# Semi-galois Categories IV:
A deformed reciprocity law for Siegel modular functions

Takeo Uramoto
Institute of Mathematics for Industry, Kyushu University

###### Abstract

This paper is a sequel to our previous work [19], where we proved the
_“modularity theorem” for algebraic Witt vectors_ over imaginary quadratic
fields. This theorem states that, in the case of imaginary quadratic fields
$K$, the algebraic Witt vectors over $K$ are precisely those generated by the
_modular vectors_ whose components are given by special values of deformation
family of Fricke modular functions; arithmetically, this theorem implies
certain congruences between special values of modular functions that are not
necessarily galois conjugate. In order to take a closer look at this
modularity theorem, the current paper extends it to the case of CM fields. The
main results include (i) a construction of algebraic Witt vectors from special
values of deformation family of Siegel modular functions on Siegel upper-half
space $\mathfrak{H}_{g}$ given by ratios of theta functions, and (ii) a
galois-theoretic characterization of which algebraic Witt vectors arise in
this modular way, intending to exemplify a general galois-correspondence
result which is also proved in this paper.

##  1 Introduction

The goal of this paper is to extend our previous work on the _modularity
theorem for algebraic Witt vectors_ over imaginary quadratic fields (Theorem
4.3.1, [19]) to the case of CM fields in particular. This theorem states that,
if $K$ is an imaginary quadratic field, the $K$-algebra $E_{K}=K\otimes
W_{O_{K}}^{a}(O_{\bar{K}})$ of _algebraic Witt vectors_ over $K$ is identical
to the $K$-algebra $M_{K}=K[\widehat{f}_{a}\mid
a\in\mathbb{Q}^{2}/\mathbb{Z}^{2}]$ generated by the _modular vectors_
$\widehat{f}_{a}$, whose components are given by special values of Fricke’s
modular functions $f_{a}:\mathfrak{H}\rightarrow\mathbb{C}$
($a\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$), cf. §2. This theorem connects two
objects (algebraic/analytic) of different nature; and technically this theorem
is proved using _Shimura’s reciprocily law_ , which relates special values of
_galois-conjugate_ modular functions (§6.8 [12]). But compared to Shimura’s
reciprocity law, our modularity theorem is characterized by the fact that it
relates, by congruences, special values of modular functions living at
different levels that are _not necessarily galois conjugate_ ; in this sense,
our modularity theorem strengthens/deforms Shimura’s reciprocity law in a way
analogous to _Hida theory_ , e.g. mod-$p$ congruences between Fourier
coefficients of non-galois-conjugate cusp forms [8]. In view of that Shimura’s
reciprocity law is now known for more general Shimura varieties [9, 10], our
modularity theorem would eventually be a facet of some more general theme. To
pursue this prospect, the current paper provides further results of that
general direction, working on the case of Siegel modular functions.

Our proof of the modularity theorem $E_{K}=M_{K}$ consisted of three major
steps; the extension to CM field in this paper will basically follow this
line. The first step is to see that we can identify the algebraic Witt vectors
$\xi\in E_{K}$ with locally constant, $K^{ab}$-valued, $G_{K}$-equivariant
functions on the _profinite Deligne-Ribet monoid_ $DR_{K}$ (cf. Corollary
3.1.6 [19]), formally, $E_{K}\simeq\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$; this
identification is a consequence of the fact that the algebraic Witt vectors
$\xi\in E_{K}$ generate all _galois objects_ [17] of the semi-galois category
$(\mathscr{C}_{K},\mathrm{F}_{K})$ of certain $\Lambda$-rings whose
fundamental monoid $\pi_{1}(\mathscr{C}_{K},\mathrm{F}_{K})$ was known
isomorphic to $DR_{K}$ [3]; this identification itself holds for arbitrary
number fields $K$ and thus is used in the current paper too. The second step
is to construct algebraic Witt vectors using special values of the family
$\\{f_{a}\\}$ of Fricke’s modular functions
$f_{a}:\mathfrak{H}\rightarrow\mathbb{C}$ indexed by
$a\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$; thanks to the first step, the second step
is reduced to constructing locally constant, $K^{ab}$-valued,
$G_{K}$-equivariant functions on $DR_{K}$ from these $f_{a}$’s, for which the
adelic description of the Deligne-Ribet monoid $DR_{K}$ by Yalkinoglu [21],
$DR_{K}\simeq\widehat{O}_{K}\times_{\widehat{O}_{K}^{\times}}(\mathbb{A}_{K,f}^{\times}/K^{\times})$,
plays an auxiliary role. The third step is to prove that these algebraic Witt
vectors constructed from $f_{a}$’s (namely the modular vectors
$\widehat{f}_{a}\in E_{K}$) actually generate $E_{K}$ over $K$; this last part
is our key step, in which we utilized the _galois-orbit decompositions_ of
finite Dligne-Ribet monoids $DR_{\mathfrak{f}}$ for conductors
$\mathfrak{f}\in I_{K}$ given as
$DR_{\mathfrak{f}}=\coprod_{\mathfrak{d}\mid\mathfrak{f}}C_{\mathfrak{f}/\mathfrak{d}}$
[21] in order to separate components of modular vectors (= special values of
Fricke functions $f_{a}$’s that may live at different levels), reducing it to
Boolean-valued problems of judging prime ramifications (cf. the proof of Lemma
4.3.2 [19] in particular).

As mentioned above too, the current work also follows basically this line; but
in the case of CM fields, major differences occur in the second and third
steps. First, we shall replace Fricke modular functions
$f_{a}:\mathfrak{H}\rightarrow\mathbb{C}$ with certain _ratios
$\theta^{k/l}_{a}$ of theta functions_, which naturally define deformation
families of modular functions on the Siegel upper-half space
$\mathfrak{H}_{g}$; nevertheless, the basic idea behind these two families of
modular functions is essentially the same, in that both families of functions
(Fricke functions and ratios of theta functions) are those equally appearing
in the construction of projective embedding of complex tori (resp. of
dimension one and higher dimension) [12, 11], which will be a central step in
our construction of modular vectors; essentially, their values (= components
of modular vectors) both represent the coordinates of torsion points on
elliptic curves and abelian varieties respectively. Our major subject in the
second step is to give the right setting of necessary constructions and a
proof that the ratios of theta functions naturally produce locally constant
$K^{ab}$-valued $G_{K}$-equivariant functions on $DR_{K}$, hence, algebraic
Witt vectors (i.e.  _modular vectors_) over CM fields $K$ (cf. §3).

Second, unlike the case of imaginary quadratic fields, these modular vectors
do not generate all algebraic Witt vectors over CM fields; thus it should be
of natural concern to characterize which algebraic Witt vectors are generated
by our modular vectors. The subject of §4 is to discuss this problem, where we
show that an algebraic Witt vector $\xi\in E_{K}$ is generated by the modular
vectors if and only if it is periodic with respect to a certain monoid
congruence $\sim_{N}$ of finite index on $DR_{K}$ for some $N\geq 1$, where
the congruences $\sim_{N}$ are given purely in an adelic term. Technically
speaking, this result is a major result of the current paper and intended to
exemplify a general _galois-theoretic correspondence_ (mentioned in Remark
3.1.8 [19] without proof) between (i) $\Lambda$-subalgebras of $E_{K}$, (ii)
profinite quotients of $DR_{K}$, and (iii) semi-galois full subcategories of
$(\mathscr{C}_{K},\mathrm{F}_{K})$; a proof of this general galois
correspondence is also provided in §4.

It would be possible to extend the results of the current paper to more
general situations, say, along the line of general Shimura varieties [9]; in
this relation, compare our results to Yalkinoglu’s [20] and Ha and Paugam’s
work [5]; in terms of general Shimura varieties the current work concerns the
case of Siegel modular varieties (= Shimura varieties for general symplectic
group) in particular. In this paper, however, we do not pursue such full
generality yet; instead we shall focus on the most basic case where the
common/different points of CM fields to the case of imaginary quadratic fields
[19] become transparent.

##### Related works

The current work is motivated by Yalkinoglu’s work [20] (2012) in particular.
In that paper, Yalkinoglu constructed an arithmetic subalgebra of the Bost-
Connes system for number fields containing CM fields using Siegel modular
functions on Siegel modular varieties (§7 [20]); and if we restrict his
arithmetic subalgebra (consisting of certain functions on a _BCM groupoid_ ;
cf. §6 [20]) to the subspace of its objects (which is isomorphic to $DR_{K}$),
we can in fact obtain a subalgebra of $E_{K}$ under the identification
$E_{K}=\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$; in other words, up to this
identification, his construction of arithmetic subalgebra can be also
understood as a construction of algebraic Witt vectors over CM fields from
special values of Siegel modular functions; this way, Yalkinoglu’s work [20]
is relevant to our major goal in the current paper.

Comparing our construction to Yalkinoglu’s [20], however, highlights the above
mentioned fact that our result can be regarded as a deformation of Shimura’s
reciprocity law. On the one hand, the algebraic Witt vectors arising from
Yalkinoglu’s construction, viewed as functions on $DR_{K}$, have their support
only on the unit group $DR_{K}^{\times}=G_{K}^{ab}\subset DR_{K}$ (§7 [20]),
while those of our construction have their support on the whole $DR_{K}$ (cf.
§3–§4). Technically speaking, it is due to this aspect that our algebraic Witt
vectors can relate special values of Siegel modular functions living at
different levels that are not necessarily galois conjugate, unlike Shimura’s
reciprocity law only relates special values of galois-conjugate functions.

To be more specific, an algebraic Witt vector $\xi\in E_{K}$, viewed as a
function on $DR_{K}$, is locally constant and thus factors through a finite
quotient $DR_{K}\twoheadrightarrow DR_{\mathfrak{f}}$ for some conductor
$\mathfrak{f}$; this finite monoid $DR_{\mathfrak{f}}$ has a natural action of
its unit group $DR_{\mathfrak{f}}^{\times}$ isomorphic to the ray class group
$C_{\mathfrak{f}}$, and then, has a $C_{\mathfrak{f}}$-orbit decomposition of
the form
$DR_{\mathfrak{f}}=\coprod_{\mathfrak{d}\mid\mathfrak{f}}C_{\mathfrak{f}/\mathfrak{d}}$;
the $G_{K}$-equivariance of $\xi$ together with $K^{ab}$-valuedness implies
that $\xi$ is a function
$\coprod_{\mathfrak{d}}C_{\mathfrak{f}/\mathfrak{d}}\rightarrow K^{ab}$ such
that, when restricted on each component
$\xi|_{C_{\mathfrak{f}/\mathfrak{d}}}:C_{\mathfrak{f}/\mathfrak{d}}\rightarrow
K^{ab}$, its value $\xi(\sigma)$ at $\sigma\in C_{\mathfrak{f}/\mathfrak{d}}$
is equal to $\xi(1)^{\sigma}$ and thus determined only by its value at the
unit $1\in C_{\mathfrak{f}/\mathfrak{d}}$. This way, the values of $\xi$
within a single component $C_{\mathfrak{f}/\mathfrak{d}}$ are correlated by
galois conjugate; but $\xi$ being an algebraic Witt vector, the values of
$\xi$ on the whole
$DR_{\mathfrak{f}}=\coprod_{\mathfrak{d}}C_{\mathfrak{f}/\mathfrak{d}}$ (i.e.
those on different components too) must be correlated so that the distribution
of the values of $\xi$ on the whole
$DR_{\mathfrak{f}}=\coprod_{\mathfrak{d}}C_{\mathfrak{f}/\mathfrak{d}}$ are
_arithmetically smooth_ (i.e. infinitely differentiable with respect to
arithmetic derivatives) up to fixed denominator (i.e. $\xi$ multiplied by some
integer gets arithmetically smooth). In this sense, an algebraic Witt vector
$\xi$ is a data consisting, generally, not only of galois-conjugate values but
connecting (or smoothly _interpolating_) non-galois conjugate values too,
where “smooth” is in the sense of arithmetic derivative. Phrased in these
specific terms, the algebraic Witt vectors arising from Yalkinoglu’s
construction consist only of galois-conjugate values (= special values of
galois conjugate Siegel modular functions) by his definition, and thus, in the
above general sense, do not give interpolation of special values of non-
galois-conjugate modular functions; on the other hand, our construction
induces such interpolations in a natural way.111That said, it should be also
recognized that Yalkinoglu’s work has his own goal that belongs to a natural
research context on Bost-Connes systems, and for this goal, his construction
was sufficient; our above argument is just to see that, reviewing Yalkinoglu’s
work in the above specific way, our research subject (interpolations of
special values of non-galois-conjugate Siegel modular functions) emerges
naturally; solidifying this subject itself, as a sequel to our previous work
[19], is also a major theme of the current work.

Technically speaking, it seems characteristic that, in our construction of
algebraic Witt vectors (i.e. modular vectors) by special values of Siegel
modular functions (living at different levels), moduli interpretations of
Siegel modular varieties (or Siegel upper-half space $\mathfrak{H}_{g}$) play
an intrinsic key role, which is also in a sharp contrast to Yalkinoglu’s work.
In fact, as mentioned above, our modular vectors over CM fields (including
imaginary quadratic fields) are constructed by choosing specific generators of
the field $\mathfrak{F}$ of modular functions, namely, Fricke functions
$f_{a}$ ($a\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$) and ratios $\theta^{k/l}_{a}$
($\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$) of theta functions (cf. §3); and these
specific modular functions have natural moduli interpretation that they
represent coordinates of torsion points of (projectively embedded) elliptic
curves and abelian varieties. Based on this choice of generator, we can deform
modular functions by deforming the indices
$a\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$ (cf. §3) of the generators (i.e. $f_{a}$
and $\theta^{k/l}_{a}$), which correspond to descending the levels of torsion
points of abelian varieties; our modular vectors are constructed by special
values of thus deformed modular functions. In a sense, the reason why the
values of our modular vectors
$\xi:\coprod_{\mathfrak{d}\mid\mathfrak{d}}C_{\mathfrak{f}/\mathfrak{d}}\rightarrow
K^{ab}$ can be correlated even among different components is that their values
are controlled uniformly by this well-behaved deformation family
$\theta^{k/l}_{a}$; in fact we can even characterize what algebraic Witt
vectors are generated by our modular vectors, by geometry of abelian
varieties, thanks to their geometric origin (cf. §4).

In the case of imaginary quadratic fields, the current work also overlaps with
our own previous work [19]; in particular, we now have two classes of modular
vectors, i.e. those by Fricke functions $f_{a}$ and those by the ratios
$\theta^{k/l}_{a}$ of theta functions. Although the latter cannot generate all
algebraic Witt vectors over CM field of higher degree in general, we can see
by our result in §4 that, at least in the case of imaginary quadratic fields,
the latter too can generate all algebraic Witt vectors, as in the former case
[19]; so, in view of the fact that the latter makes sense for any CM fields,
modular vectors by the ratios of theta functions have a theoretical
advantage/canonicity; in fact, our choice of ratios of theta functions for
general CM case is a natural consequence of Mumford’s general theory of
projective embeddings of abelian varieties [11] and Shimura’s theory of
complex multiplication [13, 14]. Also as Iwao Kimura has told us in the
context of our work [6] with Yasuhiro Ishitsuka on the decidability of
integral Witt vectors over imaginary quadratic fields, numerical computations
of special values of Fricke functions can be made fast by representing them by
theta series [7]. In view of this, the modular vectors by ratios of theta
functions may also have a computational advantage even for the case of
imaginary quadratic fields only. Also, in the case of CM fields, Streng [15]
has developed some computational results on Shimura’s reciprocity law for
Siegel modular functions, which is also relevant to the current work (§3–§4);
in particular, Streng’s result may be helpful for concrete numerical
computations on our general theory of modular vectors developed below.

##### Acknowledgements

This paper is an extended version of our talk at _Low Dimensional Topology and
Number Theory XIV_ on 27–30, March 2023. We are grateful to the organizer of
the workshop for inviting us there; we are also grateful to Yasuhiro Ishitsuka
and Iwao Kimura for several discussions on [6]. This work was supported by
JSPS KAKENHI Grant-No.22K03248.

##  2 Preliminaries

Throughout this paper, $K$ denotes a number field, and $O_{K}$ the ring of
integers in $K$; we denote by $P_{K}$ the set of non-zero prime ideals of
$O_{K}$, while $I_{K}$ denotes the monoid of non-zero integral ideals of
$O_{K}$; $N\mathfrak{a}$ denotes the absolute norm of $\mathfrak{a}\in I_{K}$.
Let $\mathbb{A}_{K}$ be the adele ring of $K$, $\mathbb{A}_{K,f}$ the finite
adeles, and $\widehat{O}_{K}$ the finite integral adeles; the Artin map is
denoted $[-]:\mathbb{A}_{K}^{\times}\rightarrow G_{K}^{ab}$, where
$G_{K}^{ab}$ is the galois group of the maximal abelian extension $K^{ab}$
over $K$; for a ring $R$ with unit, $R^{\times}$ denotes the invertible
elements in $R$. Note that this notation is different from that in [19], where
we denoted $R^{*}$ to mean the invertible elements of $R$; in this paper, the
superscript $*$ is reserved for the _reflex fields_ $E^{*}$ for CM fields with
CM types $(E,\Phi)$ (cf. §3.2).

In this paper the reader is assumed to be familiar with the terminologies,
notations, and main results in our previous works [16, 17, 18, 19]. In
particular, $W_{O_{K}}^{a}(O_{\bar{K}})$ denotes the $\Lambda$-ring of
integral Witt vectors over $O_{K}$, while $E_{K}=K\otimes
W_{O_{K}}^{a}(O_{\bar{K}})$ denotes the $\Lambda$-ring of algebraic Witt
vectors over $K$. Also, $(\mathscr{C}_{K},\mathrm{F}_{K})$ denotes the semi-
galois category of finite etale $\Lambda$-rings over $K$ having integral
models, whose fundamental monoid $\pi_{1}(\mathscr{C}_{K},\mathrm{F}_{K})$ is
isomorphic to the Deligne-Ribet monoid $DR_{K}$ [3].

###  2.1 Modular vectors

This subsection is devoted to an overview of our previous construction of
modular vectors (§4 [19]), rephrasing it in terms of elliptic curves and their
Kummer varieties in particular. This rephrase is to highlight the geometric
ingredient of modular vectors, and to compare them to the case of CM fields
given in §3, where elliptic curves (i.e. abelian varieties of dimension one)
are replaced with abelian varieties of higher dimension (cf. §2.2).

####  2.1.1 Witt vectors as functions on $DR_{K}$

As outlined in §1, the construction of modular vectors is based on the fact
that we can identify the algebraic Witt vectors $\xi\in E_{K}$ with locally
constant $K^{ab}$-valued $G_{K}$-equivariant functions on the profinite
Deligne-Ribet monoid $DR_{K}$, which holds for arbitrary number fields $K$ (§3
[19]). Since we need this identification in this paper too, let us start with
recalling some results on it:

######  Definition 2.1.1 (Deligne-Ribet monoid $DR_{K}$).

The _(profinite) Deligne-Ribet monoid_ $DR_{K}$ is defined as the inverse
limit of finite quotient monoids
$I_{K}/\sim_{\mathfrak{f}}=:DR_{\mathfrak{f}}$ given for each $\mathfrak{f}\in
I_{K}$, where $\sim_{\mathfrak{f}}$ is the monoid congruence of finite index
on the monoid $I_{K}$ defined by, for $\mathfrak{a},\mathfrak{b}\in I_{K}$,

$\displaystyle\mathfrak{a}\sim_{\mathfrak{f}}\mathfrak{b}$
$\displaystyle\Leftrightarrow$ $\displaystyle\exists t\in
K_{+}\cap(1+\mathfrak{f}\mathfrak{b}^{-1}),\hskip
5.69046pt\mathfrak{a}\mathfrak{b}^{-1}=(t),$ (2.1)

where $K_{+}$ denotes totally positive elements in $K$. Note that, for
$\mathfrak{f}\mid\mathfrak{f}^{\prime}$, we have a natural surjective monoid
homomorphism $DR_{\mathfrak{f}^{\prime}}\twoheadrightarrow DR_{\mathfrak{f}}$,
with which $DR_{\mathfrak{f}}$’s constitute an inverse system of finite
monoids; with this inverse system $\\{DR_{\mathfrak{f}}\\}_{\mathfrak{f}\in
I_{K}}$ of _finite Deligne-Ribet monoids $DR_{\mathfrak{f}}$ for conductors
$\mathfrak{f}\in I_{K}$_, the profinite Deligne-Ribet monoid $DR_{K}$ is given
by the inverse limit:

$\displaystyle DR_{K}$ $\displaystyle:=$
$\displaystyle\lim_{\leftarrow\mathfrak{f}\in I_{K}}DR_{\mathfrak{f}}$ (2.2)

By definition we have a canonical monoid homomorphism $I_{K}\rightarrow
DR_{K}$, which is in fact injective; also, it is known [4] that the unit group
$DR_{\mathfrak{f}}$ of each $DR_{\mathfrak{f}}$ is isomorphic to the strict
ray class group $C_{\mathfrak{f}}$ for the conductor $\mathfrak{f}$, namely
$DR_{\mathfrak{f}}^{\times}\simeq C_{\mathfrak{f}}$; and hence, that
$DR_{K}^{\times}\simeq\lim_{\mathfrak{f}}C_{\mathfrak{f}}\simeq G_{K}^{ab}$,
where the last isomorphism is the one given by class field theory. Therefore,
the absolute galois group $G_{K}$ naturally acts on $DR_{K}$ (say, from the
right) via the maximal abelian quotient $G_{K}\twoheadrightarrow
G_{K}^{ab}\simeq DR_{K}^{\times}\subseteq DR_{K}$, with which $DR_{K}$ forms a
profinite right $G_{K}$-set. Thanks to this fact, it makes sense to consider
locally constant $K^{ab}$-valued _$G_{K}$ -equivariant_ functions on $DR_{K}$,
the set of which we shall denote by $\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$.

This set $\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$ has a natural $K$-algebra
structure with respect to the point-wise operations, as well as the following
operation
$\psi_{\mathfrak{p}}:\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})\rightarrow\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$
for each $\mathfrak{p}\in P_{K}$, which is given by multiplication-
by-$\mathfrak{p}$, regarding each $\mathfrak{p}\in P_{K}$ as an element in
$DR_{K}$ via the above-mentioned injection $I_{K}\hookrightarrow DR_{K}$:

$\displaystyle\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$
$\displaystyle\xrightarrow{\psi_{\mathfrak{p}}}$
$\displaystyle\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$ (2.3) $\displaystyle\xi(*)$
$\displaystyle\longmapsto$ $\displaystyle\xi(\mathfrak{p}\cdot*)$ (2.4)

With these operators $\psi_{\mathfrak{p}}$, the $K$-algebra
$\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$ forms a $\Lambda$-ring over $K$, which
in fact is isomorphic to the $\Lambda$-ring $E_{K}$ of algebraic Witt vectors
in a canonical way:

######  Theorem 2.1.2 (Corollary 3.1.5, [19]).

We have a canonical isomorphism of $\Lambda$-rings over $K$:

$\displaystyle E_{K}$ $\displaystyle\simeq$
$\displaystyle\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab}).$ (2.5)

Technically, the proof of this theorem is based on the two facts (cf. §3.2
[19]) that (i) each finite set $\Xi\subseteq E_{K}$ of algebraic Witt vectors
generates a _galois object_ $X_{\Xi}$ of the semi-galois category
$\mathscr{C}_{K}$, where
$X_{\Xi}:=K[\psi_{\mathfrak{a}}\xi\mid\xi\in\Xi,\mathfrak{a}\in I_{K}]$ is the
$\Lambda$-ring generated over $K$ by the orbit of $\Xi$ under the action of
$\psi_{\mathfrak{p}}$’s; and (ii) every galois object of $\mathscr{C}_{K}$ is
isomorphic to $X_{\Xi}$ for some finite $\Xi\subseteq E_{K}$. (See loc. cit.
for more detail.)

In particular, the isomorphism from $\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$ to
$E_{K}$ is defined by restricting locally constant $K^{ab}$-valued
$G_{K}$-equivariant functions $\xi:DR_{K}\rightarrow K^{ab}$ onto the
submonoid $I_{K}\hookrightarrow DR_{K}$, inducing functions
$\xi:I_{K}\rightarrow K^{ab}$, or in other words, vectors
$\xi\in(K^{ab})^{I_{K}}$; the above theorem says that the vector
$\xi\in(K^{ab})^{I_{K}}$ induced from
$\xi\in\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$ always defines an algebraic Witt
vector, namely $\xi\in E_{K}\subset(K^{ab})^{I_{K}}$ more than
$\xi\in(K^{ab})^{I_{K}}$; and conversely, every algebraic Witt vector $\xi\in
E_{K}$ arises uniquely in this way. In the sense of this correspondence, we
can now identify algebraic Witt vectors $\xi\in E_{K}$ with locally constant
$K^{ab}$-valued $G_{K}$-equivariant functions on the profinite monoid
$DR_{K}$; in particular, to construct algebraic Witt vectors $\xi\in E_{K}$ is
equivalent to constructing such functions
$\xi\in\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$.

####  2.1.2 Modular vectors

When $K$ is an imaginary quadratic field, our previous work [19] constructed
algebraic Witt vectors $\xi\in E_{K}$ from special values of the family
$\\{f_{a}\\}$ of Fricke functions $f_{a}:\mathfrak{H}\rightarrow\mathbb{C}$,
($a\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$), which are called _modular vectors_. In
the rest of this subsection, we shall review this construction in terms of
elliptic curves and their Kummer varieties, which is suitable for extension to
abelian varieties of higher dimension (§3).

To this end, an adelic description of $DR_{K}$ essentially by Yalkinoglu [21]
plays an auxiliary role:

######  Proposition 2.1.3 (Proposition 8.2, Yalkinoglu [21]).

We have a natural isomorphism of profinite monoids:

$\displaystyle DR_{K}$ $\displaystyle\simeq$
$\displaystyle\widehat{O}_{K}\times_{\widehat{O}_{K}^{\times}}\bigl{(}\mathbb{A}_{K}^{\times}/\overline{K^{\times}\cdot
K_{\infty}^{\times,\circ}}\bigr{)}$ (2.6)

Here $K^{\times,\circ}_{\infty}$ denotes the connected component of the
archimedian part $K_{\infty}^{\times}$ of the idele group
$\mathbb{A}^{\times}_{K}$ containing the unit $1$; the overline on
$K^{\times}\cdot K^{\times,\circ}_{\infty}$ denotes the closure in
$\mathbb{A}_{K}^{\times}$. Also the multiplicative group
$\widehat{O}_{K}^{\times}$ acts on the product monoid
$\widehat{O}_{K}\times(\mathbb{A}_{K}^{\times}/\overline{K^{\times}\cdot
K^{\times,\circ}_{\infty}})$ by $u\cdot(\rho,s):=(\rho u,u^{-1}s)$, for which
we take the quotient in (2.6). In particular, when $K$ is imaginary quadratic,
we have the following simpler form:

$\displaystyle DR_{K}$ $\displaystyle\simeq$
$\displaystyle\widehat{O}_{K}\times_{\widehat{O}_{K}^{\times}}\bigl{(}\mathbb{A}_{K,f}^{\times}/K^{\times}\bigr{)}$
(2.7)

Generally, for CM fields $K$, we can represent elements of $DR_{K}$ as
$[\rho,s]$ with $\rho\in\widehat{O}_{K}$ and $s\in\mathbb{A}_{K,f}^{\times}$.

The subject here is to review geometrically our previous construction in [19]
of locally constant $K^{ab}$-valued $G_{K}$-equivariant functions on $DR_{K}$
from _Fricke modular functions_ $f_{a}:\mathfrak{H}\rightarrow\mathbb{C}$
indexed by $a\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$. Technically the Fricke
functions $f_{a}:\mathfrak{H}\rightarrow\mathbb{C}$ can be described
explicitly but in a case-by-case manner according as
$K=\mathbb{Q}(i),\mathbb{Q}(\sqrt{-3})$ or others (cf. §6.1 [12]); this is
related to the fact that the automorphism groups of elliptic curves with
complex multiplication in $K$ differ for the respective cases (cf. §4.5 [12]).
It is for this reason that our study of modular vectors was also done in a
case-by-case manner (cf. e.g. §4.3 [19]).

Geometrically, we have a more uniform description of modular vectors via
_Kummer varieties_ , which is equal to the original one. The _Kummer variety_
$V_{E}$ of an elliptic curve $E$ is the quotient of $E$ by its automorphism
group, $V_{E}=E/\mathrm{Aut}(E)$, which is again an algebraic curve; and the
quotient map $h:E\twoheadrightarrow V_{E}$ is called the _Weber function_ ;
the Kummer varieties $V_{E}$ are in fact isomorphic to the projective space
${\bf P}^{1}(\mathbb{C})$ in such a way that we have a natural geometric
interpretation of the Fricke functions $f_{a}$: That is, for each non-zero
$a=(a_{1},a_{2})\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$ and $\tau\in\mathfrak{H}$,
the special value $f_{a}(\tau)$ is (the $x$-coordinate of) the image of the
torsion point $a_{1}\tau+a_{2}\in\mathbb{C}/(\mathbb{Z}\tau+\mathbb{Z})$ under
the map $\mathbb{C}/(\mathbb{Z}\tau+\mathbb{Z})\rightarrow{\bf
P}^{1}(\mathbb{C})$ given by the following composition:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
27.40797pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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(2.8)

also, for $a=0\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$, we put $f_{0}(\tau)=j(\tau)$.

With this geometric interpretation of the Fricke functions $f_{a}$, we now
construct locally constant $K^{ab}$-valued $G_{K}$-equivariant functions
$\widehat{f}_{a}$ on $DR_{K}$, which provide our target modular vectors:

######  Definition 2.1.4 (modular vectors $\widehat{f}_{a}$).

For each $a=(a_{1},a_{2})\in\mathbb{Q}^{2}/\mathbb{Z}^{2}$, define
$\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ by the following rule: Given
each
$[\rho,s]\in\widehat{O}_{K}\times_{\widehat{O}_{K}^{\times}}(\mathbb{A}_{K,f}^{\times}/K^{\times})\simeq
DR_{K}$, the value $\widehat{f}_{a}(\rho,s)$ is the image of the torsion point
$a_{1}\tau_{K}+a_{2}\in K/O_{K}$ under the map $K/O_{K}\rightarrow{\bf
P}^{1}(\mathbb{C})$ given by the composition:

$\textstyle{K/O_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\simeq}$$\textstyle{K/O_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{\simeq}$$\textstyle{K/sO_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\subseteq}$$\textstyle{\mathbb{C}/sO_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{E_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\bf
P}^{1}(\mathbb{C})=\mathbb{C}\cup\\{\infty\\}}$$\textstyle{\mathbb{A}_{K,f}/\widehat{O}_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{\mathbb{A}_{K,f}/\widehat{O}_{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathbb{A}_{K,f}/s\widehat{O}_{K}}$

where $O_{K}=\mathbb{Z}\tau_{K}+\mathbb{Z}$ with $\tau_{K}\in\mathfrak{H}$ and
$sO_{K}:=s\widehat{O}_{K}\cap K$ for each $s\in\mathbb{A}^{\times}_{K,f}$; and
the isomorphism $\mathbb{C}/sO_{K}\simeq E_{s}$ is given by the Weierstrass
$\wp$-function $\wp(u,\Lambda_{s})$ for the lattice
$\Lambda_{s}=sO_{K}\subseteq\mathbb{C}$; also, if the value
$\widehat{f}_{a}(\rho,s)$ goes to $\infty$ by this map, we renormalize it as
$j(\Lambda_{s})$.

######  Proposition 2.1.5.

These functions $\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ are equal to
those in [19], hence define locally constant $K^{ab}$-valued
$G_{K}$-equivariant functions on $DR_{K}$, or equivalently, modular vectors
$\widehat{f}_{a}\in E_{K}$.

###### Proof.

The only slight difference from the original construction is that, in the
above construction, we did not re-scale the lattice $sO_{K}$ into the form
$\mathbb{Z}\tau+\mathbb{Z}$ with $\tau\in\mathfrak{H}$; therefore, the lattice
$sO_{K}$ is of the form $sO_{K}=\mathbb{Z}w_{1}+\mathbb{Z}w_{2}$ with
$\tau_{s}=w_{1}/w_{2}\in\mathfrak{H}$ in general. But this difference in
scaling does not affect the resulting values because the complex torus
$\mathbb{C}/sO_{K}$ is isomorphic to the rescaled one
$\mathbb{C}/(\mathbb{Z}\tau_{s}+\mathbb{Z})$; and the Weber functions on
elliptic curves are compatible with their isomorphisms (cf. §4.5 [12]). This
also means that the above construction is independent of the choice of
$s\in\mathbb{A}^{\times}_{K,f}$ that represents the class
$[s]\in\mathbb{A}^{\times}_{K,f}/K^{\times}$, hence
$\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ is well-defined. ∎

###  2.2 The case of CM fields

Now we prepare some basic results necessary to extend the above construction
of modular vectors to the case of CM fields. In this process, as outlined in
§1 too, we replace elliptic curves with abelian varieties of higher dimension;
accordingly, we need to recall projective embedding of complex tori of higher
dimension by theta functions [11], which replaces projective embedding of one-
dimensional complex tori $\mathbb{C}/(\mathbb{Z}w_{1}+\mathbb{Z}w_{2})$ by the
Weierstrass $\wp$-function $\wp$ (and $\wp^{\prime}$). Starting from a few
basic facts on CM fields, this subsection gathers some results related to
projective embedding of complex tori with complex multiplication in CM fields;
for more details on these topics, the reader is referred to e.g. [1, 9, 13,
14].

####  2.2.1 CM fields and Riemann forms

When $K$ is an imaginary quadratic field, the complex tori
$\mathbb{C}/sO_{K}\simeq K_{\infty}/sO_{K}$ are polarizable for any
$s\in\mathbb{A}^{\times}_{K,f}$, which is not the case for complex tori of
higher dimension in general. However, when $K$ is a CM field (say, of degree
$[K:\mathbb{Q}]=2g$), the complex tori $\mathbb{C}^{g}/sO_{K}\simeq
K_{\infty}/sO_{K}$ are always polarizable, hence, can be embedded into
projective space as abelian varieties.

To be precise, let $K$ be a CM field of degree $[K:\mathbb{Q}]=2g$; also, fix
a _CM type_ $\Phi=\\{\phi_{1},\cdots,\phi_{g}\\}$. That is,
$\phi_{i}:K\hookrightarrow\mathbb{C}$ are embeddings that are neither equal
nor conjugate to each other; therefore, denoting by
$\bar{\phi}_{i}:K\hookrightarrow\mathbb{C}$ the complex conjugate of
$\phi_{i}$, the set
$\\{\phi_{1},\cdots,\phi_{g},\bar{\phi}_{1},\cdots,\bar{\phi}_{g}\\}$ is
precisely that of all embeddings $K\hookrightarrow\mathbb{C}$. We abusively
use the same symbol $\Phi:K\rightarrow\mathbb{C}^{g}$ to denote also the map
$\Phi:K\ni a\mapsto(\phi_{1}(a),\cdots,\phi_{g}(a))\in\mathbb{C}^{g}$; by this
map $\Phi$, the $O_{K}$-modules $sO_{K}:=s\widehat{O}_{K}\cap K\subseteq K$
for $s\in\mathbb{A}^{\times}_{K,f}$ are embedded into $\mathbb{C}^{g}$ as
lattices $\Lambda_{s}:=\Phi(sO_{K})\subseteq\mathbb{C}^{g}$, which then induce
the complex tori $\mathbb{C}^{g}/\Lambda_{s}$. The first fact necessary in
this paper is that the complex tori $\mathbb{C}^{g}/\Lambda_{s}$ are
_polarizable_ in the following sense, for any $s\in\mathbb{A}^{\times}_{K,f}$
of the CM field $K$:

######  Definition 2.2.1 (polarization on complex tori).

For a lattice $\Lambda\subseteq\mathbb{C}^{g}$, the complex torus
$\mathbb{C}^{g}/\Lambda$ is said to be _polarizable_ if there exists an
alternating $\mathbb{R}$-bilinear form
$\psi:\mathbb{C}^{g}\times\mathbb{C}^{g}\rightarrow\mathbb{R}$ such that:

  1. 1.

$\psi(u,v)\in\mathbb{Z}$ for any $u,v\in\Lambda$;

  2. 2.

$\psi(J(u),J(v))=\psi(u,v)$ for any $u,v\in\mathbb{C}^{g}$;

  3. 3.

$\psi(u,J(u))>0$ for any non-zero $u\in\mathbb{C}^{g}$;

where $J:\mathbb{C}^{g}\rightarrow\mathbb{C}^{g}$ is the standard complex
structure. Such a form $\psi$ is called a (non-degenerate) _Riemann form_ on
the complex torus $\mathbb{C}^{g}/\Lambda$.

The fact that the complex tori $\mathbb{C}^{g}/\Lambda_{s}$ are polarizable is
based on a few structural properties of the CM field $K$ (e.g. Theorem 4, §6.2
[13]): Being a CM field, $K$ is by definition a totally imaginary quadratic
extension of its maximal totally real subfield $K_{0}$; take a totally
imaginary element $e\in K$ such that $-e^{2}$ is a totally positive element of
$K_{0}$ and $\mathrm{Im}(\phi_{i}(e))>0$ for all $\phi_{i}\in\Phi$, where
$\mathrm{Im}(z)$ denotes the imaginary part of $z\in\mathbb{C}$. With this
$e$, define $\psi_{e}:\mathbb{C}^{g}\times\mathbb{C}^{g}\rightarrow\mathbb{R}$
by:

$\displaystyle\psi_{e}(u,v)$ $\displaystyle:=$
$\displaystyle\sum_{i=1}^{g}\phi_{i}(e)\cdot(u_{i}\bar{v}_{i}-\bar{u}_{i}v_{i})$
(2.9)

for $u=(u_{1},\cdots,u_{g}),v=(v_{1},\cdots,v_{g})\in\mathbb{C}^{g}$. In
particular, for each $a,b\in K$, we have:

$\displaystyle\psi_{e}(\Phi(a),\Phi(b))$ $\displaystyle=$
$\displaystyle\mathrm{Tr}_{K/\mathbb{Q}}(e\cdot a\cdot b^{\iota}),$ (2.10)

where $\mathrm{Tr}_{K/\mathbb{Q}}$ denotes the trace over $\mathbb{Q}$; and
$\iota\in Gal(K/K_{0})$ denotes the non-trivial element of order 2. In
particular, this means that, the $O_{K}$-module $sO_{K}$ for any
$s\in\mathbb{A}^{\times}_{K,f}$ being a fractional ideal of $K$, we have
$n_{s}\cdot\psi_{e}(sO_{K},sO_{K})\subseteq\mathbb{Z}$ for some integer
$n_{s}$; and then, this $\psi:=n_{s}\cdot\psi_{e}$ gives a Riemann form on the
complex torus $\mathbb{C}^{g}/\Lambda_{s}$.

######  Remark 2.2.2 (symplectic basis and type).

Given any Riemann form $\psi$ on a complex torus $\mathbb{C}^{g}/\Lambda$ in
general, the lattice $\Lambda$ has a _symplectic basis_ , which is a basis of
the lattice $\Lambda$ over $\mathbb{Z}$ by which the Riemann form
$\psi:\mathbb{C}^{g}\times\mathbb{C}^{g}\rightarrow\mathbb{R}$ can be
expressed by the matrix of the form:

$\displaystyle\psi(u,v)$ $\displaystyle=$ $\displaystyle
u\cdot\begin{pmatrix}0&\delta\\\ -\delta&0\end{pmatrix}\cdot v^{t}$ (2.11)

where $u,v\in\mathbb{C}^{g}$ in the left-hand side are re-represented in the
right-hand side as real (row) vectors $u,v\in\mathbb{R}^{2g}$ with respect to
the symplectic basis of $\Lambda$; also $\delta\in M_{g}(\mathbb{Z})$ is a
matrix of the form:

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\begin{pmatrix}d_{1}&0&\cdots&0\\\ 0&d_{2}&&0\\\
\vdots&&\ddots&\vdots\\\ 0&0&\cdots&d_{g}\end{pmatrix}$ (2.12)

with $0<d_{i}$ and $d_{i}\mid d_{i+1}$; in the following we denote
$\delta=\mathrm{diag}[d_{1},\cdots,d_{g}]$. This $\delta$ is independent of
the choice of symplectic basis of $\Lambda$ for the Riemann form $\psi$; thus
it makes sense to say that the Riemann form $\psi$ on the complex torus
$\mathbb{C}^{g}/\Lambda$ is _of type $\delta$_.

####  2.2.2 Projective embedding

By the above fact that the complex torus $\mathbb{C}^{g}/\Lambda_{s}$ is
polarizable for any $s\in\mathbb{A}^{\times}_{K,f}$ of the CM field $K$ with a
fixed CM type $\Phi$, it follows that the complex torus
$\mathbb{C}^{g}/\Lambda_{s}$ can be isomorphically embedded as an abelian
variety in a suitable projective space ${\bf P}^{d}(\mathbb{C})$. Such an
embedding can be concretely provided by _theta functions_ [11], which depends
on a choice of Riemann form on the complex torus $\mathbb{C}^{g}/\Lambda_{s}$
and its type.

To see such embeddings and theta functions explicitly, take a lattice
$\Lambda\subseteq\mathbb{C}^{g}$ such that we have a Riemann form $\psi$ on
the complex torus $\mathbb{C}^{g}/\Lambda$ of type
$\delta=\mathrm{diag}[d_{1},\cdots,d_{g}]$; then the lattice $\Lambda$ has a
symplectic basis
$(\tau_{1},\cdots,\tau_{g},\delta_{1},\cdots,\delta_{g})=:(\tau,\delta)\in
M_{g\times 2g}(\mathbb{C})$ so that
$\Lambda=\mathbb{Z}^{g}\cdot\tau+\mathbb{Z}^{g}\cdot\delta$, where
$\mathbb{Z}^{g}$ consists of row vectors and
$\tau_{i}\in\mathbb{C}^{g},\delta_{i}\in\mathbb{Z}^{g}$ are column vectors.
Here the $g\times g$ matrix $\tau\in M_{g}(\mathbb{C})$ is in fact a member of
the _Siegel upper-half space $\mathfrak{H}_{g}$_ defined by:

$\displaystyle\mathfrak{H}_{g}$ $\displaystyle:=$
$\displaystyle\biggl{\\{}\tau\in
M_{g}(\mathbb{C})\mid\tau^{t}=\tau,\mathrm{Im}(\tau)>0\biggr{\\}}$ (2.13)

where $\mathrm{Im}(\tau)>0$ means that the imaginary part
$\mathrm{Im}(\tau)\in M_{g}(\mathbb{R})$ of $\tau$ is positive definite. We
note conversely that, given $\tau\in\mathfrak{H}_{g}$, one obtains a lattice
$\Lambda_{\tau}:=\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta$ such that the
complex torus $\mathbb{C}^{g}/\Lambda_{\tau}$ has a Riemann form $\psi_{\tau}$
of type $\delta$, given by
$\psi_{\tau}(u,v):=\mathrm{Im}(u\cdot(\mathrm{Im}(\tau))^{-1}\cdot\bar{v}^{t})$;
then $(\tau,\delta)$ gives a symplectic basis of $\Lambda_{\tau}$ for this
Riemann form $\psi_{\tau}$. Therefore, it follows that the complex tori
$\mathbb{C}^{g}/\Lambda$ with Riemann forms $\psi$ of type $\delta$ are
precisely those of the form
$\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)$ for some
$\tau\in\mathfrak{H}_{g}$ (cf. e.g. §8.1 [1]).

Now we are ready for projective embeddings of the complex tori
$\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)$ with Riemann forms
$\psi$ of type $\delta$ by the following _theta functions_ :

######  Definition 2.2.3 (theta function $\theta^{k}$).

For each $k\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$, define
$\theta^{k}:\mathbb{C}^{g}\times\mathfrak{H}_{g}\rightarrow\mathbb{C}$ as
follows:

$\displaystyle\theta^{k}(u;\tau)$ $\displaystyle:=$ $\displaystyle\sum_{w\in
k+\mathbb{Z}^{g}}{\bf e}(\frac{w^{t}\cdot\tau\cdot w}{2}+u\cdot w),$ (2.14)

where ${\bf e}(t):=e^{2\pi it}$; also $u\in\mathbb{C}^{g}$ are row vectors,
while $k\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ and $w\in
k+\mathbb{Z}^{g}$ are supposed to be column vectors.

The theta functions $\theta^{k}$ define projective embedding of complex tori
of type $\delta=\mathrm{diag}[d_{1},\cdots,d_{g}]$ when $3\leq d_{1}$ in the
following manner:

######  Definition 2.2.4 (projective embedding).

Put $d:=1+|\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}|=1+d_{1}d_{2}\cdots
d_{g}$; then for each $\tau\in\mathfrak{H}_{g}$, we can define
$\Theta_{\tau}:\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)\rightarrow{\bf
P}^{d}(\mathbb{C})$ by:

$\displaystyle\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)$
$\displaystyle\xrightarrow{\Theta_{\tau}}$ $\displaystyle{\bf
P}^{d}(\mathbb{C})$ (2.15) $\displaystyle u$ $\displaystyle\longmapsto$
$\displaystyle[\theta^{k}(u;\tau)]_{k\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}}$
(2.16)

where $[\theta^{k}]_{k\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}}$ denotes
the homogeneous coordinate on the projective space ${\bf P}^{d}(\mathbb{C})$;
by our assumption on $\delta$, this map
$\Theta_{\tau}:\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)\rightarrow{\bf
P}^{d}(\mathbb{C})$ defines a projective embedding, whose image
$\Theta_{\tau}(\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta))\subseteq{\bf
P}^{d}(\mathbb{C})$ is an abelian variety and will be denoted
$A_{\tau}\subseteq{\bf P}^{d}(\mathbb{C})$, thus
$\Theta_{\tau}:\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)\xrightarrow{\sim}A_{\tau}$.

####  2.2.3 Complex multiplication

In the case when $K$ is an imaginary quadratic field, the special value
$f(\tau)$ of a modular function on $\mathfrak{H}$ (rational over
$\mathbb{Q}^{ab}$) at $\tau\in K\cap\mathfrak{H}$ is abelian over $K$;
Shimura’s reciprocity law then shows that the galois conjugate
$(f(\tau))^{[s]^{-1}}$ of this special value $f(\tau)$ for
$s\in\mathbb{A}_{K,f}^{\times}$ is equal to the special value
$f^{q_{\tau}(s)}(\tau)$ of a suitably defined galois conjugate modular
function $f^{q_{\tau}(s)}$ (cf. Theorem 6.18 [12]). In terms of elliptic
curves, this reciprocity law is proved using the following _main theorem of
complex multiplication_ :

######  Theorem 2.2.5 (main theorem of complex multiplication, Theorem 5.4
[12]).

Let $\mathfrak{a}$ be a fractional ideal of an imaginary quadratic field
$K\subseteq\mathbb{C}$; and $\Theta:\mathbb{C}/\mathfrak{a}\rightarrow
E_{\mathfrak{a}}$ an isomorphism of the complex torus
$\mathbb{C}/\mathfrak{a}$ to an elliptic curve $E_{\mathfrak{a}}$. Then, for
any automorphism $\sigma:\mathbb{C}\rightarrow\mathbb{C}$ over $K$ with
$\sigma|_{K^{ab}}=[s]^{-1}$ for some $s\in\mathbb{A}^{\times}_{K,f}$, there
exists a unique isomorphism
$\Theta^{\prime}:\mathbb{C}/s\mathfrak{a}\rightarrow E^{\sigma}$ such that the
following diagram commutates:

$\textstyle{K/\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathbb{C}/\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta}$$\textstyle{E_{\mathfrak{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{K/s\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{C}/s\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{\prime}}$$\textstyle{E_{\mathfrak{a}}^{\sigma}}$
(2.17)

In the case when $K$ is a CM field of higher degree in general, a similar
theorem still holds, where elliptic curves are replaced with abelian varieties
of higher dimension; but in order to state it in the case of degree greater
than $2$, we need to consider not only $K$ itself but also its _reflex field_
$K^{*}$ for a CM type of $K$:

######  Definition 2.2.6 (reflex field).

Let $K$ be a CM field of degree $[K:\mathbb{Q}]=2g$ and
$\Phi=\\{\phi_{1},\cdots,\phi_{g}\\}$ a CM type of $K$. Then the reflex field
$K^{*}$ of the CM type $(K,\Phi)$ is defined as follows:

$\displaystyle K^{*}$ $\displaystyle:=$
$\displaystyle\mathbb{Q}\biggl{(}\sum_{i=1}^{g}\phi_{i}(e)\mid e\in
K\biggr{)}.$ (2.18)

######  Remark 2.2.7 (reflex norm).

For the CM type $(K,\Phi)$, the reflex field $K^{*}$ is again a CM field;
also, we have the _reflex norm_ for $(K,\Phi)$, which is a multiplicative
monoid morphism $\eta:\mathbb{A}_{K^{*}}\rightarrow\mathbb{A}_{K}$; this
morphism restricts to
$\eta:\mathbb{A}_{K^{*},f}^{\times}\rightarrow\mathbb{A}_{K,f}^{\times}$ and
$\eta:\widehat{O}_{K^{*}}\rightarrow\widehat{O}_{K}$; for more details the
reader is referred to e.g. §18.5 [13] or §9.4 [14].

######  Theorem 2.2.8 (main theorem of complex multiplication, Theorem 18.6
[13]).

Let $\mathfrak{a}$ be a fractional ideal of a CM field $K$; and
$\Theta:\mathbb{C}^{g}/\Phi(\mathfrak{a})\rightarrow A_{\mathfrak{a}}$ an
isomorphism of the complex torus $\mathbb{C}^{g}/\Phi(\mathfrak{a})$ to an
abelian variety $A_{\mathfrak{a}}$ (§2.2.2). Then, for any automorphism
$\sigma:\mathbb{C}\rightarrow\mathbb{C}$ over $K^{*}$ with
$\sigma|_{(K^{*})^{ab}}=[s]^{-1}$ for some
$s\in\mathbb{A}_{K^{*},f}^{\times}$, there exists a unique isomorphism
$\Theta^{\prime}:\mathbb{C}^{g}/\Phi(\eta(s)\mathfrak{a})\rightarrow
A_{\mathfrak{a}}^{\sigma}$ such that the following diagram commutes:

$\textstyle{E/\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\eta(s)}$$\textstyle{\mathbb{C}^{g}/\Phi(\mathfrak{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
8.5359pt\Theta}$$\textstyle{A_{\mathfrak{a}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{E/\eta(s)\mathfrak{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Phi(\eta(s)\mathfrak{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
11.38092pt\Theta^{\prime}}$$\textstyle{A_{\mathfrak{a}}^{\sigma}}$ (2.19)

##  3 Modular vectors over CM fields

The subject of this section is to construct _modular vectors_ in the case when
$K$ is the reflex field of a CM type; to be compatible with notations in §1,
let $(E,\Phi)$ be a CM type of degree $[E:\mathbb{Q}]=2g$, and put $K=E^{*}$;
by the identification discussed in §2.2.1, it suffices to construct locally
constant $K^{ab}$-valued $G_{K}$-equivariant functions on $DR_{K}$. Thus, we
divide this section into two sub-sections for (i) a construction of suitable
functions on $DR_{K}$ from modular functions on the Siegel upper-half space
$\mathfrak{H}_{g}$ (§3.1), and (ii) a proof that they are indeed locally
constant, $K^{ab}$-valued, and $G_{K}$-equivariant (§3.2).

###  3.1 Construction of modular vectors

Throughout this section, we shall fix a CM field $E$ of degree
$[E:\mathbb{Q}]=2g$ together with a CM type
$\Phi=\\{\phi_{1},\cdots,\phi_{g}\\}$ of $E$; let $K=E^{*}$ be the reflex
field of $(E,\Phi)$. Also we fix a Riemann form on the complex torus
$\mathbb{C}^{g}/\Phi(O_{E})$,
$\psi:\mathbb{C}^{g}\times\mathbb{C}^{g}\rightarrow\mathbb{R}$, so that the
ring $O_{E}$ of integers has a symplectic basis
$(\tau_{1},\cdots,\tau_{g},\delta_{1},\cdots,\delta_{g})$ for $\psi$ with
which $\psi$ is of type $\delta=\mathrm{diag}[d_{1},\cdots,d_{g}]$ and $3\leq
d_{i}\mid d_{i+1}$. Such a Riemann form
$\psi:\mathbb{C}^{g}\times\mathbb{C}^{g}\rightarrow\mathbb{R}$ always exists;
the condition on its type $\delta$ is necessary to guarantee that the theta
functions $\theta^{k}$ induce projective embeddings of complex tori
$\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau+\mathbb{Z}^{g}\delta)$ of type $\delta$
for each $\tau\in\mathfrak{H}_{g}$ (cf. §2.2.2); for a technical reason (cf.
Proposition 8.10.2 [1]), we further suppose that $4\leq d_{1}$ and $2\mid
d_{1}$ or $3\mid d_{1}$.

In particular, by construction, the ring $O_{E}$ of integers in $E$ is mapped
to a lattice $\Phi(O_{E})\subseteq\mathbb{C}^{g}$ by
$\Phi:E\rightarrow\mathbb{C}^{g}$, which can be written by a matrix
$\tau_{E}\in\mathfrak{H}_{g}$ as in the following form:

$\displaystyle\Phi(O_{E})$ $\displaystyle=$
$\displaystyle\mathbb{Z}^{g}\tau_{E}+\mathbb{Z}^{g}\delta$ (3.1)

Therefore, as discussed in §2.2.3, the complex torus
$\mathbb{C}^{g}/\Phi(O_{E})=\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau_{E}+\mathbb{Z}^{g}\delta)$
is isomorphic to an abelian variety, which we shall denote $A_{E}\subseteq{\bf
P}^{d}(\mathbb{C})$, through the following projective embedding
$\Theta_{E}:=\Theta_{\tau_{E}}:\mathbb{C}^{g}/\Phi(O_{E})\hookrightarrow{\bf
P}^{d}(\mathbb{C})$ given by (cf. Definition 2.2.4):

$\displaystyle\mathbb{C}^{g}/\Phi(O_{E})$
$\displaystyle\xrightarrow{\Theta_{E}}$ $\displaystyle{\bf P}^{d}(\mathbb{C})$
(3.2) $\displaystyle u$ $\displaystyle\longmapsto$
$\displaystyle[\theta^{k}(u;\tau_{E})]_{k\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}},$
(3.3)

where $d=1+d_{1}d_{2}\cdots d_{g}$. More generally, for any
$s\in\mathbb{A}_{K,f}^{\times}$, the $O_{E}$-module
$\eta(s)O_{E}:=\eta(s)\widehat{O}_{E}\cap E$ is mapped to the lattice
$\Phi(\eta(s)O_{E})\subseteq\mathbb{C}^{g}$, where
$\eta:\mathbb{A}_{K,f}^{\times}=\mathbb{A}_{E^{*},f}^{\times}\rightarrow\mathbb{A}_{E,f}^{\times}$
denotes the reflex norm (cf. §2.2.3); in the following we denote this lattice
$\Phi(\eta(s)O_{E})$ by $\Lambda_{s}$ for each
$s\in\mathbb{A}_{K,f}^{\times}$, in particular,
$\Lambda_{E}:=\Lambda_{1}=\Phi(O_{E})$.

The main theorem of complex multiplication (Theorem 2.2.8) shows that, for
each automorphism $\sigma:\mathbb{C}\rightarrow\mathbb{C}$ over $K=E^{*}$ with
$\sigma|_{K^{ab}}=[s]^{-1}$, we have a unique isomorphism
$\Theta^{\prime}:\mathbb{C}/\Lambda_{s}\rightarrow A_{E}^{\sigma}$ such that:

$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\eta(s)}$$\textstyle{\mathbb{C}^{g}/\Lambda_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
8.5359pt\Theta_{E}}$$\textstyle{A_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
11.38092pt\Theta^{\prime}}$$\textstyle{A_{E}^{\sigma}}$ (3.4)

We will use these $\Theta^{\prime}$ for our construction of modular vectors;
but note that, while the isomorphism
$\Theta_{E}:\mathbb{C}^{g}/\Lambda_{E}\rightarrow A_{E}$ is given explicitly
by the theta functions $\theta^{k}(*;\tau_{E})$ as above, the isomorphism
$\Theta^{\prime}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow A_{E}^{\sigma}$ is not
yet quite explicit for $s\in\mathbb{A}^{\times}_{K,f}$. Therefore, our task
here is first to construct modular vectors over $K$ using these abstract
$\Theta^{\prime}$; and then to describe the resulting modular vectors more
explicitly.

Concerning this first task, notice that, while
$\Theta^{\prime}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow A_{E}^{\sigma}$ itself
depends on $\sigma$ rather than $\sigma|_{K^{ab}}=[s]^{-1}$ (hence, is not
actually determined only by $s$), the composition
$\Theta^{\prime}\circ\Phi:E/\eta(s)O_{E}\hookrightarrow\mathbb{C}^{g}/\Lambda_{s}\rightarrow
A_{E}^{\sigma}\subseteq{\bf P}^{d}(\mathbb{C})$ does not because the image of
$E/\eta(s)O_{E}$ are the torsion points of $A_{E}^{\sigma}$ and the action of
$\sigma$ on the torsion points is determined by $[s]$. Therefore, it makes
sense to denote this composition as $\Theta_{s}:E/\eta(s)O_{E}\rightarrow
A_{E}^{\sigma}\subseteq{\bf P}^{d}(\mathbb{C})$ for each
$s\in\mathbb{A}_{K,f}^{\times}$. With this in mind, we abusively denote as
$A_{E}^{\sigma}=A_{s}$, and also
$\Theta_{s}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow A_{s}$.

Now we construct modular vectors using these (still abstract) isomorphisms
$\Theta_{s}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow A_{s}$. To this end, we note
that each abelian variety $A_{s}$ is a priori a subvariety of the ambient
projective space ${\bf P}^{d}(\mathbb{C})$; thus the rational functions on
${\bf P}^{d}(\mathbb{C})$ uniformly restrict to each subvariety $A_{s}$. More
explicitly, in terms of the homogeneous coordinates
$[\theta^{k}]_{k\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}}$ of ${\bf
P}^{d}(\mathbb{C})$, the rational functions on ${\bf P}^{d}(\mathbb{C})$ are
of the form $f(\theta^{k})=g(\theta^{k})/h(\theta^{k})$, where $g,h$ are
homogeneous polynomials of $\theta^{k}$’s of the same degree; in particular,
we mean by _rational functions (over $K$)_ those
$f(\theta^{k})=g(\theta^{k})/h(\theta^{k})$ whose coefficients are in $K$.
Also, for each $a=(a_{1},a_{2})\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$, we say
that a rational function $f(\theta^{k})$ is _defined at $a$_ if it is defined
on the torsion point of $A_{E}$ corresponding to $a_{1}\tau_{E}+a_{2}\delta\in
E/O_{E}$ under the map
$\Theta_{E}\circ\Phi:E/O_{E}\hookrightarrow\mathbb{C}^{g}/\Lambda_{E}\simeq
A_{E}$; also we say that $f(\theta^{k})$ is _defined for $a$_ if
$f(\theta^{k})$ is defined at $a\eta(\rho)$ for any $\rho\in\widehat{O}_{K}$.

######  Definition 3.1.1 (modular vector).

Let $a\in N^{-1}\mathbb{Z}^{2g}/\mathbb{Z}^{2g}$ and $f$ a rational function
defined for $a$. Then the _modular vector_
$\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ is defined by the following
rule: Given each $[\rho,s]\in DR_{K}$, the value $\widehat{f}_{a}(\rho,s)$ is
the image of the torsion point $a_{1}\tau_{E}+a_{2}\delta\in E/O_{E}$ (at the
upper-left corner in (3.5)) under the map $E/O_{E}\rightarrow\mathbb{C}$ given
by the following composition:

$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\rho)}$$\scriptstyle{\simeq}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\simeq}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{s}}$$\textstyle{A_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{A_{s}}}$$\textstyle{\mathbb{C}}$$\textstyle{\mathbb{A}_{E,f}/\widehat{O}_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\rho)}$$\textstyle{\mathbb{A}_{E,f}/\widehat{O}_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\textstyle{\mathbb{A}_{E,f}/\eta(s)\widehat{O}_{E}}$
(3.5)

######  Remark 3.1.2 (rational functions over $\mathbb{Q}$).

In the definition of modular vectors, we used rational functions $f$ defined
over $K$; but it suffices for our purpose in §4 to consider only rational
functions defined over $\mathbb{Q}$. In the latter case, nevertheless, the set
of modular vectors itself does not form a $K$-algebra; in that case, we need
to consider the $K$-algebra generated (over $K$) by modular vectors.

######  Lemma 3.1.3.

The above modular vectors $\widehat{f}_{a}$ are well-defined as
$\mathbb{C}$-valued functions on $DR_{K}$.

###### Proof.

We need to check two points for the well-definedness of $\widehat{f}_{a}$: The
first is that the value $\widehat{f}_{a}(\rho,s)$ is indeed finite; the second
is that it depends only on the class $[\rho,s]\in DR_{K}$ rather than
$(\rho,s)$. To see these, consider the following commutative diagram:

$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\rho)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{E}}$$\textstyle{A_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[s]^{-1}}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{s}}$$\textstyle{A_{s}}$
(3.6)

Since the right-most vertical isomorphism $[s]^{-1}:A_{E}\rightarrow A_{s}$
(on the torsion points) depends only on the class $[s]$, it first follows that
the value $\widehat{f}_{a}(\rho,s)$ depends on $[s]$ rather than $s$ itself;
also, the torsion point $a_{1}\tau_{E}+a_{2}\delta\in N^{-1}O_{E}/O_{E}$ is
mapped again to $N^{-1}O_{E}/O_{E}$ by $\eta(\rho):E/O_{E}\rightarrow
E/O_{E}$; therefore, the assumption that $f$ is now defined for $a$ implies
that the value $\widehat{f}_{a}(\rho,s)$ is finite. Finally the equality
$\widehat{f}_{a}(\rho u,u^{-1}s)=\widehat{f}_{a}(\rho,s)$ for any
$u\in\widehat{O}_{K}^{\times}$ follows from the fact that the composition
$\eta(u^{-1}s)\circ\eta(\rho u):E/O_{E}\rightarrow
E/\eta(u^{-1}s)O_{E}=E/\eta(s)O_{E}$ is equal to
$\eta(s)\circ\eta(\rho):E/O_{E}\rightarrow E/\eta(s)O_{E}$ since the reflex
norm $\eta$ is multiplicative. This completes the proof. ∎

###  3.2 Basic properties

We prove here that the modular vectors
$\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ are indeed locally constant
$K^{ab}$-valued and $G_{K}$-equivariant, hence define algebraic Witt vectors
$\widehat{f}_{a}\in E_{K}$ over $K$. Moreover, as mentioned above too, the
construction of modular vectors depends on the abstract isomorphisms
$\Theta_{s}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow A_{s}$; thus, we shall also
give a more explicit description of modular vectors in terms of the theta
functions
$\theta^{k}:\mathbb{C}^{g}\times\mathfrak{H}_{g}\rightarrow\mathbb{C}$ of
§2.2.2 at a concrete $\tau_{s}\in\mathfrak{H}_{g}$.

We start with proving the fact that the modular vectors define algebraic Witt
vectors:

######  Proposition 3.2.1.

Take a rational function $f$ and $a\in N^{-1}\mathbb{Z}^{2g}/\mathbb{Z}^{2g}$
as in Definition 3.1.1; then the modular vector
$\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ is locally constant
$K^{ab}$-valued and $G_{K}$-equivariant.

###### Proof.

We first see that $\widehat{f}_{a}:DR_{K}\rightarrow\mathbb{C}$ is
$K^{ab}$-valued. To this end, note the following commutative diagram:

$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\rho)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{E}}$$\textstyle{A_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{f|_{A_{E}}}$$\textstyle{\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{s}}$$\textstyle{A_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{A_{s}}}$$\textstyle{\mathbb{C}}$
(3.7)

where $\sigma|_{K^{ab}}=[s]^{-1}$. By definition of $\widehat{f}_{a}$, this
diagram shows that we have:

$\displaystyle\widehat{f}_{a}(\rho,s)$ $\displaystyle=$ $\displaystyle
f(\theta^{k}(a_{1}^{\prime}\tau_{E}+a_{2}^{\prime}\delta;\tau_{E}))^{\sigma}$
(3.8)

where $a^{\prime}=(a_{1}^{\prime},a_{2}^{\prime})\in
N^{-1}\mathbb{Z}^{2g}/\mathbb{Z}^{2g}$ is defined as
$a^{\prime}=a\cdot\eta(\rho)$. Since
$f(\theta^{k})=g(\theta^{k})/h(\theta^{k})$ is now a rational function over
$K$ with homogeneous polynomials $g,h$ of the same degree, it follows that
this value $\widehat{f}_{a}(\rho,s)$ can be expressed as an algebraic
combination of ratios of theta values in the following form for some
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$:

$\displaystyle\biggl{(}\frac{\theta^{k}(a_{1}^{\prime}\tau_{E}+a_{2}^{\prime}\delta;\tau_{E})}{\theta^{l}(a_{1}^{\prime}\tau_{E}+a_{2}^{\prime}\delta;\tau_{E})}\biggr{)}^{\sigma}$
(3.9)

In fact, these ratios are abelian over $K$ since the following meromorphic
function $\theta^{k/l}_{a}$ on $\mathfrak{H}_{g}$ defines a modular function
on $\mathfrak{H}_{g}$ for all $a\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$ and
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ (cf. Theorem 27.13, [13]):

$\displaystyle\theta^{k/l}_{a}(\tau)$ $\displaystyle:=$
$\displaystyle\frac{\theta^{k}(a_{1}\tau+a_{2}\delta;\tau)}{\theta^{l}(a_{1}\tau+a_{2}\delta;\tau)}$
(3.10)

Therefore, the action of $\sigma$ on these values at $\tau_{E}$ is in fact
equal to $[s]^{-1}$; and we eventually have:

$\displaystyle\widehat{f}_{a}(\rho,s)$ $\displaystyle=$ $\displaystyle
f(\theta^{k}(a_{1}^{\prime}\tau_{E}+a_{2}^{\prime}\delta;\tau_{E}))^{[s]^{-1}}\in
K^{ab}.$ (3.11)

From this expression, we also easily deduce that
$\widehat{f}_{a}:DR_{K}\rightarrow K^{ab}$ is $G_{K}$-equivariant. (Recall
that the action of $G_{K}$ on $DR_{K}$ factors through
$G_{K}\twoheadrightarrow G_{K}^{ab}$, and the action of $[t]\in G_{K}^{ab}$ on
$[\rho,s]$ is equal to $[\rho,t^{-1}s]$; see e.g. Lemma 4.2.3 [19].)

Finally, we see that $\widehat{f}_{a}:DR_{K}\rightarrow K^{ab}$ is locally
constant; in fact, to be more specific, $\widehat{f}_{a}$ factors through the
finite quotient $DR_{K}\twoheadrightarrow DR_{NO_{K}}$ for the conductor
$NO_{K}$ for $a\in N^{-1}\mathbb{Z}^{2g}/\mathbb{Z}^{2g}$. To see this, note
first that the action of $\eta(\rho)\in\widehat{O}_{E}$ (by multiplication) on
$N^{-1}O_{E}/O_{E}$ is determined by the class
$[\rho]\in\widehat{O}_{K}/N\widehat{O}_{K}$; this follows from the fact that
the reflex norm
$\eta:\widehat{O}_{K}=\widehat{O}_{E^{*}}\rightarrow\widehat{O}_{E}$ maps
$N\widehat{O}_{K}$ into $N\widehat{O}_{E}$. Moreover, the values
$\widehat{f}_{a}(\rho,s)$ belong at largest to the ray class field
$K_{NO_{K}}$ of the conductor $NO_{K}$ since these values are given by special
values of $f(\theta^{k})$ at $N$-torsion points of $A_{E}$; thus the action of
$[s]^{-1}$ on these values also factor through the quotient
$G_{K}^{ab}\twoheadrightarrow C_{NO_{K}}$. Combining these with the
isomorphism
$DR_{NO_{K}}\simeq(O_{K}/NO_{K})\times_{(O_{K}/NO_{K})^{\times}}C_{NO_{K}}$
(cf. Proposition 7.2 [21]), we deduce that $\widehat{f}_{a}$ indeed factors
through $DR_{K}\twoheadrightarrow DR_{NO_{K}}$. This completes the proof. ∎

As seen above, the proof of the fact that the modular vectors
$\widehat{f}_{a}$ are locally constant $K^{ab}$-valued $G_{K}$-equivariant was
possible even with the abstract isomorphisms
$\Theta_{s}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow A_{s}$. But our next task is
to give a more explicit description of the values $\widehat{f}_{a}(\rho,s)$ by
special values of ratios of theta functions $\theta^{k}(*;\tau_{s})$ at an
_explicit_ $\tau_{s}\in\mathfrak{H}_{g}$ rather than the galois conjugates
$\theta^{k}(*;\tau_{E})^{[s]^{-1}}$ as given in the right hand side of (3.11);
technically, this task is done by finding $\tau_{s}\in\mathfrak{H}_{g}$ such
that the lattice $\Lambda_{s}=\Phi(\eta(s)O_{E})$ is isomorphic to
$\mathbb{Z}^{g}\tau_{s}+\mathbb{Z}^{g}\delta$.

To this end, note first that, for each $s\in\mathbb{A}_{K,f}^{\times}$, the
$\mathbb{A}_{\mathbb{Q},f}$-linear map $\mathbb{A}_{E,f}\ni
t\mapsto\eta(s)\cdot t\in\mathbb{A}_{E,f}$ can be expressed as a $2g\times 2g$
matrix in $GL_{2g}(\mathbb{A}_{\mathbb{Q},f})$ with respect to the symplectic
basis of $O_{E}$ for the Riemann form $\psi$, which we abusively denote also
as $\eta(s)\in GL_{2g}(\mathbb{A}_{\mathbb{Q},f})$; this adelic matrix
$\eta(s)$ has the following standard matrix decomposition, with which we will
construct the target $\tau_{s}\in\mathfrak{H}_{g}$:

######  Lemma 3.2.2 (cf. strong approximation theorem).

Put $\tilde{\eta}(s)\in GL_{2g}(\mathbb{A}_{\mathbb{Q},f})$ as follows:

$\displaystyle\tilde{\eta}(s)$ $\displaystyle:=$
$\displaystyle\begin{pmatrix}1&0\\\
0&\delta\end{pmatrix}^{-1}\cdot\eta(s)\cdot\begin{pmatrix}1&0\\\
0&\delta\end{pmatrix}$ (3.12)

Then we have $\tilde{\eta}(s)=\tilde{u}_{s}\cdot\tilde{\alpha}_{s}$ with some
$\tilde{u}_{s}\in GSp_{2g}(\mathbb{A}_{\mathbb{Q},f})$ and
$\tilde{\alpha}_{s}\in GSp^{+}_{2g}(\mathbb{Q})$ such that we have the
equality
$(\widehat{\mathbb{Z}}^{g}+\widehat{\mathbb{Z}}^{g}\delta)\cdot\tilde{u}_{s}=\widehat{\mathbb{Z}}^{g}+\widehat{\mathbb{Z}}^{g}\delta$.

###### Proof.

First, by the basic property of the reflex norm (cf. Remark 2.2.7), the
$\mathbb{A}_{\mathbb{Q},f}$-linear map $\eta(s)$ in fact preserves the Riemann
form
$\psi=\psi_{\mathbb{A}_{\mathbb{Q},f}}:\mathbb{A}_{E,f}\times\mathbb{A}_{E,f}\rightarrow\mathbb{A}_{\mathbb{Q},f}$
up to a fixed scalar over $\mathbb{A}_{\mathbb{Q},f}$. That is, for any
$u,v\in\mathbb{A}_{E,f}$:

$\displaystyle\psi(u\cdot\eta(s),v\cdot\eta(s))$ $\displaystyle=$
$\displaystyle\psi(u,v\cdot\eta(s)\eta(s)^{\iota})$ (3.13) $\displaystyle=$
$\displaystyle\psi(u,v\cdot N_{K/\mathbb{Q}}(s))$ (3.14) $\displaystyle=$
$\displaystyle N_{K/\mathbb{Q}}(s)\cdot\psi(u,v)$ (3.15)

By definition of symplectic basis for the Riemann form $\psi$, this means
that, in terms of the matrix $\eta(s)\in GL_{2g}(\mathbb{A}_{\mathbb{Q},f})$:

$\displaystyle\eta(s)\cdot\begin{pmatrix}0&\delta\\\
-\delta&0\end{pmatrix}\cdot\eta(s)^{t}$ $\displaystyle=$ $\displaystyle
N_{K/\mathbb{Q}}(s)\cdot\begin{pmatrix}0&\delta\\\ -\delta&0\end{pmatrix}$
(3.16)

Note also that we have:

$\displaystyle\begin{pmatrix}0&\delta\\\ -\delta&0\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}1&0\\\
0&\delta\end{pmatrix}\cdot\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\cdot\begin{pmatrix}1&0\\\ 0&\delta\end{pmatrix}$ (3.17)

From this, we easily deduce $\tilde{\eta}(s)\in
GSp_{2g}(\mathbb{A}_{\mathbb{Q},f})$. The decomposition in the claim is a
consequence of the strong approximation theorem for
$GSp_{2g}(\mathbb{A}_{\mathbb{Q},f})$ (cf. Lemma 8.3 and Lemma 9.5 [14]). ∎

For each $s\in\mathbb{A}_{K,f}^{\times}$, let $u_{s}$ and $\alpha_{s}$ be the
similar transformations of $\tilde{u}_{s}$ and $\tilde{\alpha}_{s}$ as the
above $\eta(s)$ of $\tilde{\eta}(s)$, whence $\eta(s)=u_{s}\cdot\alpha_{s}$
and
$(\widehat{\mathbb{Z}}^{g}+\widehat{\mathbb{Z}}^{g})u_{s}=\widehat{\mathbb{Z}}^{g}+\widehat{\mathbb{Z}}^{g}$;
also, partition $\alpha_{s}$ into the following form:

$\displaystyle\alpha_{s}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}a_{s}&b_{s}\\\ c_{s}&d_{s}\end{pmatrix}.$ (3.18)

We then define $\tau_{s}:=\alpha_{s}(\tau_{E})$ as
$\tau_{s}:=(a_{s}\tau_{E}+b_{s}\delta)\cdot(c_{s}\tau_{E}+d_{s}\delta)^{-1}\delta$,
which is indeed a member of $\mathfrak{H}_{g}$ by
$\tau_{E}\in\mathfrak{H}_{g}$ and $\tilde{\alpha}_{s}\in
GSp^{+}_{2g}(\mathbb{Q})$; we see that this $\tau_{s}\in\mathfrak{H}_{g}$ is
our target one. In the following we often identify
$a=(a_{1},a_{2})\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$ with
$a_{1}\tau_{E}+a_{2}\delta\in E/O_{E}$ via the symplectic basis of $O_{E}$.

######  Proposition 3.2.3 (see also Theorem 2.4, Streng [15]).

With notations as in Definition 3.1.1, we have the following equation:

$\displaystyle\widehat{f}_{a}(\rho,s)$ $\displaystyle=$ $\displaystyle
f(\theta^{k}(a^{\prime\prime}_{1}\tau_{s}+a_{2}^{\prime\prime}\delta;\tau_{s})).$
(3.19)

where $\tau_{s}=\alpha_{s}(\tau_{E})$ and
$a^{\prime\prime}=(a_{1}^{\prime\prime},a_{2}^{\prime\prime})\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$
is defined as the image of $a\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$ under the
following map:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.1164pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Q}^{2g}/\mathbb{Z}^{2g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
20.84677pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\eta(\rho)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 42.1164pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.1164pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Q}^{2g}/\mathbb{Z}^{2g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
84.40808pt\raise 4.9375pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.07639pt\hbox{$\scriptstyle{u_{s}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.34918pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.34918pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Q}^{2g}/\mathbb{Z}^{2g}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$
(3.20)

That is, $a^{\prime\prime}=a\cdot\eta(\rho)u_{s}\in
N^{-1}\mathbb{Z}^{2g}/\mathbb{Z}^{2g}$.

###### Proof.

Again recall the following diagram defining the value of
$\widehat{f}_{a}(\rho,s)$:

$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\rho)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{E}}$$\textstyle{A_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{f|_{A_{E}}}$$\textstyle{\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{s}}$$\textstyle{A_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{A_{s}}}$$\textstyle{\mathbb{C}}$
(3.21)

Here the lattice $\Lambda_{s}=\Phi(\eta(s)O_{E})$ is equal to
$(\mathbb{Z}^{g},\mathbb{Z}^{g})\cdot
u_{s}\alpha_{s}\cdot(\tau_{E},\delta)^{t}=(\mathbb{Z}^{g},\mathbb{Z}^{g})\cdot\alpha_{s}\cdot(\tau_{E},\delta)^{t}$
by our construction (Lemma 3.2.2), which is further isomorphic to
$(\mathbb{Z}^{g},\mathbb{Z}^{g})\cdot(\tau_{s},\delta)^{t}=\mathbb{Z}^{g}\tau_{s}+\mathbb{Z}^{g}\delta$
by the right action of $\chi_{s}:=(c_{s}\tau_{E}+d_{s}\delta)^{-1}\delta$.
With this in mind, we consider yet another diagram:

$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\rho)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{E}}$$\textstyle{A_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[s]^{-1}}$$\scriptstyle{f|_{A_{E}}}$$\textstyle{\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi_{s}}$$\scriptstyle{\simeq}$$\textstyle{\mathbb{C}^{g}/(\mathbb{Z}^{g}\tau_{s}+\mathbb{Z}^{g}\delta)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76228pt\simeq}$$\scriptstyle{\hskip
28.45274pt\Theta_{\tau_{s}}}$$\textstyle{A_{\tau_{s}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f|_{A_{\tau_{s}}}}$$\textstyle{\mathbb{C}}$
(3.22)

By the defining property of $\Theta_{s}:\mathbb{C}^{g}/\Lambda_{s}\rightarrow
A_{s}$, in order to prove (3.19), it suffices to prove that this diagram
commutes.

We see that this commutativity follows from the fact that the ratios
$\theta^{k/l}_{a}$ (3.10) of homogeneous coordinates $[\theta^{k}]$ at torsion
points of $A_{E}\subseteq{\bf P}^{d}(\mathbb{C})$ define modular functions on
$\mathfrak{H}_{g}$ (cf. Theorem 27.13 [13]), together with Shimura’s
reciprocity law (Theorem 9.6 [14]). That is, to be more specific, first note
that since $f(\theta^{k})=g(\theta^{k})/h(\theta^{k})$ is a rational function
of homogeneous degree $0$, it can be in fact composed of the ratios
$\theta^{k}/\theta^{l}$ of theta functions $\theta^{k}$; thus,
$f(\theta^{k}(a_{1}\tau+a_{2}\delta;\tau))$ also defines a modular function on
$\mathfrak{H}_{g}$ for each fixed $a\in\mathbb{Q}^{2g}/\mathbb{Z}^{2g}$, which
we may denote as $f(\theta^{k/l}_{a}(\tau))$. Moreover, since $f$ is defined
over $K$ and $\sigma$ fixes $K$, we have
$f(\theta_{a}^{k/l}(\tau))^{\sigma}=f((\theta_{a}^{k/l}(\tau))^{[s]^{-1}})$.
Then applying Shimura’s reciprocity law for these modular functions
$\theta^{k/l}_{a}$ and $[s]\in G_{K}^{ab}$, we deduce that the middle
rectangle commutes; in fact see Lemma 8.6 and Theorem 8.10 [14] for a careful
comparison of the definition of galois conjugate
$(\theta^{k/l}_{a})^{\eta(s)}$ of the modular functions $\theta^{k/l}_{a}$
with our composition of the above maps; in particular we have
$(\theta^{k/l}_{a})^{\eta(s)}(\tau_{E})=\theta^{k/l}_{au_{s}}(\alpha_{s}(\tau_{E}))=\theta^{k/l}_{au_{s}}(\tau_{s})$.
This completes the proof. ∎

##  4 Galois correspondence

Unlike the case of imaginary quadratic fields, the modular vectors constructed
above do not generate all algebraic Witt vectors over a CM field $K$ of higher
degree; therefore, it becomes reasonable to ask what algebraic Witt vectors
are generated by our modular vectors. Denoting by $M_{K}$ the set of modular
vectors, we can see by definition that $M_{K}$ is in fact a subalgebra of
$E_{K}$ over $K$; moreover, it can be easily seen that $M_{K}$ is a
$\Lambda$-subalgebra of $E_{K}$ (i.e. closed under the action of
$\psi_{\mathfrak{p}}$’s), which follows from the same argument as Lemma 4.3.5
[19]. Therefore, the _general galois correspondence_ mentioned in Remark 3.1.8
[19] implies that we have a dual characterization of modular vectors in terms
of profinite quotient of $DR_{K}$.

This section is devoted to this problem and divided into two subsections. In
the first subsection (§4.1), we give a proof of the general galois
correspondence mentioned in Remark 3.1.8 [19] without proof, which claims a
canonical bijective correspondence between (i) $\Lambda$-subalgebras of
$E_{K}$, (ii) profinite quotients of $DR_{K}$, and (iii) semi-galois full
subcategories of $\mathscr{C}_{K}$. This general correspondence, however, is
not quite satisfactory in itself in that it just tells us only an existence of
such a quotient but not its explicit structure. To remedy this issue, the
second subsection (§4.2) is devoted to an explicit characterization of modular
vectors, where we give an adelic description of the profinite quotient of
$DR_{K}$ that corresponds to the $\Lambda$-subalgebra of modular vectors
$M_{K}\subseteq E_{K}$ under our general galois correspondence.

###  4.1 General galois correspondence

In this subsection, we allow $K$ to be an arbitrary number field; our goal
here is to prove the following general galois correspondence mentioned in
Remark 3.1.8 [19]:

######  Proposition 4.1.1 (galois correspondence).

There exists a canonical bijective correspondence between the following three
classes of objects:

  1. 1.

$\Lambda$-subalgebras of $E_{K}$;

  2. 2.

profinite quotients of $DR_{K}$; and

  3. 3.

semi-galois full subcategories of $\mathscr{C}_{K}$.

Note that the correspondence between the second and the third is just a direct
consequence of the general duality theorem between profinite monoids and semi-
galois categories [16, 17] together with the fact that the fundamental monoid
$\pi_{1}(\mathscr{C}_{K},\mathrm{F}_{K})$ of $\mathscr{C}_{K}$ is isomorphic
to $DR_{K}$ [2, 3]; therefore, in order to prove this proposition, it suffices
to show a bijective correspondence between the first and the second only. For
this proof, however, the concept of _galois objects_ in the semi-galois
category $\mathscr{C}_{K}$ plays an auxiliary key role.

We start with the correspondence from the first to the second; and to this
end, we need to recall a few basic concepts (with notations slightly different
from those in [19]).

######  Definition 4.1.2 (monoid congruence on $I_{K}$).

Let $\Xi\subseteq E_{K}$ be a finite subset of $E_{K}$. Then, define the
monoid congruence $\mathfrak{a}\sim_{\Xi}\mathfrak{b}$ for
$\mathfrak{a},\mathfrak{b}\in I_{K}$ as follows:

$\displaystyle\mathfrak{a}\sim_{\Xi}\mathfrak{b}$
$\displaystyle\Leftrightarrow$ $\displaystyle\forall\xi\in\Xi,\hskip
5.69046pt\psi_{\mathfrak{a}}\xi=\psi_{\mathfrak{b}}\xi.$ (4.1)

Also, denote by $D_{\Xi}$ the finite quotient monoid
$D_{\Xi}:=I_{K}/\sim_{\Xi}$.

Concerning this monoid congruence, note that, for two finite subsets
$\Xi,\Xi^{\prime}\subseteq E_{K}$ with $\Xi\subseteq\Xi^{\prime}$, we have a
canonical surjective monoid homomorphism
$r^{\Xi^{\prime}}_{\Xi}:D_{\Xi^{\prime}}\twoheadrightarrow D_{\Xi}$;
therefore, with respect to the inclusion relation $\Xi\subseteq\Xi^{\prime}$
of finite subsets of $E_{K}$, we have an inverse system
$\\{D_{\Xi},r^{\Xi^{\prime}}_{\Xi}\\}$ of finite monoids $D_{\Xi}$. The fact
that $E_{K}$ is the union of the galois objects $X_{\Xi}$ of $\mathscr{C}_{K}$
(§3.1 [19]) implies in particular that $DR_{K}$ is isomorphic to the inverse
limit of this system $\\{D_{\Xi},r^{\Xi^{\prime}}_{\Xi}\\}$ when $\Xi\subseteq
E_{K}$ runs over all finite subsets of $E_{K}$; thus by restricting the range
where $\Xi\subseteq E_{K}$ are allowed to run, we naturally obtain a quotient
of $DR_{K}$:

######  Definition 4.1.3 (dual to $\Lambda$-subalgebra of $E_{K}$).

Let $E\subseteq E_{K}$ be a $\Lambda$-subalgebra of $E_{K}$; then define
$D_{E}$ as the following profinite monoid:

$\displaystyle D_{E}$ $\displaystyle:=$
$\displaystyle\lim_{\leftarrow\Xi\subseteq E}D_{\Xi}.$ (4.2)

Conversely, given a profinite quotient monoid $DR_{K}\twoheadrightarrow D$, we
construct a $\Lambda$-subalgebra $E_{D}\subseteq E_{K}$ so that this
construction $D\mapsto E_{D}$ gives the inverse of the above construction
$E\mapsto D_{E}$. To this end, recall (§2.1) that we can identify $E_{K}$ with
the $\Lambda$-ring $\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})$ of locally constant
$K^{ab}$-valued $G_{K}$-equivariant functions of $DR_{K}$; under this
identification, we can naturally describe the target correspondence $D\mapsto
E_{D}$ as follows:

######  Definition 4.1.4 (dual to quotient of $DR_{K}$).

Let $DR_{K}\twoheadrightarrow D$ be a profinite quotient monoid of $DR_{K}$;
then we define a $\Lambda$-subalgebra $E_{D}\subseteq E_{K}$ as follows:

$\displaystyle E_{D}$ $\displaystyle:=$
$\displaystyle\biggl{\\{}\xi\in\mathrm{Hom}_{G_{K}}(DR_{K},K^{ab})\mid\textrm{$\xi$
factors through $DR_{K}\twoheadrightarrow D$}\biggr{\\}}.$ (4.3)

######  Lemma 4.1.5.

For any $\Lambda$-subalgebra $E\subseteq E_{K}$ we have the identity
$E=E_{D_{E}}$.

###### Proof.

Given $\xi\in E$, the finite monoid $D_{\xi}:=D_{\\{\xi\\}}$ is by definition
a quotient of $D_{E}$. Since $\xi$, when viewed as a function on $DR_{K}$,
factors through $D_{\xi}$, it follows that $\xi$ also factors through $D_{E}$,
hence $\xi\in E_{D_{E}}$, concluding $E\subseteq E_{D_{E}}$.

Conversely, given $\xi\in E_{D_{E}}$ that is viewed as a function on $DR_{K}$,
the definition of $E_{D_{E}}$ implies that $\xi:DR_{K}\rightarrow K^{ab}$
factors through $DR_{K}\twoheadrightarrow D_{E}$; therefore, $D_{\xi}$ is a
finite quotient of $D_{E}$. Then, by definition of $D_{E}$, this also implies
that there exists a finite subset $\Xi\subseteq E$ such that $D_{\xi}$ is a
finite quotient of $D_{\Xi}$. Since $\mathrm{Hom}_{G_{K}}(D_{\Xi},K^{ab})$ is
equal to the galois object $X_{\Xi}=K[I_{K}\Xi]$ (§2.1), we obtain $\xi\in
X_{\Xi}$. But since $E$ is now a $\Lambda$-subalgebra thus closed under
$\psi_{\mathfrak{p}}$’s, we also have $X_{\Xi}\subseteq E$, hence $\xi\in
X_{\Xi}\subseteq E$, concluding $E_{D_{E}}\subseteq E$. This completes the
proof. ∎

######  Lemma 4.1.6.

For any profinite quotient monoid $DR_{K}\twoheadrightarrow D$, we have a
canonical isomorphism $D\simeq D_{E_{D}}$.

###### Proof.

First assume that two ideals $\mathfrak{a},\mathfrak{b}\in
I_{K}\hookrightarrow DR_{K}$ go into the same element in $D$ under the
quotient $DR_{K}\twoheadrightarrow D$; we show that
$\mathfrak{a},\mathfrak{b}$ go to the same element in $D_{E_{D}}$ as well
under the quotient $DR_{K}\twoheadrightarrow D_{E_{D}}$. Now the assumption
implies that the operators $\psi_{\mathfrak{a}},\psi_{\mathfrak{b}}$ define
the same action on $E_{D}=\mathrm{Hom}_{G_{K}}(D,K^{ab})$ by definition (cf.
§2.1); therefore, so they do on $X_{\Xi}\subseteq E_{D}$ for any finite subset
$\Xi\subseteq E_{D}$; in particular, we have
$\psi_{\mathfrak{a}}\xi=\psi_{\mathfrak{b}}\xi$ for any $\xi\in\Xi\subseteq
X_{\Xi}$, which means that $\mathfrak{a}\sim_{\Xi}\mathfrak{b}$. Now since
$\Xi$ is any finite subset of $E_{D}$, this eventually deduces that
$\mathfrak{a},\mathfrak{b}$ are mapped to the same element in $D_{E_{D}}$.

Conversely, assume that $\mathfrak{a},\mathfrak{b}$ are mapped onto two
different elements in $D$, whence we have a finite quotient
$D\twoheadrightarrow H$ so that $\mathfrak{a},\mathfrak{b}$ are still
different in $H$ as well; then $\psi_{\mathfrak{a}}$ and $\psi_{\mathfrak{b}}$
define different endomorphisms on the galois object
$X_{H}:=\mathrm{Hom}_{G_{K}}(H,K^{ab})$ in $\mathscr{C}_{K}$, thus we have
$\psi_{\mathfrak{a}}\xi\neq\psi_{\mathfrak{b}}\xi$ for some $\xi\in X_{H}$.
Since $X_{H}$ is now a $\Lambda$-subalgebra of $E_{D}$, this means that
$\mathfrak{a}$ and $\mathfrak{b}$ are not congruent with respect to
$\sim_{\\{\xi\\}}$ for the singleton $\\{\xi\\}\subseteq E_{D}$; that is,
$\mathfrak{a}\neq\mathfrak{b}$ in the finite monoid $D_{\\{\xi\\}}$. Again, by
definition of $D_{E_{D}}=\lim_{\Xi\subseteq E_{D}}D_{\Xi}$, this finite monoid
$D_{\\{\xi\\}}$ is a quotient of $D_{E_{D}}$. Since
$\mathfrak{a}\neq\mathfrak{b}\in D_{\xi}$, it follows that
$\mathfrak{a}\neq\mathfrak{b}\in DR_{E_{D}}$ too.

The above arguments show that the two quotients $DR_{K}\twoheadrightarrow D$
and $DR_{K}\twoheadrightarrow D_{E_{D}}$ have the same kernel on the discrete
submonoid $I_{K}$; but since $I_{K}$ is dense in $DR_{K}$, it follows that
they have the same kernel on $DR_{K}$ as well. Thus they must be isomorphic. ∎

Basically, we use the above galois correspondence in the following form:

######  Corollary 4.1.7 (dual characterization).

Let $E\subseteq E_{K}$ be a $\Lambda$-subalgebra such that $E$ corresponds to
a profinite quotient $DR_{K}\twoheadrightarrow D$ under the above galois
correspondence. Then $\xi\in E_{K}$ is a member of $E$ if and only if
$D_{\xi}$ is a quotient of $D$.

###  4.2 Characterization of modular vectors

Since the modular vectors constitute a $\Lambda$-subalgebra $M_{K}$ of
$E_{K}$, the general galois correspondence (Proposition 4.1.1) guarantees that
we have a profinite quotient $DR_{K}\twoheadrightarrow D_{M_{K}}$ so that an
algebraic Witt vector $\xi\in E_{K}$ is modular if and only if $D_{\xi}$ is a
finite quotient of $D_{M_{K}}$; that said, nevertheless, the general galois
correspondence gives us only a somewhat tautological description of
$D_{M_{K}}$, i.e. $D_{M_{K}}=\lim_{\Xi\subseteq M_{K}}D_{\Xi}$. In view of
this issue, this subsection gives in a step-by-step manner a purely adelic
description of this profinite monoid $D_{M_{K}}$, which is intrinsic for $K$
and independent of modular vectors $\Xi\subseteq M_{K}$.

As the first step, we define finite quotients $DR_{K}\twoheadrightarrow D_{N}$
for $N\geq 1$ using projective embeddings of abelian varieties so that
$D_{N}$’s constitute an inverse system of finite monoids whose inverse limit
is isomorphic to our target profinite monoid $D_{M_{K}}$ (Lemma 4.2.3); this
description of $D_{M_{K}}$ is not yet intrinsic for $K$ but we utilize it to
study the structure of $D_{M_{K}}$ via geometry of abelian varieties. Based on
this, as the second step, we describe the structure of these finite monoids
$D_{N}$ in a purely adelic term, which provides us a $K$-intrinsic description
of $D_{M_{K}}\simeq\lim_{N}D_{N}$ and concludes our dual ($K$-intrinsic)
characterization of modular vectors (cf. Theorem 4.2.8).

######  Definition 4.2.1 (congruence $\sim_{N}$).

For $[\rho,s],[\lambda,t]\in DR_{K}$, we define $[\rho,s]\sim_{N}[\lambda,t]$
if and only if the following two parallel arrows $E/O_{E}\rightarrow{\bf
P}^{d}(\mathbb{C})$ coincide on the subset $N^{-1}O_{E}/O_{E}\subseteq
E/O_{E}$:

$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{s}}$$\textstyle{A_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\lambda)}$$\scriptstyle{\eta(\rho)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\eta(t)}$$\textstyle{{\bf
P}^{d}(\mathbb{C})}$$\textstyle{E/\eta(t)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{t}}$$\textstyle{A_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$
(4.4)

######  Lemma 4.2.2.

The congruence $\sim_{N}$ is a monoid congruence on $DR_{K}$ of finite order.

###### Proof.

First to see that $\sim_{N}$ is a monoid congruence, it suffices to prove that
$[\rho,s]\sim_{N}[\lambda,t]$ implies $[\rho\nu,sr]\sim_{N}[\lambda\nu,tr]$
for any $[\nu,r]\in DR_{K}$. But this follows from the following larger
commutative diagram, where we used Shimura’s reciprocity law (cf. §2.2.3) for
the fractional ideals $\mathfrak{a}=\eta(s)O_{E}$ and
$\mathfrak{a}=\eta(t)O_{E}$ in particular; note also that
$\eta(\nu):E/O_{E}\rightarrow E/O_{E}$ maps the subset $N^{-1}O_{E}/O_{E}$ to
itself:

$\textstyle{E/\eta(sr)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{sr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{sr}}$$\textstyle{A_{sr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$$\textstyle{E/\eta(s)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\eta(r)}$$\textstyle{\mathbb{C}^{g}/\Lambda_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{s}}$$\textstyle{A_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[r]^{-1}}$$\scriptstyle{\subseteq}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\nu)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(\lambda)}$$\scriptstyle{\eta(\rho)}$$\textstyle{E/O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta(s)}$$\scriptstyle{\eta(t)}$$\scriptstyle{\eta(sr)}$$\scriptstyle{\eta(tr)}$$\textstyle{{\bf
P}^{d}(\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[r]^{-1}}$$\textstyle{{\bf
P}^{d}(\mathbb{C})}$$\textstyle{E/\eta(t)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\scriptstyle{\eta(r)}$$\textstyle{\mathbb{C}^{g}/\Lambda_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{t}}$$\textstyle{A_{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[r]^{-1}}$$\scriptstyle{\subseteq}$$\textstyle{E/\eta(tr)O_{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{\mathbb{C}^{g}/\Lambda_{tr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta_{tr}}$$\textstyle{A_{tr}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subseteq}$
(4.5)

Also, the fact that the monoid congruence $\sim_{N}$ is of finite index
follows from the same argument in the last part of the proof of Proposition
3.2.1. This completes the proof. ∎

For $N\mid N^{\prime}$, it is obvious from the definition that
$[\rho,s]\sim_{N^{\prime}}[\lambda,t]$ implies $[\rho,s]\sim_{N}[\lambda,t]$;
therefore, writing $D_{N}:=DR_{K}/\sim_{N}$, we have a surjective homomorphism
$r^{N^{\prime}}_{N}:D_{N^{\prime}}\twoheadrightarrow D_{N}$ of finite monoids
for $N\mid N^{\prime}$. This way we obtain an inverse system
$\\{D_{N},r^{N^{\prime}}_{N}\\}$ of finite monoids $D_{N}$, whose inverse
limit is in fact isomorphic to $D_{M_{K}}$.

######  Lemma 4.2.3.

We have a canonical isomorphism of profinite monoids,
$D_{M_{K}}\simeq\lim_{N\geq 1}D_{N}$.

###### Proof.

Let $X_{N}:=\\{\widehat{f}_{a}\mid a\in N^{-1}O_{E}/O_{E},\textrm{$f$ is
rational over $K$ and defined for $a$}\\}\subseteq M_{K}$; since $M_{K}$ is
the union of $X_{N}$’s, it suffices to prove that $X_{N}$ is the galois object
of $\mathscr{C}_{K}$ dual to $D_{N}$. First, by the fact that all
$\widehat{f}_{a}$ in $X_{N}$ as functions on $DR_{K}$ factor through
$DR_{NO_{K}}$, it follows that $X_{N}$ is finite etale over $K$; also, by the
same argument of Lemma 4.3.5 [19], it can be readily seen that $X_{N}$’s are
closed under $\psi_{\mathfrak{p}}$’s. From these we deduce that
$X_{N}=X_{\Xi(N)}$ for some finite set $\Xi(N)\subseteq X_{N}\subseteq M_{K}$,
hence $X_{N}$ is a galois object of $\mathscr{C}_{K}$.

The rest task is to prove that this galois object $X_{N}$ is dual to $D_{N}$;
since we already know that $X_{N}$ is galois, this is equivalent to proving
that for any $[\rho,s],[\lambda,t]\in DR_{K}$, we have
$[\rho,s]\sim_{N}[\lambda,t]$ if and only if the equality
$\widehat{f}_{a}(\rho,s)=\widehat{f}_{a}(\lambda,t)$ holds for all $a\in
N^{-1}O_{E}/O_{E}$ and rational functions $f$ over $K$ defined for $a$.

The only-if part is almost trivial: notice from the definition that
$[\rho,s]\sim_{N}[\lambda,t]$ means that each $N$-torsion point
$a_{1}\tau_{E}+a_{2}\delta\in N^{-1}O_{E}/O_{E}$ is mapped to the same point
in ${\bf P}^{d}(\mathbb{C})$ under the two maps $E/O_{E}\rightarrow{\bf
P}^{d}(\mathbb{C})$ given in (4.4); therefore, their images of
$a_{1}\tau_{E}+a_{2}\delta$ cannot be distinguished by any rational function
$f$ defined on such torsion points, which mean that we have
$\widehat{f}_{a}(\rho,s)=\widehat{f}_{a}(\lambda,t)$ for any $a\in
N^{-1}O_{E}/O_{E}$ and rational functions $f$ over $K$ defined for $a$. (Hence
$X_{N}$ is contained in the galois object $\mathrm{Hom}_{G_{K}}(D_{N},K^{ab})$
dual to $D_{N}$.)

Conversely, suppose $\widehat{f}_{a}(\rho,s)=\widehat{f}_{a}(\lambda,t)$ for
any $a\in N^{-1}O_{E}/O_{E}$ and rational functions $f$ over $K$ defined for
$a$; this means that each of the $N$-torsion points
$a_{1}\tau_{E}+a_{2}\delta\in N^{-1}O_{E}/O_{E}$ is mapped to the points, say
$p_{\rho,s}(a),p_{\lambda,t}(a)\in{\bf P}^{d}(\mathbb{C})$, by the two maps in
(4.4) respectively so that they cannot be distinguished by any such rational
functions $f$. We shall prove that, if these points
$p_{\rho,s}(a),p_{\lambda,t}(a)$ are distinct for some $a\in
N^{-1}O_{E}/O_{E}$, they can be distinguished by some rational function $f$
over $K$ defined for $a$: Since we suppose $p_{\rho,s}(a)\neq
p_{\lambda,t}(a)\in{\bf P}^{d}(\mathbb{C})$, we can take some rational
function $f_{1}:{\bf P}^{d}(\mathbb{C})\rightarrow{\bf P}^{1}(\mathbb{C})$
defined over $\mathbb{Q}$ such that $f_{1}(p_{\rho,s}(a))\neq
f_{1}(p_{\lambda,t}(a))\in{\bf P}^{1}(\mathbb{C})$. Then consider the finite
subset $S\subseteq{\bf P}^{1}(\mathbb{C})$ consisting of
$f_{1}(p_{\rho,s}(a^{\prime}))$ and $f_{1}(p_{\lambda,t}(a^{\prime}))$ for all
$a^{\prime}\in N^{-1}O_{E}/O_{E}$, which (of course) contains
$f_{1}(p_{\rho,s}(a))$ and $f_{1}(p_{\lambda,t}(a))$. If $S$ does not contain
$\infty\in{\bf P}^{1}(\mathbb{C})$, the original $f_{1}$ already gives our
desired one; otherwise, take a rational function $h:{\bf
P}^{1}(\mathbb{C})\rightarrow{\bf P}^{1}(\mathbb{C})$ over $\mathbb{Q}$ so
that all the points in $S\subseteq{\bf P}^{1}(\mathbb{C})$ are mapped
injectively into ${\bf P}^{1}(\mathbb{C})\setminus\\{\infty\\}$; then $h\circ
f_{1}:{\bf P}^{d}(\mathbb{C})\rightarrow{\bf P}^{1}(\mathbb{C})$ is our
desired one. This completes the proof. ∎

From this, it follows that an algebraic Witt vector $\xi\in E_{K}$ is modular
(i.e. $\xi\in M_{K}$) if and only if the finite monoid $D_{\xi}$ is a quotient
of $D_{N}$ for some $N\geq 1$. Recall that $D_{\xi}$ is the quotient finite
monoid $D_{\xi}:=I_{K}/\sim_{\xi}$, where the congruence $\sim_{\xi}$ is
defined as $\mathfrak{a}\sim_{\xi}\mathfrak{b}$ if and only if
$\psi_{\mathfrak{a}}\xi=\psi_{\mathfrak{b}}\xi$, or in more elementary words,
$\xi_{\mathfrak{a}\mathfrak{c}}=\xi_{\mathfrak{b}\mathfrak{c}}$ for any
$\mathfrak{c}\in I_{K}$. That is, the congruence $\sim_{\xi}$ on $I_{K}$
concerns only the pattern of whether or not the components
$\xi_{\mathfrak{a}\mathfrak{c}},\xi_{\mathfrak{b}\mathfrak{c}}$ coincide for
all $\mathfrak{c}$, and does not concern their actual values. The real
ingredient of this dual characterization is that whether $\xi\in E_{K}$ is
realized as a modular vector $\xi=\widehat{f}_{a}$ analytically can be
determined only by such a pattern of the equality between the components
$\xi_{\mathfrak{a}}$ of $\xi$.

Nevertheless, the definition of the congruence $\sim_{N}$ depends on theta
functions $\Theta_{s}$ and thus the finite monoid $D_{N}=I_{K}/\sim_{N}$ is
not yet intrinsic for $K$. The subject of the rest of this section is to
remedy this issue, giving a purely adelic description of the structure of
$D_{N}$. As in the case of our previous work [19], we study the structure of
$D_{N}$’s by looking at their _galois-orbit decompositions_ : being a finite
quotient of $DR_{K}$, the finite monoid $D_{N}$ has an action of $G_{K}$ to
form a finite $G_{K}$-set; therefore, we have its $G_{K}$-orbit decomposition
$D_{N}=\coprod_{i}C_{N}^{(i)}$, where each component $C_{N}^{(i)}$ is a finite
quotient of $G_{K}$ (or actually, of $G_{K}^{ab}$); dually, the finite
$G_{K}$-set $D_{N}$ is dual to the galois object $X_{N}$, where the
$G_{K}$-orbit decomposition $\coprod_{i}C_{N}^{(i)}$ corresponds to the
decomposition $X_{N}=\prod_{i}K_{N}^{(i)}$ as a finite product of finite
(abelian) extensions $K_{N}^{(i)}/K$ over $K$, where
$C_{N}^{(i)}=Gal(K_{N}^{(i)}/K)$. Technically we describe $D_{N}$’s by
determining the structure of these decompositions
$D_{N}=\coprod_{i}C_{N}^{(i)}$ (or dually, $X_{N}=\prod_{i}K_{N}^{(i)}$).

Concerning this problem, the following lemma is our key result:

######  Lemma 4.2.4.

For any $[\rho,s],[\lambda,t]\in DR_{K}$, we have
$[\rho,s]\sim_{N}[\lambda,t]$ if and only if there exists some
$u\in\widehat{O}_{K}^{\times}$ satisfying the following two conditions:

  1. 1.

$\eta(\rho)\eta(u)=\eta(\lambda)$ in $O_{E}/NO_{E}$;

  2. 2.

$[s]=[ut]$ in $Gal(K_{S_{\rho}}/K)$;

where $S_{\rho}\subseteq N^{-1}O_{E}/O_{E}$ denotes the image of the map
$\eta(\rho):N^{-1}O_{E}/O_{E}\rightarrow N^{-1}O_{E}/O_{E}$; also, for any
subset $S\subseteq N^{-1}O_{E}/O_{E}$, we denote by $K_{S}$ the finite abelian
extension of $K$ by the coordinates of the torsion points in $A_{E}$
corresponding to some $a\in S$.

###### Proof.

We first prove the if part; suppose that the above two conditions hold for
$[\rho,s],[\lambda,t]\in DR_{K}$ for some $u\in\widehat{O}_{K}^{\times}$, from
which we deduce $[\rho,s]\sim_{N}[\lambda,t]$. Since
$\eta(\rho)\cdot\eta(u)=\eta(\lambda)\in O_{E}/NO_{E}$, we have the following
equalities for any $k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ and $a\in
N^{-1}O_{E}/O_{E}$:

$\displaystyle\theta^{k/l}_{a}(\lambda,t)$ $\displaystyle=$
$\displaystyle\biggl{(}\theta^{k/l}_{a\eta(\lambda)}(\tau_{E})\biggr{)}^{[t]^{-1}}$
(4.6) $\displaystyle=$
$\displaystyle\biggl{(}\theta^{k/l}_{a\eta(\rho)}(\tau_{E})\biggr{)}^{[ut]^{-1}}$
(4.7)

But by $a\eta(\rho)\in S_{\rho}$ and by the second condition $[s]=[ut]$, this
last one is equal to $\theta^{k/l}_{a}(\rho,s)$, thus:

$\displaystyle\theta^{k/l}_{a}(\lambda,t)$ $\displaystyle=$
$\displaystyle\theta^{k/l}_{a}(\rho,s).$ (4.8)

Since $a\in N^{-1}O_{E}/O_{E}$ and
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ are arbitrary, this implies
that $[\rho,s]\sim_{N}[\lambda,t]$.

Conversely, we prove the only-if part: suppose that we have
$[\rho,s]\sim_{N}[\lambda,t]$, from which we shall construct
$u\in\widehat{O}_{K}^{\times}$ satisfying the above two conditions. To this
end, we note that the first condition almost immediately implies the second:
in fact, if we have $\eta(\rho u)=\eta(\lambda)$, the above argument shows
$(\theta^{k/l}_{a\eta(\rho)}(\tau_{E}))^{[s]^{-1}}=(\theta^{k/l}_{a\eta(\rho)}(\tau_{E}))^{[ut]^{-1}}$
for any $k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ and $a\in
N^{-1}O_{E}/O_{E}$; this means that $[s^{-1}ut]\in G_{K}^{ab}$ fixes the
generators of the field $K_{S_{\rho}}/K$; thus, $[s]=[ut]$ in
$Gal(K_{S_{\rho}}/K)$. Therefore, it suffices to construct
$u\in\widehat{O}_{K}^{\times}$ satisfying the first condition.

We prove this by essentially the same idea of our argument in §4.3 [19]: Note
that $[\rho,s]\sim_{N}[\lambda,t]$ implies the equality
$(\eta(\rho)O_{E},NO_{E})=(\eta(\lambda)O_{E},NO_{E})$ of the greatest common
divisors. Indeed, by the above characterization (cf. Lemma 4.2.3) of the
congruence $\sim_{N}$, the assumption $[\rho,s]\sim_{N}[\lambda,t]$ implies
the following equalities for all $a\in N^{-1}O_{E}/O_{E}$ and
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$:

$\displaystyle\theta^{k/l}_{a\eta(\rho)}(\tau_{E})$ $\displaystyle=$
$\displaystyle\biggl{(}\theta^{k/l}_{a\eta(\lambda)}(\tau_{E})\biggr{)}^{[st^{-1}]}.$
(4.9)

In particular, applying this equality to the case of $a=0$ and using a
decomposition $\eta(st^{-1})=u\cdot\alpha$ (cf. Lemma 3.2.2) together with
Shimura’s reciprocity law, we have:

$\displaystyle\theta^{k/l}_{0}(\tau_{E})$ $\displaystyle=$
$\displaystyle\theta^{k/l}_{0}(\alpha(\tau_{E})),$ (4.10)

for all $k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$. With this in mind,
suppose now that $a\eta(\lambda)=0$ for some $a\in N^{-1}O_{E}/O_{E}$; then we
can show $a\eta(\rho)=0$ for this $a$ as well. In fact, for any
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$:

$\displaystyle\theta^{k/l}_{a\eta(\rho)}(\tau_{E})$ $\displaystyle=$
$\displaystyle\theta^{k/l}_{0}(\alpha(\tau_{E}))$ (4.11) $\displaystyle=$
$\displaystyle\theta^{k/l}_{0}(\tau_{E});$ (4.12)

thus the injectivity of
$\Theta_{\tau_{E}}:\mathbb{C}^{g}/\Lambda_{E}\hookrightarrow{\bf
P}^{d}(\mathbb{C})$ implies $a\eta(\rho)=0$. The converse argument then shows
that $a\eta(\rho)=0$ if and only if $a\eta(\lambda)=0$ for all $a\in
O_{E}/NO_{E}$, which means that
$(\eta(\rho)O_{E},NO_{E})=(\eta(\lambda)O_{E},NO_{E})$ as desired.

Finally, it is easy to see that
$(\eta(\rho)O_{E},NO_{E})=(\eta(\lambda)O_{E},NO_{E})$ is equivalent to that
there exists some
$w\in(O_{E}/NO_{E})^{\times}=(\widehat{O}_{E}/N\widehat{O}_{E})^{\times}$ such
that $\eta(\rho)w=\eta(\lambda)$ in $O_{E}/NO_{E}$. If either of
$\eta(\rho),\eta(\lambda)$ is zero in $O_{E}/NO_{E}$, both of them are zero,
whence any $u\in\widehat{O}_{K}^{\times}$ satisfies the first condition; thus
suppose that neither of $\eta(\rho),\eta(\lambda)$ is zero in $O_{E}/NO_{E}$.
Now for each rational prime $p\mid N$, consider the $p$-part
$\eta_{p}:K_{p}^{\times}\rightarrow E_{p}^{\times}$ of
$\eta:\mathbb{A}_{K,f}^{\times}\rightarrow\mathbb{A}_{E,f}^{\times}$, where
$K_{p},E_{p}$ denote the completions at $p$; similarly, let
$\rho_{p},\lambda_{p}$ be the $p$-part of the non-zero integral ideles
$\rho,\lambda$, as well as $w_{p}$ for $w\in\widehat{O}_{E}$. The equality
$\eta(\rho)w=\eta(\lambda)$ in $O_{E}/NO_{E}$ implies that we have
$\eta_{p}(\rho_{p})w_{p}=\eta_{p}(\lambda_{p})+p^{v_{p}(N)}\mu_{p}$ for some
integral adele $\mu_{p}$ over $p$. Since $w_{p}$ is coprime to $p$, this shows
that $\eta_{p}(\rho_{p})$ is divisible by $p^{v_{p}(N)}$ if and only if so is
$\eta_{p}(\lambda_{p})$; otherwise, $\eta_{p}(\rho_{p}^{-1}\lambda_{p})$ is an
integral unit, whence $\rho_{p}^{-1}\lambda_{p}$ is also an integral unit.
Thus, define $u\in\widehat{O}_{K}^{\times}$ by $u_{p}=1$ if $p$ is coprime to
$N$ or $\eta_{p}(\rho_{p}),\eta_{p}(\lambda_{p})$ are divisible by
$p^{v_{p}(N)}$; otherwise, $u_{p}=\rho_{p}^{-1}\lambda_{p}$; then by
construction we have $\eta(\rho u)=\eta(\lambda)$ in $O_{E}/NO_{E}$. This
completes the proof. ∎

######  Remark 4.2.5 (components of $D_{N}$).

The above lemma implies that $[\rho,s],[\lambda,t]\in D_{N}$ belong to the
same galois-orbit component if and only if
$(\eta(\rho)O_{E},NO_{E})=(\eta(\lambda)O_{E},NO_{E})$; this is also
equivalent to that $S_{\rho}=S_{\lambda}$, where $S_{\rho}\subseteq
O_{E}/NO_{E}$ denotes the image of the map $\eta(\rho):O_{E}/NO_{E}\rightarrow
O_{E}/NO_{E}$. Also, by the second condition in the above lemma, the galois-
orbit component containing $[\rho,s]\in D_{N}$ is isomorphic to
$Gal(K_{S_{\rho}}/K)$ as a finite set. We use this decomposition to prove our
target adelic description of $D_{N}$, which should be compared to the
decomposition $DR_{NO_{K}}=\coprod_{\mathfrak{d}\mid
NO_{K}}C_{NO_{K}/\mathfrak{d}}$ that we utilized for our proof of the
modularity theorem in the case of imaginary quadratic fields $K$ (cf. e.g.
Lemma 4.3.3 [19]).

######  Corollary 4.2.6 (components of $X_{N}$).

The galois object $X_{N}\in\mathscr{C}_{K}$ dual to $D_{N}$ is of the
following form as a finite etale $K$-algebra:

$\displaystyle X_{N}$ $\displaystyle\simeq$
$\displaystyle\prod_{[\rho]_{N}}K_{S_{\rho}}$ (4.13)

where $[\rho]_{N}$ runs over the finitely many equivalence classes of
$\rho\in\widehat{O}_{K}$ with respect to the equivalence relation
$\rho\simeq_{N}\lambda$ given by
$(\eta(\rho)O_{E},NO_{E})=(\eta(\lambda)O_{E},NO_{E})$, or equivalently
$S_{\rho}=S_{\lambda}\subseteq O_{E}/NO_{E}$.

So far, we have observed that the finite monoid $D_{N}$ has a galois-orbit
decomposition of the form $\coprod Gal(K_{S_{\rho}}/K)$; therefore, an adelic
description of $D_{N}$ is now reduced to that of each component
$Gal(K_{S_{\rho}}/K)$, which is the very subject of classical theory of
complex multiplication and Shimura’s reciprocity law. In terms of ideal
groups, a description of $Gal(K_{S_{N}}/K)$ was already given in Streng
(Theorem 2.5 [15]), where $S_{N}:=N^{-1}O_{E}/O_{E}$; following his idea as
well as Shimura’s (Corollary 18.9 [13]), we prove it for any
$Gal(K_{S_{\rho}}/K)$ in the form that we need; for the following we suppose
that our CM type $(E,\Phi)$ is _primitive_ , whence
$\mathrm{End}_{\mathbb{Q}}(A_{E})\simeq E$ (cf. §8.2 [14]):

######  Proposition 4.2.7 (cf. Theorem 2.5, Streng [15]; Corollary 18.9,
Shimura [13]).

For each $\rho\in\widehat{O}_{K}$, the galois group $Gal(K_{S_{\rho}}/K)$ is
isomorphic to the quotient of $\mathbb{A}_{K,f}^{\times}$ by the following
subgroup $T_{\rho}\subseteq\mathbb{A}_{K,f}^{\times}$:

$T_{\rho}:=\bigl{\\{}s\in\mathbb{A}_{K,f}^{\times}\mid\exists r\in
E^{\times},\hskip 2.84544ptrr^{\iota}=N(sO_{K}),\hskip
2.84544ptrO_{E}=\eta(s)O_{E},\hskip 2.84544ptr\cdot a=a\cdot\eta(s)\hskip
2.84544pt(\forall a\in S_{\rho})\bigr{\\}}.$ (4.14)

###### Proof.

First, suppose that $s\in\mathbb{A}_{K,f}^{\times}$ fixes $K_{S_{\rho}}/K$, or
equivalently,
$(\theta^{k/l}_{a}(\tau_{E}))^{[s]^{-1}}=\theta^{k/l}_{a}(\tau_{E})$ for any
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$ and $a\in S_{\rho}$. Then
applying this to the case of $a=0$ in particular, we obtain:

$\displaystyle\theta_{0}^{k/l}(\alpha_{s}(\tau_{E}))$ $\displaystyle=$
$\displaystyle\theta^{k/l}_{0}(\tau_{E}),$ (4.15)

for any $k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$. By Theorem 8.10.2
[1] that the theta null values preserve the isomorphism classes of abelian
varieties, we see that $A_{\tau_{E}}$ and $A_{\tau_{s}}$ are isomorphic. Thus
we obtain an isomorphism
$h:\mathbb{C}^{g}/\Phi(O_{E})\rightarrow\mathbb{C}^{g}/\Phi(\eta(s)O_{E})$ of
complex tori, preserving their polarizations, which is given by the
multiplication of some $r\in E^{\times}$ such that $rO_{E}=\eta(s)O_{E}$ and
$\Theta_{E}(\Phi(a^{\prime}))=\Theta_{s}(\Phi(ra^{\prime}))$ for all
$a^{\prime}\in E/O_{E}$. Now for any $a\in S_{\rho}\subseteq
N^{-1}O_{E}/O_{E}\subseteq E/O_{E}$ in particular, the assumption on $s$
implies the equality
$\Theta_{s}(\Phi(ra))=\Theta_{E}(\Phi(a))=\Theta_{E}(\Phi(a))^{[s]^{-1}}=\Theta_{s}(\Phi(a\cdot\eta(s)))$;
therefore, by the injectivity of
$\Theta_{s}:\mathbb{C}^{g}/\Phi(\eta(s)O_{E})\rightarrow A_{s}$, we have
$r\cdot a=a\cdot\eta(s)$. Moreover, since the isomorphism $h$ preserves the
polarizations, it follows by Theorem 18.6 [13] that $rr^{\iota}=N(sO_{K})$,
hence $s\in T_{\rho}$. Conversely it is immediate to show that $s\in T_{\rho}$
fixes $\theta^{k/l}_{a}(\tau_{E})$ for any $a\in S_{\rho}$ and
$k,l\in\delta^{-1}\mathbb{Z}^{g}/\mathbb{Z}^{g}$. This completes the proof. ∎

Now we conclude our target adelic description of the finite monoids $D_{N}$:

######  Theorem 4.2.8 (adelic description of $D_{N}$).

For any $[\rho,s],[\lambda,t]\in DR_{K}$, we have
$[\rho,s]\sim_{N}[\lambda,t]$ if and only if there exists some
$u\in\widehat{O}_{K}^{\times}$ satisfying the following two conditions:

  1. 1.

$\eta(\rho u)=\eta(\lambda)$ in $N^{-1}O_{E}/O_{E}$;

  2. 2.

$s^{-1}ut\in T_{\rho}=T_{\lambda}$.

###### Proof.

This follows from Lemma 4.2.4 and Proposition 4.2.7. ∎

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