c}\rho\colon f_{1}(\rho)=0\\\
\rho\neq\beta_{0}\end{subarray}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}\leq
4\sum_{\begin{subarray}{c}\rho\colon\zeta_{\widetilde{E^{*}_{\Phi}}}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\\\
\rho\neq\beta_{0}\end{subarray}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}.$
(9)

Then consider the function
$f_{2}(s)\coloneqq\frac{(\xi_{\widetilde{E^{*}_{\Phi}}}(s))^{2}}{\Lambda(s,\chi,\mathbb{Q})\Lambda(s,\overline{\chi},\mathbb{Q})}$.
Similar to the case of $f_{1}$, we have

$\begin{split}&\sum_{\begin{subarray}{c}\rho\colon f_{2}(\rho)=0\\\
\rho\neq\beta_{0}\end{subarray}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}\\\
=&\Biggl{(}2\frac{\xi^{\prime}_{\widetilde{E^{*}_{\Phi}}}(1)}{\xi_{\widetilde{E^{*}_{\Phi}}}(1)}-\frac{2\delta_{\widetilde{E^{*}_{\Phi}}}}{1-\beta_{0}}\Biggr{)}-\Biggl{(}\frac{\Lambda^{\prime}(1,\chi,\mathbb{Q})}{\Lambda(1,\chi,\mathbb{Q})}+\frac{\Lambda^{\prime}(1,\overline{\chi},\mathbb{Q})}{\Lambda(1,\overline{\chi},\mathbb{Q})}-\frac{2\delta_{\chi}}{1-\beta_{0}}\Biggr{)},\end{split}$
(10)

and

$0\leq\sum_{\begin{subarray}{c}\rho\colon f_{2}(\rho)=0\\\
\rho\neq\beta_{0}\end{subarray}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}\leq
4\sum_{\begin{subarray}{c}\rho\colon\zeta_{\widetilde{E^{*}_{\Phi}}}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\\\
\rho\neq\beta_{0}\end{subarray}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}.$
(11)

By logarithmically differentiating the functional equation of
$\Lambda(s,\chi,\mathbb{Q})\Lambda(s,\overline{\chi},\mathbb{Q})$ at $s=1$, we
have

$\frac{\Lambda^{\prime}(0,\chi,\mathbb{Q})}{\Lambda(0,\chi,\mathbb{Q})}+\frac{\Lambda^{\prime}(0,\overline{\chi},\mathbb{Q})}{\Lambda(0,\overline{\chi},\mathbb{Q})}=-\Biggl{(}\frac{\Lambda^{\prime}(1,\chi,\mathbb{Q})}{\Lambda(1,\chi,\mathbb{Q})}+\frac{\Lambda^{\prime}(1,\overline{\chi},\mathbb{Q})}{\Lambda(1,\overline{\chi},\mathbb{Q})}\Biggr{)}.$

The result then follows from subtracting Equation (8) by Equation (10) and the
following subsection 4.1. ∎

######  Lemma .

Let $K$ be a number field such that $K\neq\mathbb{Q}$. Denote as $\beta_{0}$
the (necessarily real and simple) zero of $\zeta_{K}(s)$ in the region

$1-\frac{1}{4\log|\mathrm{disc}(K)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(K)|}$

(if it exists). Then we have

$\begin{split}0\leq{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\\\
\rho\neq\beta_{0}\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}<\frac{75}{2}\log|\mathrm{disc}(K)|.\end{split}$

###### Proof.

For any $\rho\in\mathbb{C}$ such that
$\mathrm{Re}(\rho)<1-\frac{1}{4\log|\mathrm{disc}(K)|}$, we have

$\begin{split}0<&\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}<25\cdot\biggl{(}\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\rho}+\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\overline{\rho}}\biggr{)}.\end{split}$

For any $\rho\in\mathbb{C}$ such that
$1-\frac{1}{4\log|\mathrm{disc}(K)|}\leq\mathrm{Re}(\rho)<1$ and
$|\operatorname{Im}(\rho)|>\frac{1}{4\log|\mathrm{disc}(K)|}$, we have

$\begin{split}0<\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}<5\cdot\biggl{(}\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\rho}+\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\overline{\rho}}\biggr{)}.\end{split}$

Therefore, we have

$\begin{split}0\leq&{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\\\
\rho\neq\beta_{0}\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}\\\
<&25{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\\\
\rho\neq\beta_{0}\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\rho}+\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\overline{\rho}}\biggr{)}\\\
\leq&25{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\rho}+\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\overline{\rho}}\biggr{)}.\end{split}$

By the proof of Lemma 3 of [Sta74], we have

$0\leq{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{s-\rho}+\frac{1}{s-\overline{\rho}}\biggr{)}<\frac{1}{s-1}+\frac{1}{2}\log|\mathrm{disc}(K)|,$
(12)

for any $s$ real with $1<s<2$.

Taking $s=1+\frac{1}{\log|\mathrm{disc}(K)|}$ in Equation (12), we get:

$\begin{split}0\leq&{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\rho}+\frac{1}{1+\frac{1}{\log|\mathrm{disc}(K)|}-\overline{\rho}}\biggr{)}\\\
<&\frac{3}{2}\log|\mathrm{disc}(K)|.\end{split}$

Therefore, we have

$\begin{split}0\leq{\sum_{\begin{subarray}{c}\rho\colon\zeta_{K}(\rho)=0\\\
0<\mathrm{Re}(\rho)<1\\\
\rho\neq\beta_{0}\end{subarray}}}\frac{1}{2}\biggl{(}\frac{1}{1-\rho}+\frac{1}{1-\overline{\rho}}\biggr{)}<\frac{75}{2}\log|\mathrm{disc}(K)|.\end{split}$

∎

######  Corollary .

Let $c$ be a real number such that $0<c\leq\frac{1}{4}$. Suppose that for any
nontrivial odd irreducible character $\chi$ of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ such that
$m_{(E,\Phi)}(\chi)\neq 0$, there is no zero of $L(s,\chi,\mathbb{Q})$ in the
region

$1-\frac{c}{\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|},$

then we have

$\begin{split}-Z_{(E,\Phi)}-\frac{1}{2}\mu_{(E,\Phi)}<&\frac{1}{4}g\cdot(75+2c^{\prime})(2g)!\log|\mathrm{disc}(E^{*}_{\Phi})|+\frac{1}{4}g\biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}-\log(\pi)\biggr{)},\end{split}$
(13)

where $c^{\prime}$ is defined to be $\frac{1}{c}$ if $c<\frac{1}{4}$, and $0$
if $c=\frac{1}{4}$.

###### Proof.

By Lemma 2 and Section 2 of [Col98], for any
$\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))$,
$m_{(E,\Phi)}(\chi)$ is a non-negative real number and
$m_{(E,\Phi)}(\chi)=m_{(E,\Phi)}(\overline{\chi})$. Hence, we have

$\begin{split}&-Z_{(E,\Phi)}-\frac{1}{2}\mu_{(E,\Phi)}\\\
=&\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
odd}\end{subarray}}m_{(E,\Phi)}(\chi)\biggl{(}-\frac{L^{\prime}(0,\chi,\mathbb{Q})}{L(0,\chi,\mathbb{Q})}-\frac{1}{2}\log(\mathfrak{f}(\chi,\mathbb{Q}))\biggr{)}\\\
=&\frac{1}{2}\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
odd}\end{subarray}}m_{(E,\Phi)}(\chi)\Biggl{(}-\biggl{(}\frac{L^{\prime}(0,\chi,\mathbb{Q})}{L(0,\chi,\mathbb{Q})}+\frac{L^{\prime}(0,\overline{\chi},\mathbb{Q})}{L(0,\overline{\chi},\mathbb{Q})}\biggr{)}-\log(\mathfrak{f}(\chi,\mathbb{Q}))\Biggr{)}\\\
<&\frac{1}{2}\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
odd}\end{subarray}}m_{(E,\Phi)}(\chi)\Biggl{(}(75+2c^{\prime})\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|+\chi(1)\biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}-\log(\pi)\biggr{)}\Biggr{)},\end{split}$
(14)

by Equation (6).

By the definition of ${A_{(E,\Phi)}^{0}}$ we have

$\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
odd}\end{subarray}}m_{(E,\Phi)}(\chi)\chi(1)={A_{(E,\Phi)}^{0}}(1)-\frac{1}{2}g=\frac{1}{2}g.$

Since for any
$\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))$,
$m_{(E,\Phi)}(\chi)$ is a non-negative real number and $\chi(1)\geq 1$, we
have

$\sum_{\begin{subarray}{c}\chi\in\mathrm{Irr}(\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q}))\\\
\chi\neq\mathbf{1}\\\ \chi\mbox{
odd}\end{subarray}}m_{(E,\Phi)}(\chi)\leq\frac{1}{2}g.$

By the following Equation (17) we have

$\begin{split}\frac{1}{[\widetilde{E^{*}_{\Phi}}:\mathbb{Q}]}\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|\leq\log|\mathrm{disc}(E^{*}_{\Phi})|.\end{split}$

The reflex field $E^{*}_{\Phi}$ is contained in the Galois closure
$\widetilde{E}$ of the extension $E/\mathbb{Q}$, and so
$\widetilde{E^{*}_{\Phi}}$ is also contained in $\widetilde{E}$. Thus, we have
$[\widetilde{E^{*}_{\Phi}}:\mathbb{Q}]\leq(2g)!$.

Hence, we get our claim. ∎

######  Lemma .

Let $K_{1}$ and $K_{2}$ be number fields. Let $K_{1}K_{2}$ be the compositum
of $K_{1}$ and $K_{2}$. Then we have

$|\mathrm{disc}(K_{1}K_{2})|^{1/[K_{1}K_{2}:\mathbb{Q}]}\leq|\mathrm{disc}(K_{1})|^{1/[K_{1}:\mathbb{Q}]}|\mathrm{disc}(K_{2})|^{1/[K_{2}:\mathbb{Q}]},$
(15)

and

$|\mathrm{disc}(K_{1}K_{2})|\leq|\mathrm{disc}(K_{1})|^{[K_{2}:\mathbb{Q}]}|\mathrm{disc}(K_{2})|^{[K_{1}:\mathbb{Q}]}.$
(16)

In particular, let $K$ be a number field and let $\widetilde{K}$ be the Galois
closure of the extension $K/\mathbb{Q}$. Then

$|\mathrm{disc}(\widetilde{K})|^{1/[\widetilde{K}:\mathbb{Q}]}\leq|\mathrm{disc}(K)|,$
(17)

and

$|\mathrm{disc}(\widetilde{K})|\leq|\mathrm{disc}(K)|^{[K:\mathbb{Q}]!}.$ (18)

###### Proof.

This is Lemma 7 of [Sta74]. ∎

###  4.2 Sufficient conditions for the nonexistence of the zero near $1$ of
the Artin $L$-function

By Theorem 3 of [Sta74], we have the following theorem.

######  Theorem .

Let $L/K$ be a finite Galois extension of number fields. Let
$s_{0}\in\mathbb{C}$ be a simple zero of $\zeta_{L}(s)$.

(1) For any irreducible character $\chi$ of $\mathrm{Gal}(L/K)$, $L(s,\chi,K)$
is defined at $s=s_{0}$. There is a (unique) irreducible character
$\mathcal{X}_{s_{0},L/K}$ of $\mathrm{Gal}(L/K)$ such that for any irreducible
character $\chi$ of $\mathrm{Gal}(L/K)$, $L(s_{0},\chi,K)=0$ if and only if
$\chi=\mathcal{X}_{s_{0},L/K}$. $\mathcal{X}_{s_{0},L/K}$ is a linear
character of $\mathrm{Gal}(L/K)$ (so $\mathcal{X}_{s_{0},L/K}$ is a group
homomorphism from $\mathrm{Gal}(L/K)$ to $\mathbb{C}^{\times}$).

(2) There is a (unique) subfield $\mathcal{K}_{s_{0},L/K}$ of $L$ containing
$K$ such that for any field $K^{\prime}$ containing $K$ and contained in $L$,
$\zeta_{K^{\prime}}(s_{0})=0$ if and only if $K^{\prime}$ contains
$\mathcal{K}_{s_{0},L/K}$. The extension $\mathcal{K}_{s_{0},L/K}/K$ is
cyclic.

(3) $\mathcal{K}_{s_{0},L/K}$ is the fixed field of the kernel of
$\mathcal{X}_{s_{0},L/K}$.

(4) Suppose further that $s_{0}$ is real. Then exactly one of the following
holds:

  1. 1.

$\mathcal{K}_{s_{0},L/K}$ is equal to $K$ and $\mathcal{X}_{s_{0},L/K}$ is the
trivial character.

  2. 2.

$\mathcal{K}_{s_{0},L/K}$ is quadratic over $K$ and $\mathcal{X}_{s_{0},L/K}$
is the group homomorphism from $\mathrm{Gal}(L/K)$ to $\mathbb{C}^{\times}$
with kernel $\mathrm{Gal}(L/\mathcal{K}_{s_{0},L/K})$ and image $\\{\pm 1\\}$.
In particular, $\mathcal{X}_{s_{0},L/K}$ is a nontrivial real linear
character.

For the rest of this subsection $E$ is an arbitrary CM-field and $\Phi$ is an
arbitrary CM-type of $E$. We denote as $E^{*}_{\Phi}$ the reflex field of
$(E,\Phi)$. We denote as $\widetilde{E^{*}_{\Phi}}$ the Galois closure of the
extension $E^{*}_{\Phi}/\mathbb{Q}$.

######  Corollary .

Suppose that one (or two, or all) of the following conditions hold:

  1. 1.

The Galois closure $\widetilde{E}$ of the extension $E/\mathbb{Q}$ does not
contain any complex quadratic subfields.

  2. 2.

$\widetilde{E^{*}_{\Phi}}$ does not contain any complex quadratic subfields.

  3. 3.

There does not exist a nontrivial irreducible real linear character $\chi$ of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ such that
$m_{(E,\Phi)}(\chi)\neq 0$ and the homomorphism $\chi$ from
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ to $\mathbb{C}^{\times}$
has image $\\{\pm 1\\}$ and kernel $\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/K)$
for some complex quadratic subfield $K$ of $\widetilde{E^{*}_{\Phi}}$.

(Note that Condition 1 implies Condition 2 since
$E^{*}_{\Phi}\subset\widetilde{E}$, and Condition 2 implies Condition 3.)

Then for any nontrivial odd irreducible character $\chi$ of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$, there is no zero of
$L(s,\chi,\mathbb{Q})$ in the region

$1-\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}.$

###### Proof.

Let $\chi$ be a nontrivial odd irreducible character of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ such that such a zero
exists. Denote this zero as $\beta_{0}$. Then $\beta_{0}$ must be real and
$\beta_{0}$ is also a simple zero of $\zeta_{\widetilde{E^{*}_{\Phi}}}(s)$.

Therefore, by subsection 4.2, $\chi$ is a real linear character of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$, and the homomorphism
$\chi$ from $\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ to
$\mathbb{C}^{\times}$ has image $\\{\pm 1\\}$ and kernel
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/K)$ for some quadratic subfield $K$ of
$\widetilde{E^{*}_{\Phi}}$.

Since $\chi$ is an odd character, we have $\chi(\iota)=-\chi(1)$, where
$\iota$ is the element in $\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$
induced by complex conjugation, and so $K/\mathbb{Q}$ must be a complex
quadratic extension.

Therefore, our claim follows. ∎

Since the compositum of two CM-fields is also a CM-field, the Galois closure
of a CM-field (viewed as an extension over $\mathbb{Q}$) is also a CM-field.
We know that the reflex field $E^{*}_{\Phi}$ of $(E,\Phi)$ is a CM-field.
Therefore, $\widetilde{E^{*}_{\Phi}}$ is also a CM-field. We denote as
$(\widetilde{E^{*}_{\Phi}})_{+}$ the maximal totally real subfield of
$\widetilde{E^{*}_{\Phi}}$.

######  Proposition .

Let $c$ be a real number such that $0<c\leq\frac{1}{4}$. Suppose that the
function
$L(s,\chi_{\widetilde{E^{*}_{\Phi}}/(\widetilde{E^{*}_{\Phi}})_{+}})=\frac{\zeta_{\widetilde{E^{*}_{\Phi}}}}{\zeta_{(\widetilde{E^{*}_{\Phi}})_{+}}}$
has no zero in the region

$1-\frac{c}{\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}.$

Then for any nontrivial odd irreducible character $\chi$ of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$, there is no zero of
$L(s,\chi,\mathbb{Q})$ in the above region either.

###### Proof.

Let $\chi$ be a nontrivial odd irreducible character of
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ such that such a zero
exists. Denote this zero as $\beta_{0}$. Then $\beta_{0}$ must be real and
$\beta_{0}$ is also a simple zero of $\zeta_{\widetilde{E^{*}_{\Phi}}}(s)$. By
our assumption on
$L(s,\chi_{\widetilde{E^{*}_{\Phi}}/(\widetilde{E^{*}_{\Phi}})_{+}})$,
$\beta_{0}$ cannot be a zero of
$L(s,\chi_{\widetilde{E^{*}_{\Phi}}/(\widetilde{E^{*}_{\Phi}})_{+}})$.
Therefore, $\beta_{0}$ is a zero of
$\zeta_{(\widetilde{E^{*}_{\Phi}})_{+}}(s)$. Therefore, the field
$\mathcal{K}_{\beta_{0},\widetilde{E^{*}_{\Phi}}/\mathbb{Q}}$ in subsection
4.2 must be contained in the field $(\widetilde{E^{*}_{\Phi}})_{+}$, and so
$\mathcal{K}_{\beta_{0},\widetilde{E^{*}_{\Phi}}/\mathbb{Q}}$ is a real
quadratic field. By subsection 4.2, since $L(\beta_{0},\chi,\mathbb{Q})=0$,
$\chi$ is a group homomorphism from
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ to $\mathbb{C}^{\times}$
with kernel
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathcal{K}_{\beta_{0},\widetilde{E^{*}_{\Phi}}/\mathbb{Q}})$,
and so $\chi(\iota)=\chi(1)=1$, where $\iota$ is the element in
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ induced by complex
conjugation. This is a contradiction since the character $\chi$ is assumed to
be odd. ∎

###  4.3 Proofs of section 1 and section 1

###### Proof of section 1.

Let $g$ be a positive integer. Let $E$ be a CM-field with maximal totally real
subfield $F$ of degree $[F:\mathbb{Q}]=g$. Let
$(A,i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K}(A))$ be a CM abelian
variety over a number field $K$ and let $\Phi$ be the CM-type of $(A,i)$. Then
the field $K$ contains the reflex field $E^{*}_{\Phi}$. Thus, we have

$\begin{split}\frac{1}{[K:\mathbb{Q}]}\log|\mathrm{disc}(K)|&\geq\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|\\\
&\geq\frac{1}{(2g)!}\log|\mathrm{disc}(E^{*}_{\Phi})|,\end{split}$

where the last inequality follows from the fact that the reflex field
$E^{*}_{\Phi}$ is contained in the Galois closure $\widetilde{E}$ of the
extension $E/\mathbb{Q}$.

By Lemma 8 and Lemma 9 of [Sta74], suppose that there is a (necessarily real
and simple) zero $\beta_{0}$ of
$L(s,\chi_{\widetilde{E^{*}_{\Phi}}/(\widetilde{E^{*}_{\Phi}})_{+}})$ in the
range

$1-\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|},$

then there exists a complex quadratic subfield $K$ of
$\widetilde{E^{*}_{\Phi}}$ such that $\zeta_{K}(\beta_{0})=0$ also. Since the
Riemann zeta function $\zeta_{\mathbb{Q}}(s)$ has no real zeros in the range
$0<s<1$, this means that $\beta_{0}$ is a zero of the function
$L(s,\chi_{K/\mathbb{Q}})=\frac{\zeta_{K}(s)}{\zeta_{\mathbb{Q}}(s)}$. Since
$K$ is contained in $\widetilde{E^{*}_{\Phi}}$, we have
$|\mathrm{disc}(\widetilde{E^{*}_{\Phi}})|\geq|\mathrm{disc}(K)|$. Therefore,
$\beta_{0}$ is a Siegel zero of $L(s,\chi_{K/\mathbb{Q}})$.

The result then follows from subsection 4.2 and subsection 4.1. ∎

It is proved by Colmez ([Col93]) and Obus ([Obu13]) that the Colmez conjecture
is true when the CM-field is abelian:

######  Theorem .

Let $E$ be a CM-field such that the extension $E/\mathbb{Q}$ is Galois with
abelian Galois group. Then we have

$h_{(E,\Phi)}^{\mathrm{Falt}}=-Z_{(E,\Phi)}-\frac{1}{2}\mu_{(E,\Phi)}$

for any CM-type $\Phi$ of $E$.

As a corollary, we can prove an unconditional analogue of section 1.

###### Proof of section 1.

Similar to the above proof of section 1, the statement follows from the above-
mentioned Lemma 8 and Lemma 9 of [Sta74], subsection 4.2, subsection 4.1, and
subsection 4.3. ∎

##  5 The (proved) averaged Colmez conjecture

Although the formula $-Z_{(E,\Phi)}-\frac{1}{2}\mu_{(E,\Phi)}$ in the Colmez
conjecture appears very complicated, the average over all CM-types $\Phi$ of a
CM-field $E$ is much simpler: As is conjectured in Page 634 of [Col93] and
proved in [YZ18] and [AGHMP18], we have the following proposition.

######  Proposition .

Let $E$ be a CM-field with maximal totally real subfield $F$. Then we have

$\frac{1}{2^{[F:\mathbb{Q}]}}\sum_{\Phi}\biggl{(}-Z_{(E,\Phi)}-\frac{1}{2}\mu_{(E,\Phi)}\biggr{)}=-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|),$

where the sum on the left-hand-side is over all CM-types $\Phi$ of $E$.

In other words, the Colmez conjecture implies the The (proved) averaged Colmez
conjecture stated below.

######  Theorem  ((Proved) averaged Colmez conjecture).

Let $E$ be a CM-field with maximal totally real subfield $F$. Then we have

$\frac{1}{2^{[F:\mathbb{Q}]}}\sum_{\Phi}h_{(E,\Phi)}^{\mathrm{Falt}}=-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|),$
(19)

where the sum on the left-hand-side is over all CM-types $\Phi$ of $E$.

This is proved independently by Yuan–Zhang [YZ18] and
Andreatta–Goren–Howard–Madapusi-Pera [AGHMP18].

In the following, we use the proved averaged Colmez conjecture to prove
averaged analogues of section 1 and section 1.

######  Proposition .

Let $g$ be a positive integer. Suppose that there exists some effectively
computable constant $C_{\mathrm{zero}}(g)\in\mathbb{R}_{\geq 4}$ depending
only on $g$ such that for any CM-field $E$ with maximal totally real subfield
$F$ such that $[F:\mathbb{Q}]=g$, the function $L(s,\chi_{E/F})$ has no zeros
in the region

$1-\frac{1}{C_{\mathrm{zero}}(g)\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|},$

then there exist effectively computable constants $C_{9}(g)>0$,
$C_{10}(g)\in\mathbb{R}$ depending only on $g$ such that

$h_{\mathrm{Falt}}^{\mathrm{st}}(A)\leq
C_{9}(g)\log|\mathrm{disc}(E)|+C_{10}(g)$

for any CM-field $E$ of degree $[E:\mathbb{Q}]=2g$ and for any abelian variety
$A$ over a number field with complex multiplication by $\mathcal{O}_{E}$.

###### Proof.

Let $E$ be any CM-field with maximal totally real subfield $F$ such that
$[F:\mathbb{Q}]=g$. Denote as $\beta_{0}$ the (necessarily real and simple)
zero of $L(s,\chi_{E/F})$ in the region

$1-\frac{1}{4\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|}$

(if it exists).

We define $\delta_{\chi_{E/F}}$ to be $1$ if $\beta_{0}$ exists, and we define
$\delta_{\chi_{E/F}}$ to be $0$ otherwise. By an argument similar to the proof
of subsection 4.1, we have

$\begin{split}&-\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{\delta_{\chi_{E/F}}}{1-\beta_{0}}\\\
\geq&\frac{1}{2}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)+\frac{g}{2}\biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}-\log(\pi)\biggr{)},\end{split}$

and

$\begin{split}&-\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{\delta_{\chi_{E/F}}}{1-\beta_{0}}\\\
<&\frac{75}{2}\log|\mathrm{disc}(E)|+\frac{1}{2}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)+\frac{g}{2}\biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}-\log(\pi)\biggr{)}.\end{split}$

By our assumption, we then have

$\begin{split}&-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\\\
<&\frac{1}{2}\Biggl{(}\frac{75}{2}\log|\mathrm{disc}(E)|+C_{\mathrm{zero}}(g)\log|\mathrm{disc}(E)|+\frac{g}{2}\biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}-\log(\pi)\biggr{)}\Biggr{)}.\end{split}$

Let $(A,i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K}(A))$ be any CM
abelian variety over a number field $K$. Let $\Phi_{0}$ be the CM-type of
$(A,i)$. By section 5, we have

$h_{\mathrm{Falt}}^{\mathrm{st}}(A)=-\sum_{\Phi\neq\Phi_{0}}h_{(E,\Phi)}^{\mathrm{Falt}}+2^{g}\Biggl{(}-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\Biggr{)}.$

Let $C_{\mathrm{lower}}>0$ be as in section 2. Then by section 2 we have

$\begin{split}h_{\mathrm{Falt}}^{\mathrm{st}}(A)&=-\sum_{\Phi\neq\Phi_{0}}h_{(E,\Phi)}^{\mathrm{Falt}}+2^{g}\Biggl{(}-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\Biggr{)}\\\
&\leq(2^{g}-1)gC_{\mathrm{lower}}\\\
&+2^{g}\cdot\frac{1}{2}\Biggl{(}\frac{75}{2}\log|\mathrm{disc}(E)|+C_{\mathrm{zero}}(g)\log|\mathrm{disc}(E)|+\frac{g}{2}\biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}-\log(\pi)\biggr{)}\Biggr{)}.\end{split}$

∎

######  Proposition .

For any $g\in\mathbb{Z}_{\geq 1}$, there exist constants $C_{11}(g)>0$,
$C_{12}(g)\in\mathbb{R}$ depending only on $g$ such that

$h_{\mathrm{Falt}}^{\mathrm{st}}(A)\leq
C_{11}(g)\log|\mathrm{disc}(E)|+C_{12}(g)$

for any CM-field $E$ of degree $[E:\mathbb{Q}]=2g$ such that $E$ has no
complex quadratic subfields and for any abelian variety $A$ over a number
field with complex multiplication by $\mathcal{O}_{E}$.

###### Proof.

Let $g$ be a positive integer. Let $E$ be a CM-field with maximal totally real
subfield $F$ with degree $[F:\mathbb{Q}]=g$. By Lemma 9 of [Sta74], suppose
that there exists a (necessarily real and simple) zero $\beta_{0}$ of
$L(s,\chi_{E/F})$ in the range

$1-\frac{1}{16g!\log|\mathrm{disc}(E)|}\leq\mathrm{Re}(s)<1,|\operatorname{Im}(s)|\leq\frac{1}{4\log|\mathrm{disc}(E)|},$

then there exists a complex quadratic subfield $K$ of $E$ such that
$\zeta_{K}(\beta_{0})=0$ as well. So if $E$ does not contain any complex
quadratic fields, then there is no such zero.

The rest of the proof is similar to that of section 5. ∎

######  Lemma .

Let $E$ be a CM-field with maximal totally real subfield $F$ of degree
$[F:\mathbb{Q}]=g$. Let $\Phi_{1},\Phi_{2}$ be CM-types of $E$ such that
$|\Phi_{1}\cap\Phi_{2}|=g-1$. Let $\varphi_{0}$ be the unique element in
$\mathrm{Hom}_{\mathbb{Q}}(F,\mathbb{R})$ such that the element $\phi_{1}$ in
$\Phi_{1}$ lying above $\varphi_{0}$ is not equal to the element $\phi_{2}$ in
$\Phi_{2}$ lying above $\varphi_{0}$. We have $\phi_{1}=\phi_{2}\circ\iota$,
where $\iota$ is the nontrivial element of $\mathrm{Gal}(E/F)$. It is easy to
see that the subfield $\phi_{1}(E)$ of $\mathbb{C}$ is equal to the subfield
$\phi_{2}(E)$ of $\mathbb{C}$. Let $E^{*}_{\Phi_{1}},E^{*}_{\Phi_{2}}$ be the
reflex fields of $(E,\Phi_{1}),(E,\Phi_{2})$, respectively. Then the
compositum of fields $E^{*}_{\Phi_{1}}E^{*}_{\Phi_{2}}$ contains the field
$\phi_{1}(E)=\phi_{2}(E)$.

###### Proof.

Since $E$ is a totally complex quadratic extension of the totally real field
$F$, we can write $E=F[\sqrt{-\alpha_{E}}]$ for some totally positive
$\alpha_{E}\in F$, where $\sqrt{-\alpha_{E}}$ is any square root of
$-\alpha_{E}$ in $\overline{\mathbb{Q}}$. Thus,
$\sum_{\phi\in\Phi_{1}}\phi(\sqrt{-\alpha_{E}})\in E^{*}_{\Phi_{1}}$ and
$\sum_{\phi\in\Phi_{2}}\phi(\sqrt{-\alpha_{E}})\in E^{*}_{\Phi_{2}}$. By our
assumption on $\Phi_{1},\Phi_{2}$ and $\varphi_{0}$, we have

$\begin{split}\sum_{\phi\in\Phi_{1}}\phi(\sqrt{-\alpha_{E}})-\sum_{\phi\in\Phi_{2}}\phi(\sqrt{-\alpha_{E}})&=\phi_{1}(\sqrt{-\alpha_{E}})-\phi_{2}(\sqrt{-\alpha_{E}})\\\
&=2\phi_{1}(\sqrt{-\alpha_{E}})\\\
&=-2\phi_{2}(\sqrt{-\alpha_{E}}).\end{split}$

Therefore, the compositum of fields $E^{*}_{\Phi_{1}}E^{*}_{\Phi_{2}}$
contains the element
$\phi_{1}(\sqrt{-\alpha_{E}})=-\phi_{2}(\sqrt{-\alpha_{E}})$.

Let $\alpha_{F}$ be an element of $F$ such that $F=\mathbb{Q}[\alpha_{F}]$.
Then similar to above, since

$\begin{split}\sum_{\phi\in\Phi_{1}}\phi(\alpha_{F}\sqrt{-\alpha_{E}})-\sum_{\phi\in\Phi_{2}}\phi(\alpha_{F}\sqrt{-\alpha_{E}})&=\phi_{1}(\alpha_{F}\sqrt{-\alpha_{E}})-\phi_{2}(\alpha_{F}\sqrt{-\alpha_{E}})\\\
&=\varphi_{0}(\alpha_{F})\phi_{1}(\sqrt{-\alpha_{E}})-\varphi_{0}(\alpha_{F})\phi_{2}(\sqrt{-\alpha_{E}})\\\
&=2\varphi_{0}(\alpha_{F})\phi_{1}(\sqrt{-\alpha_{E}})\\\
&=-2\varphi_{0}(\alpha_{F})\phi_{2}(\sqrt{-\alpha_{E}}),\end{split}$

the compositum of fields $E^{*}_{\Phi_{1}}E^{*}_{\Phi_{2}}$ contains the
element
$\varphi_{0}(\alpha_{F})\phi_{1}(\sqrt{-\alpha_{E}})=-\varphi_{0}(\alpha_{F})\phi_{2}(\sqrt{-\alpha_{E}})$.

Combined with above, we have: the compositum of fields
$E^{*}_{\Phi_{1}}E^{*}_{\Phi_{2}}$ contains the element
$\varphi_{0}(\alpha_{F})$ and the element
$\phi_{1}(\sqrt{-\alpha_{E}})=-\phi_{2}(\sqrt{-\alpha_{E}})$, and so it
contains the field $\phi_{1}(E)=\phi_{2}(E)$. ∎

######  Remark .

The CM-types $\Phi_{1},\Phi_{2}$ in section 5 is a pair of “nearby” CM-types
considered in [YZ18].

######  Corollary .

Let $E$ be a CM-field with maximal totally real subfield $F$. Let
$\Phi_{1},\Phi_{2}$ be CM-types of $E$ such that $|\Phi_{1}\cap\Phi_{2}|=g-1$.
Then we have

$\begin{split}|\mathrm{disc}(E^{*}_{\Phi_{1}})|^{1/[E^{*}_{\Phi_{1}}:\mathbb{Q}]}|\mathrm{disc}(E^{*}_{\Phi_{2}})|^{1/[E^{*}_{\Phi_{2}}:\mathbb{Q}]}\geq|\mathrm{disc}(E)|^{1/[E:\mathbb{Q}]},\end{split}$

where $E^{*}_{\Phi_{1}},E^{*}_{\Phi_{2}}$ are the reflex fields of
$(E,\Phi_{1}),(E,\Phi_{2})$, respectively.

###### Proof.

This follows from section 5 and Equation (15). ∎

###### Proof of section 1.

This follows from section 5, section 5, and the fact that the field of
definition of any CM abelian variety contains the reflex field. ∎

###### Proof of section 1.

This follows from section 5, section 5, and the fact that the field of
definition of any CM abelian variety contains the reflex field. ∎

##  6 Field of everywhere good reduction of CM abelian varieties

We know that any abelian variety over a number field with complex
multiplication by a CM-field has potential good reduction everywhere. In this
section, we show that the logarithm of the root discriminant of the field of
everywhere good reduction can be small compared with the logarithm of the
discriminant of the CM-field.

######  Lemma .

Let $A$ be an abelian variety over a number field $K$. Let $L_{1},L_{2}$ be
number fields containing $K$. If the abelian variety $A_{L_{1}}/L_{1}$ and the
abelian variety $A_{L_{2}}/L_{2}$ both have everywhere good reduction, then
the abelian variety $A_{L_{1}\cap L_{2}}/L_{1}\cap L_{2}$ has everywhere good
reduction.

###### Proof.

This follows from the Neron-Ogg-Shafarevich criterion. ∎

By part (b) of Corollary 2 to Theorem 2 of [ST68], we have the following
theorem:

######  Theorem .

Let $A$ be an abelian variety over a number field $K$. Let $\mathfrak{p}$ be a
prime ideal of $\mathcal{O}_{K}$. Let $p$ be the characteristic of the residue
field $\mathcal{O}_{K}/\mathfrak{p}$. Suppose that $A/K$ has potential good
reduction at $\mathfrak{p}$. Let $m$ be any integer $\geq 3$ and prime to $p$.
Let $K(A[m])$ be the minimal field of definition of the set of $m$-torsion
points $A[m]$ of $A$. The following are equivalent:

(a) The extension $K(A[m])/K$ is unramified at $\mathfrak{p}$.

(b) The abelian variety $A/K$ has good reduction at $\mathfrak{p}$.

######  Corollary .

Let $K$ be a number field. Let $A$ be an abelian variety over $K$ with
potential good reduction everywhere. Let $S_{A/K}$ be the set of all prime
ideals of $\mathcal{O}_{K}$ where the abelian variety $A$ over $K$ does not
have good reduction. There exists a finite Galois extension $L/K$, $L/K$
unramified at all primes $\mathfrak{p}$ of $\mathcal{O}_{K}$ with
$\mathfrak{p}\notin S_{A/K}$, such that the abelian variety $A_{L}/L$ has good
reduction everywhere.

###### Proof.

We first fix a prime $p_{1}$ such that the abelian variety $A/K$ has good
reduction at every prime ideal $\mathfrak{p}_{1}$ of $\mathcal{O}_{K}$ above
$p_{1}$. Let $L_{1}\coloneqq K(A[p_{2}])$ be the minimal field of definition
of the set of $p_{1}$-torsion points $A[p_{1}]$ of $A$. By section 6, we can
show that $L_{1}/K$ is a finite Galois extension unramified at any prime ideal
$\mathfrak{p}$ of $\mathcal{O}_{K}$ such that $\mathfrak{p}\notin S_{A/K}$ and
the characteristic of the residue field $\mathcal{O}_{K}/\mathfrak{p}$ is not
equal to $p_{1}$, and the abelian variety $A_{L_{1}}/L_{1}$ has everywhere
good reduction.

Next, we fix a prime $p_{2}$ not equal to $p_{1}$ such that the abelian
variety $A/K$ has good reduction at every prime ideal $\mathfrak{p}_{2}$ of
$\mathcal{O}_{K}$ above $p_{2}$. Let $L_{2}\coloneqq K(A[p_{2}])$ be the
minimal field of definition of the set of $p_{2}$-torsion points $A[p_{2}]$ of
$A$. Again by section 6, we can show that $L_{2}/K$ is a finite Galois
extension unramified at any prime ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$
such that $\mathfrak{p}\notin S_{A/K}$ and the characteristic of the residue
field $\mathcal{O}_{K}/\mathfrak{p}$ is not equal to $p_{2}$, and the abelian
variety $A_{L_{2}}/L_{2}$ has everywhere good reduction.

Now consider the extension $L_{1}\cap L_{2}$ of $K$. It is a finite Galois
extension unramified at any prime ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$
such that $\mathfrak{p}\notin S_{A/K}$ (since $p_{1}\neq p_{2}$). Since the
abelian varieties $A_{L_{1}}/L_{1}$ and $A_{L_{2}}/L_{2}$ both have everywhere
good reduction, by section 6, the abelian variety $A_{L_{1}\cap
L_{2}}/L_{1}\cap L_{2}$ also has everywhere good reduction. Taking
$L=L_{1}\cap L_{2}$, we get our claim. ∎

The following lemma shows that in terms of unramifiedness, the extension $L/K$
in section 6 is the “best possible”.

######  Lemma .

Let $K$ be a number field. Let $K^{\prime}/K$ be a finite extension. Let
$\mathfrak{p}^{\prime}$ be a prime ideal of $\mathcal{O}_{K^{\prime}}$, lying
above a prime ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$. Let $A$ be an abelian
variety over $K$. Suppose that the extension $K^{\prime}/K$ is unramified at
$\mathfrak{p}^{\prime}$, and the abelian variety $A_{K^{\prime}}/K^{\prime}$
has good reduction at $\mathfrak{p}^{\prime}$, then the abelian variety $A/K$
has good reduction at $\mathfrak{p}$.

###### Proof.

This follows from the Neron-Ogg-Shafarevich criterion. ∎

By Theorem 7 and the remarks before Theorem 7 of [ST68], we have the following
theorem:

######  Theorem .

Let $K$ be a number field. Let $E$ be a CM-field. Let $A$ be an abelian
variety over $K$ with complex multiplication by $E$. Let $\mu(E)$ be the group
of all roots of unity in $E$. There exists a cyclic extension $C$ of $K$ of
degree $[C:K]\leq 2\cdot\\#\mu(E)$, such that the abelian variety $A_{C}$ over
$C$ has everywhere good reduction.

######  Corollary .

Let $K$ be a number field. Let $E$ be a CM-field. Let $A$ be an abelian
variety over $K$ with complex multiplication by $E$. Let $\mu(E)$ be the group
of all roots of unity in $E$. Let $S_{A/K}$ be the set of all prime ideals of
$\mathcal{O}_{K}$ where the abelian variety $A$ over $K$ does not have good
reduction. There exists a cyclic extension $K^{\prime}$ of $K$ of degree
$[K^{\prime}:K]\leq 2\cdot\\#\mu(E)$, $K^{\prime}/K$ unramified at any prime
ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$ such that $\mathfrak{p}\notin
S_{A/K}$, such that the abelian variety $A_{K^{\prime}}$ over $K^{\prime}$ has
everywhere good reduction.

###### Proof.

Let $C/K$ be the finite cyclic extension in section 6 and let $L/K$ be the
finite Galois extension in section 6. Let $K^{\prime}=C\cap L$. Then
$K^{\prime}/K$ is a cyclic extension of degree $[K^{\prime}:K]\leq
2\cdot\\#\mu(E)$ and $K^{\prime}/K$ is unramified at any prime ideal
$\mathfrak{p}$ of $\mathcal{O}_{K}$ such that $\mathfrak{p}\notin S_{A/K}$. By
section 6, the abelian variety $A_{K^{\prime}}/K^{\prime}$ has everywhere good
reduction. Hence we get our claim. ∎

In order to prove section 1 and section 1, we will also need the following
theorem, which is a combination of Corollary A.4.6.5, Theorem A.4.5.1 and
Remark A.4.5.2 of [CCO14].

######  Theorem .

Let $E$ be a CM-field and let $\Phi$ be a CM-type of $E$. Let $E^{*}_{\Phi}$
be the reflex field of $(E,\Phi)$ and let $M$ be the field of moduli for the
reflex norm of $(E,\Phi)$ ($M$ is an everywhere unramified finite abelian
extension of $E^{*}_{\Phi}$).

There exists a prime $p$ and a CM abelian variety
$(A,i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{M}(A))$ over $M$ of CM-
type $\Phi$ such that $A$ has good reduction at every prime ideal of
$\mathcal{O}_{M}$ outside $p$.

Moreover, we can choose $p$ such that

$p\leq 2\cdot|\mathrm{disc}(E(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|^{C_{\mathrm{prime}}},$ (20)

where $C_{\mathrm{prime}}$ is an effectively computable absolute constant in
$\mathbb{R}_{>0}$, for any positive integer $n$ $\mu_{n}$ denotes a primitive
$n$-th root of unity, $m$ is the order of the group $\mu(E)$ of all roots of
unity in $E$, and $p_{1},p_{2},\cdots,p_{s}$ are the distinct prime divisors
of $m$.

Assuming the Generalized Riemann Hypothesis, the above bound on $p$ can be
improved to

$p\leq 70\cdot\biggl{(}\log|\mathrm{disc}(E(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|\biggr{)}^{2}.$ (21)

###### Proof of section 1.

Assume the Generalized Riemann Hypothesis.

Let $g$ be a positive integer. Let $E$ be a CM-field such that
$[E:\mathbb{Q}]=2g$. Let $\Phi$ be a CM-type of $E$. Let $E^{*}_{\Phi}$ be the
reflex field of $(E,\Phi)$ and let the field $M$, the abelian variety $A$ over
$M$ and the prime $p$ be as in section 6, such that the upper bound on $p$ is
given by Equation (21).

Denote $K\coloneqq M$. As in section 6, let $S_{A/K}$ be the set of all prime
ideals of $\mathcal{O}_{K}$ where the abelian variety $A$ over $K$ does not
have good reduction. By our choice of $A$, for any $\mathfrak{p}\in S_{A/K}$,
$\mathfrak{p}$ lies above the prime $p$. Let $K^{\prime}$ be the cyclic
extension of $K$ in section 6 of degree $[K^{\prime}:K]\leq 2\cdot\\#\mu(E)$,
$K^{\prime}/K$ unramified at any prime ideal $\mathfrak{p}$ of
$\mathcal{O}_{K}$ such that $\mathfrak{p}\notin S_{A/K}$, such that the
abelian variety $A_{K^{\prime}}$ over $K^{\prime}$ has everywhere good
reduction.

Therefore, the extension $K^{\prime}/K$ is ramified only at the prime ideals
$\mathfrak{q}$ of $\mathcal{O}_{K^{\prime}}$ such that $\mathfrak{q}$ lies
above the prime $p$. Let $\mathcal{D}_{K^{\prime}/K}$ be the different of the
extension $K^{\prime}/K$. By Chapter 3, Section 6 of [Ser79], we have

$e_{\mathfrak{q}/\mathfrak{p}}-1\leq\mathrm{val}_{\mathfrak{q}}(\mathcal{D}_{K^{\prime}/K})\leq
e_{\mathfrak{q}/\mathfrak{p}}-1+\mathrm{val}_{\mathfrak{q}}(e_{\mathfrak{q}/\mathfrak{p}}),$

for any prime ideal $\mathfrak{q}$ of $\mathcal{O}_{K^{\prime}}$ lying above a
prime ideal $\mathfrak{p}$ of $\mathcal{O}_{K}$, where
$e_{\mathfrak{q}/\mathfrak{p}}$ is the ramification index of
$\mathfrak{q}|\mathfrak{p}$.

This means that we have

$\mathrm{val}_{\mathfrak{q}}(\mathcal{D}_{K^{\prime}/K})\leq
2e_{\mathfrak{q}/p}e_{\mathfrak{q}/\mathfrak{p}}\leq
2e_{\mathfrak{q}/p}[K^{\prime}:K],$

where $e_{\mathfrak{q}/p}$ is the ramification index of the prime ideal
$\mathfrak{q}$ of $\mathcal{O}_{K^{\prime}}$ lying above the prime ideal $(p)$
of $\mathbb{Z}$.

Therefore, we have

$\mathcal{D}_{K^{\prime}/K}\Biggl{|}\prod_{\begin{subarray}{c}\mathfrak{q}\subset\mathcal{O}_{K^{\prime}}\\\
\mathfrak{q}|p\end{subarray}}\mathfrak{q}^{e_{\mathfrak{q}/p}\cdot
2\cdot\\#\mu(E)},$

where the product is over the prime ideals $\mathfrak{q}$ of
$\mathcal{O}_{K^{\prime}}$ above $p$.

Thus, we have

$\begin{split}\log\biggl{(}\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{K^{\prime}/K})\biggr{)}&\leq\log\Biggl{(}\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}\biggl{(}\prod_{\begin{subarray}{c}\mathfrak{q}\subset\mathcal{O}_{K^{\prime}}\\\
\mathfrak{q}|p\end{subarray}}\mathfrak{q}^{e_{\mathfrak{q}/p}\cdot
2\cdot\\#\mu(E)}\biggr{)}\Biggr{)}\\\ &\leq
2\cdot\\#\mu(E)\sum_{\begin{subarray}{c}\mathfrak{q}\subset\mathcal{O}_{K^{\prime}}\\\
\mathfrak{q}|p\end{subarray}}e_{\mathfrak{q}/p}\cdot\log\biggl{(}\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathfrak{q})\biggr{)}\\\
&=2\cdot\\#\mu(E)\sum_{\begin{subarray}{c}\mathfrak{q}\subset\mathcal{O}_{K^{\prime}}\\\
\mathfrak{q}|p\end{subarray}}e_{\mathfrak{q}/p}f_{\mathfrak{q}/p}\log(p)\\\
&=2\cdot\\#\mu(E)[K^{\prime}:\mathbb{Q}]\log(p),\end{split}$ (22)

where $f_{\mathfrak{q}/p}$ is the residue degree of the prime ideal
$\mathfrak{q}$ of $\mathcal{O}_{K^{\prime}}$ lying above the prime ideal $(p)$
of $\mathbb{Z}$.

Since the extension $K/E^{*}_{\Phi}$ is unramified, the different
$\mathcal{D}_{K/E^{*}_{\Phi}}$ of the extension $K/E^{*}_{\Phi}$ is equal to
the unit ideal of $\mathcal{O}_{K}$. Thus, we have

$\begin{split}&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log|\mathrm{disc}(K)|\\\
=&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log(\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{K^{\prime}/\mathbb{Q}}))\\\
=&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log(\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{K^{\prime}/K}\mathcal{D}_{K/E^{*}_{\Phi}}\mathcal{D}_{E^{*}_{\Phi}/\mathbb{Q}}))\\\
=&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log(\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{K^{\prime}/K}\mathcal{D}_{E^{*}_{\Phi}/\mathbb{Q}}))\\\
=&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log(\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{K^{\prime}/K}))+\frac{1}{[K^{\prime}:\mathbb{Q}]}\log(\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{E^{*}_{\Phi}/\mathbb{Q}}))\\\
=&\frac{1}{[K^{\prime}:\mathbb{Q}]}\log(\mathrm{Norm}_{K^{\prime}/\mathbb{Q}}(\mathcal{D}_{K^{\prime}/K}))+\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|\\\
\leq&2\cdot\\#\mu(E)\log(p)+\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|,\end{split}$
(23)

where the last inequality follows from Equation (22).

By our assumption on $p$, we have

$p\leq 70\cdot\biggl{(}\log|\mathrm{disc}(E(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|\biggr{)}^{2}.$

Thus, we have

$\begin{split}\log(p)&\leq\log(70)+2\log\log|\mathrm{disc}(E(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|\\\ &\leq\log(70)+2\log\log\biggl{(}((4g)^{4})^{(4g)^{4}\cdot
2g}(|\mathrm{disc}(E)|)^{(4g)^{4}}\biggr{)},\end{split}$ (24)

where the last inequality is by the following section 6.

By the following section 6, we have

$\begin{split}\mu(E)\leq(4g)^{2}.\end{split}$ (25)

Plugging Equation (24) and Equation (25) into Equation (23), we get our claim.
∎

######  Lemma .

Let $K$ be a number field of degree $[K:\mathbb{Q}]=n$. Let $\mu(K)$ be the
group of all roots of unity in $K$. Then $\mu(K)$ is a finite cyclic group of
order less than or equal to $(2n)^{2}$.

###### Proof.

Since $[K:\mathbb{Q}]$ is finite, $\mu(K)$ is a finite group. It is easy to
see that $\mu(K)$ is a cyclic group. Let $m$ be the order of the group
$\mu(K)$, and let $p_{1},p_{2},\cdots,p_{s}$ be the distinct prime divisors of
$m$. Denote as $\mu_{m}$ the primitive $m$-th root of unity. Then $K$ contains
the $m$-th cyclotomic field $\mathbb{Q}(\mu_{m})$. Since
$[\mathbb{Q}(\mu_{m}):\mathbb{Q}]=\\#(\mathbb{Z}/m\mathbb{Z})^{\times}=\frac{m(p_{1}-1)(p_{2}-1)\cdots(p_{s}-1)}{p_{1}p_{2}\cdots
p_{s}}$ and $[K:\mathbb{Q}]=n$, this means that

$\frac{m(p_{1}-1)(p_{2}-1)\cdots(p_{s}-1)}{p_{1}p_{2}\cdots p_{s}}\leq n.$

For any prime $p\neq 2$, we have $\sqrt{p}\leq p-1$. Thus, we have

$\begin{split}\frac{m(p_{1}-1)(p_{2}-1)\cdots(p_{s}-1)}{p_{1}p_{2}\cdots
p_{s}}&\geq\frac{m\sqrt{p_{1}}\sqrt{p_{2}}\cdots\sqrt{p_{s}}}{2p_{1}p_{2}\cdots
p_{s}}\\\ &=\frac{m}{2\sqrt{p_{1}}\sqrt{p_{2}}\cdots\sqrt{p_{s}}}\\\
&\geq\frac{m}{2\sqrt{m}}\\\ &=\sqrt{m}/2.\end{split}$

Therefore, we have

$m\leq(2n)^{2}.$

∎

######  Lemma .

Let $K$ be a number field of degree $[K:\mathbb{Q}]=n$. Then we have

$|\mathrm{disc}(K(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|\leq((2n)^{4})^{(2n)^{4}\cdot n}(|\mathrm{disc}(K)|)^{(2n)^{4}}.$

where for any positive integer $k$ $\mu_{k}$ denotes a primitive $k$-th root
of unity, $m$ is the order of the group $\mu(K)$ of all roots of unity in $K$,
and $p_{1},p_{2},\cdots,p_{s}$ are the distinct prime divisors of $m$.

###### Proof.

For any $k\in\mathbb{Z}_{\geq 3}$, the $k$-th cyclotomic field
$\mathbb{Q}(\mu_{k})$ has degree
$[\mathbb{Q}(\mu_{k}):\mathbb{Q}]=\\#(\mathbb{Z}/k\mathbb{Z})^{\times}$. By
[MP05], we have:

$\mathrm{disc}(\mathbb{Q}(\mu_{k}))=(-1)^{\varphi(k)/2}\frac{k^{\varphi(k)}}{\prod_{p|k}p^{\varphi(k)/(p-1)}},$

where $\varphi(k)=\\#(\mathbb{Z}/k\mathbb{Z})^{\times}$ is Euler’s totient
function, and the product in the denominator on the right-hand-side is over
primes $p$ dividing $k$.

Thus, we have

$|\mathrm{disc}(\mathbb{Q}(\mu_{k}))|\leq k^{k}.$

By section 6, we have

$m\leq(2n)^{2}.$

Thus, the $mp_{1}p_{2}\cdots p_{s}$-th cyclotomic field
$\mathbb{Q}(\mu_{mp_{1}p_{2}\cdots p_{s}})$ has degree

$\begin{split}[\mathbb{Q}(\mu_{mp_{1}p_{2}\cdots
p_{s}}):\mathbb{Q}]&=\\#(\mathbb{Z}/mp_{1}p_{2}\cdots
p_{s}\mathbb{Z})^{\times}\\\ &=m(p_{1}-1)(p_{2}-1)\cdots(p_{s}-1)\\\ &\leq
m^{2}\\\ &\leq(2n)^{4}.\end{split}$

Moreover, $mp_{1}p_{2}\cdots p_{s}\leq m^{2}\leq(2n)^{4}$. Thus, we have

$|\mathrm{disc}(\mathbb{Q}(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|\leq((2n)^{4})^{(2n)^{4}}.$

We know that $K(\mu_{mp_{1}p_{2}\cdots p_{s}})$ is equal to the compositum of
$K$ and $\mathbb{Q}(\mu_{mp_{1}p_{2}\cdots p_{s}})$. Thus, by Equation (16),
we have

$\begin{split}|\mathrm{disc}(K(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|&\leq|\mathrm{disc}(K)|^{[\mathbb{Q}(\mu_{mp_{1}p_{2}\cdots
p_{s}}):\mathbb{Q}]}|\mathrm{disc}(\mathbb{Q}(\mu_{mp_{1}p_{2}\cdots
p_{s}}))|^{[K:\mathbb{Q}]}\\\ &\leq((2n)^{4})^{(2n)^{4}\cdot
n}(|\mathrm{disc}(K)|)^{(2n)^{4}}.\end{split}$

∎

###### Proof of section 1.

The proof is similar to that of section 1 (using section 6 and section 6). The
difference is that since we do not assume the Generalized Riemann Hypothesis,
we use Equation (20) in section 6 to bound the prime $p$ instead of Equation
(21). The term
$\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|$ of
Equation (1) is submerged into the term $C_{15}(g)\log|\mathrm{disc}(E)|$ of
Equation (2) by the fact that the reflex field $E^{*}_{\Phi}$ is contained in
the Galois closure $\widetilde{E}$ of the extension of $E/\mathbb{Q}$ (and so
$\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|$ is less
than or equal to
$\frac{1}{[\widetilde{E}:\mathbb{Q}]}\log(|\mathrm{disc}(\widetilde{E})|)$)
and the fact that
$\frac{1}{[\widetilde{E}:\mathbb{Q}]}\log(|\mathrm{disc}(\widetilde{E})|)\leq\log|\mathrm{disc}(E)|$
by Equation (17). ∎

######  Remark .

In comparison with section 1, one might ask how the right-hand-side of the
formula in The (proved) averaged Colmez conjecture behaves under the
Generalized Riemann Hypothesis.

The right-hand-side of Equation (19) is equal to

$\begin{split}&-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\\\
=&\frac{1}{2}\frac{L^{\prime}(1,\chi_{E/F})}{L(1,\chi_{E/F})}+\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\\\
&+\frac{1}{2}\cdot\frac{g}{2}\Biggl{(}\frac{\Gamma^{\prime}\bigl{(}\frac{1}{2}\bigr{)}}{\Gamma\bigl{(}\frac{1}{2}\bigr{)}}+\frac{\Gamma^{\prime}(1)}{\Gamma(1)}-2\log(\pi)\Biggr{)},\end{split}$

where $g\coloneqq[F:\mathbb{Q}]$. The equality follows from logarithmically
differentiating the functional equation of $L(s,\chi_{E/F})$ at $s=0$.

Assume the Generalized Riemann Hypothesis. Then for any $g\in\mathbb{Z}_{\geq
1}$, there exist constants
$C_{\mathrm{GRH},1}(g)>0,C_{\mathrm{GRH},2}(g)\in\mathbb{R}$ depending only on
$g$ such that

$\biggl{|}\frac{L^{\prime}(1,\chi_{E/F})}{L(1,\chi_{E/F})}\biggr{|}\leq
C_{\mathrm{GRH},1}(g)\log\log|\mathrm{disc}E|+C_{\mathrm{GRH},2}(g)$

for any CM-field $E$ with maximal totally real subfield $F$ such that
$[F:\mathbb{Q}]=g$. Then it is easy to see that for any $g\in\mathbb{Z}_{\geq
1}$, for any $\epsilon>0$, there exists a constant $c(g,\epsilon)>0$ depending
only on $g$ and $\epsilon$ such that

$\begin{split}&\Biggl{|}\biggl{(}-\frac{1}{2}\frac{L^{\prime}(0,\chi_{E/F})}{L(0,\chi_{E/F})}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\biggr{)}-\frac{1}{4}\log(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)\Biggr{|}\\\
<&\epsilon\log|\mathrm{disc}(E)|,\end{split}$

for any CM-field $E$ with maximal totally real subfield $F$ such that
$[E:\mathbb{Q}]=2g$ and $|\mathrm{disc}(E)|\geq c(g,\epsilon)$.

Since
$|\mathrm{disc}(E)|/|\mathrm{disc}(F)|\leq|\mathrm{disc}(E)|\leq(|\mathrm{disc}(E)|/|\mathrm{disc}(F)|)^{2}$,
this means that assuming the Generalized Riemann Hypothesis, the right-hand-
side of Equation (19) is “approximately some constant times
$\log|\mathrm{disc}(E)|$”.

######  Remark .

One might wonder whether there is a lower bound for
$|\mathrm{disc}(E^{*}_{\Phi})|$ in terms of $|\mathrm{disc}(E)|$ and
$[E:\mathbb{Q}]$. The following example shows that the answer is no:

Let $F$ be any totally real number field. Let $-d\in\mathbb{Z}_{\leq-2}$ be
any fundamental discriminant (so $\mathrm{disc}(\mathbb{Q}(\sqrt{-d}))=d$)
such that $-d$ is prime to $\mathrm{disc}(F)$. (For any totally real number
field $F$, there are infinitely many such $-d$.) Let $E$ be the compositum of
the fields $F$ and $\mathbb{Q}(\sqrt{-d})$.

Then $E$ is a CM-field with maximal totally real subfield $F$. Let $\Phi$ be
the CM-type defined as follows:

For any $\varphi_{0}\in\mathrm{Hom}_{\mathbb{Q}}(F,\mathbb{R})$, the element
$\phi\colon E\to\mathbb{C}$ in $\Phi$ lying above $\varphi_{0}$ always sends
$\sqrt{-d}$ to $\sqrt{-d}$.

Then it is easy to see that $E^{*}_{\Phi}=\mathbb{Q}(\sqrt{-d})$. Thus,
$\mathrm{disc}(E^{*}_{\Phi})=\mathrm{disc}(\mathbb{Q}(\sqrt{-d}))=d$.

Since $\mathrm{disc}(\mathbb{Q}(\sqrt{-d}))=d$ is coprime to
$\mathrm{disc}(F)$, by Theorem 4.26 of [Nar90], for example, we have

$\begin{split}|\mathrm{disc}(E)|=d^{[F:\mathbb{Q}]}|\mathrm{disc}(F)|^{2}.\end{split}$

Therefore, for any fixed $g\in\mathbb{Z}_{\geq 2}$, the quotient

$\frac{\log|\mathrm{disc}(E^{*}_{\Phi})|}{\log|\mathrm{disc}(E)|},$

where $E$ is a CM-field of degree $[E:\mathbb{Q}]=2g$ and $\Phi$ is a CM-type
of $E$, can be arbitrarily small.

Combining section 6 with section 1, we have shown the following:

######  Proposition .

Assume the Generalized Riemann Hypothesis. For any $g\in\mathbb{Z}$ such that
$g\geq 2$, for any $\epsilon>0$, there exists a CM-field $E$ with
$[E:\mathbb{Q}]=2g$, a CM-type $\Phi$ of $E$, a number field $K^{\prime}$ and
a CM abelian variety
$(A,i\colon\mathcal{O}_{E}\hookrightarrow\mathrm{End}_{K^{\prime}}(A))$ over
$K^{\prime}$ of CM-type $\Phi$ such that the abelian variety $A$ over
$K^{\prime}$ has everywhere good reduction and

$\begin{split}\frac{1}{[K^{\prime}:\mathbb{Q}]}\log|\mathrm{disc}(K^{\prime})|\leq\epsilon\log|\mathrm{disc}(E)|.\end{split}$

######  Remark .

In view of section 6 and section 6, we cannot remove the “average” condition
in section 1 and section 1—Using only the The (proved) averaged Colmez
conjecture, we can only prove averaged analogues of section 1 and section 1.

######  Remark .

In Theorem 6(i) of [Col98], Colmez has proved that there exist effectively
computable absolute constants $C_{\mathrm{Col},3}>0$,
$C_{\mathrm{Col},4}\in\mathbb{R}$ such that for any CM-field $E$ of degree
$[E:\mathbb{Q}]=2g$ and any CM-type $\Phi$ of $E$ such that the following
hold:

  1. 1.

$(E,\Phi)$ satisfies the Colmez conjecture,

  2. 2.

For any irreducible Artin character $\chi$ such that $m_{(E,\Phi)}(\chi)\neq
0$, the Artin conjecture for $\chi$ holds (i.e. the Artin $L$-function
$L(s,\chi,\mathbb{Q})$ is holomorphic everywhere except possibly for a simple
pole at $s=1$),

we have

$\begin{split}h_{(E,\Phi)}^{\mathrm{Falt}}\geq
C_{\mathrm{Col},3}\cdot\mu_{(E,\Phi)}+gC_{\mathrm{Col},4}.\end{split}$

Let $E$ be a CM-field of degree $[E:\mathbb{Q}]=2g$ and let $\Phi$ be a CM-
type of $E$. It is easy to see that for the function ${A_{(E,\Phi)}^{0}}$ from
$\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$ to $\mathbb{C}$, for any
$\sigma\in\mathrm{Gal}(\widetilde{E^{*}_{\Phi}}/\mathbb{Q})$,
${A_{(E,\Phi)}^{0}}(\sigma)=g$ if and only if $\sigma=1$. Therefore, some
calculations using the definition of the Artin conductor of Artin characters
show that for any $g\in\mathbb{Z}_{\geq 1}$, there exist effectively
computable constants $C_{\mu,1}(g)>0$, $C_{\mu,2}(g)\in\mathbb{R}$ such that

$\mu_{(E,\Phi)}\geq
C_{\mu,1}(g)\frac{1}{[E^{*}_{\Phi}:\mathbb{Q}]}\log|\mathrm{disc}(E^{*}_{\Phi})|+C_{\mu,2}(g)$

for any CM-field $E$ of degree $[E:\mathbb{Q}]=2g$ and any CM-type $\Phi$ of
$E$. We can compare this to section 1.

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