# Thermal state quantum key distribution

A Walton1,3, A Ghesquière1, G. Brumpton1, D Jennings1,2 and B Varcoe1 1 School
of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom
2 Department of Physics, Imperial College London, London, SW7 2AZ, United
Kingdom  3 Email<EMAIL_ADDRESS>
(27th August 2024)

###### Abstract

We analyse a central broadcast continuous variable quantum key distribution
protocol in which a beam produced by a thermal source is used to create a
secret key between two parties, Alice and Bob. A beam splitter divides the
initial beam into a pair of output beams, which are sent to Alice and Bob,
with Eve intercepting Bob’s beam. We investigate the protocol in detail,
calculating mutual informations through a pair of analytic methods and
comparing the results to the outputs of a Monte Carlo simulation of the
protocol. In a lossless system, we find that a lower bound on the key rate
remains positive in the protocol under a beam splitter attack, provided Bob
receives a nonzero proportion of the beam initially sent to him. This suggests
that the thermal state protocol could be used experimentally to produce secure
keys.

Keywords: Thermal states, QKD, Key distribution, Continuous variables,
Correlation, Computing, Quantum Key Distribution

##  1 Introduction

In quantum key distribution (QKD), two parties, Alice and Bob, want to
communicate in a secure fashion despite the presence of Eve, who is
eavesdropping on their communication channel. They do this through
establishing a cryptographic key that is known only to them and no one else
[1, 2]. However, Alice cannot simply send Bob a key over their communication
channel, as Eve will also learn the key by eavesdropping. Therefore, protocols
are needed which can distribute an identical key to Alice and Bob over an
insecure channel, without Eve discovering it.

Currently, protocols exist that can accomplish this, though many methods of
classical encryption base their security on the fact that certain mathematical
operations, such as factorising large semiprime numbers, are very difficult to
perform using current technology [3]. However there is no reason to assume
that solving these problems within a reasonable timeframe will continue to be
difficult in the future as computing power increases and new algorithms are
created.

In quantum protocols on the other hand, we make the assumption that Eve has
access to arbitrarily large amounts of computing power while still being able
to establish secure communication between Alice and Bob. This is done by
basing security on restrictions imposed by the laws of quantum mechanics [2],
such as the inability to measure a quantum state without affecting the system.
This cannot be overcome through any amount of computing power.

Currently, most QKD protocols use coherent light, produced by lasers, as a
method of generating secure keys. An example of this is the Gaussian Modulated
Coherent State (GMCS) protocol [4, 5, 6], where the key is encoded in randomly
chosen quadratures of a beam described by randomly distributed coherent
states. However, recently more analysis has been done concerning the use of
thermal states in QKD [7, 8, 9]. These involve splitting a beam emitted by a
thermal source at a beam splitter and sending the outputs to Alice and Bob
respectively. Previous work concerning thermal states showed that they exhibit
Hanbury Brown and Twiss correlations [10] when split at a beam splitter, and
quantum discord, a requirement for quantum key distribution [11].

One of the main factors limiting thermal methods is that noise and
thermalisation of states are seen as detrimental for QKD protocols [12],
however work in this area is valuable due to the widespread use of microwaves
in wireless modern communication, such as in WiFi and Bluetooth, in which
thermal state QKD could be applied. Coherent state QKD is not suitable for
these applications as the devices involved do not broadcast such states.

Here, we analyse a central broadcast protocol using a thermal input, with Eve
intercepting the beam sent to Bob in order to eavesdrop. Monte Carlo
simulations of the protocol are performed to produce sample bit strings,
setting up for future experimental work using microwave sources. The paper
begins with a brief overview of thermal states in Section 2, followed by
Section 3, which describes the setup that will be simulated, while Sections
4-6 describe the measurements and workings.

##  2 Thermal states

When written in the Fock basis, with $\hat{a}^{\dagger}$ denoting the creation
operator and
$|n\rangle=\frac{\left(\hat{a}^{\dagger}\right)^{n}}{\sqrt{n!}}|0\rangle$
describing an n-photon state, thermal states are given in the form
$\rho_{\mbox{\tiny Th}}=\sum_{n=0}^{\infty}p_{n}|n\rangle\langle n|$. Here,
$p_{n}=\frac{\exp\left(-n\beta\hbar\omega\right)}{1-\exp\left(-\beta\hbar\omega\right)}$
describes a thermal distribution where $\beta=(k_{B}T)^{-1}$ is the
thermodynamic beta. When a beam from a thermal source is input into a beam
splitter, correlations are observed in intensity measurements performed on the
output beams [7, 10, 13] which are not present when a coherent source is used.

These correlations exist due to the bunched nature of photons in thermal
light. When detecting light from a thermal source, photons are not detected in
random intervals, but are instead detected in clusters [13]. High variance in
the intensity of thermal light, which is not present with a coherent source,
is the result of this bunching.

We aim to take advantage of the correlations produced by this phenomenon to
devise a QKD protocol which produces correlated bit strings between Alice and
Bob using microwave sources. These bit strings can then be used to create a
secure key to allow private communication in the presence of an eavesdropper.

The use of thermal states differentiates this protocol from similar versions
involving modulated coherent states. Using a thermal source lets us carry out
the protocol with common microwave-based wireless communication equipment
instead of relying on fibre.

Additionally, the output of a thermal source and a Gaussian modulated coherent
source are statistically equivalent. This allows the application of security
proofs for GMCS protocols to thermal protocols. An important distinction to
note is that coherent states are superpositions of Fock states, whereas
thermal states are a mixture. This allows a Monte Carlo simulation to be used
as an appropriate method to model the protocol, through random sampling of
Fock states. Here, we will compare the outputs of such a simulation to mutual
information values predicted through two separate analytic methods.

##  3 The QKD Protocol

Figure 1: Protocol Schematic. A beam produced by a thermal source provides the
initial state $\rho$. A series of beam splitters are used to direct the beam
to Alice and Bob, with Eve performing a beam splitter attack on the channel
leading to Bob. Eve’s beam splitter has unknown transmittance and reflectance,
$\tau$ and $\mu$, while each other beam splitter is 50:50.

In the QKD protocol to be analysed, as shown in Figure 1, we use a central
broadcast system in which light from a thermal source is incident on a 50:50
beam splitter. The output beams from this splitter are sent to Alice and Bob.
An eavesdropper, Eve, uses a beam splitter attack, intercepting the beam sent
to Bob using their own beam splitter of unknown transmittance. The part of the
beam transmitted by Eve’s beam splitter continues to Bob.

Alice is considered to be in control of the initial source, the first beam
splitter, and the channels between the source and her measurement apparatus,
while Bob is in control of their beam splitter and its output channels. The
channel between the initial beam splitter and Bob is not under Alice or Bob’s
control, giving a point in the protocol where Eve may interfere with the
system.

When each person receives their beam, they use a 50:50 beam splitter to divide
the incoming signal into two outputs. Double homodyne detection is employed in
order to measure the X quadrature of one beam, and the P quadrature of the
second beam as shown in Figure 2. Each person cannot simply measure the X and
P quadratures of the single beam they receive as the quadrature operators do
not commute. This replaces the common method of measurement in QKD, in which
the variable to measure [14] (or the measurement basis [2]) is randomly
switched in order to ensure security. This method of performing measurements
in QKD without random basis switching has been previously used with success
for continuous variable QKD protocols [15].

Repeated measurements yield an array of X and P quadrature measurements for
each person. For each pair of quadrature measurement outcomes
$\left\\{x_{i},p_{i}\right\\}$, Alice, Bob and Eve each calculate
$z_{i}=\sqrt{x_{i}^{2}+p_{i}^{2}}$, producing a distribution of $z$
measurements for each person. This is converted into a bit string by having
Alice, Bob, and Eve each find the median value of their distribution, and
recording a 0 or a 1 for each $z$ value depending on if it above or below the
median. Due to the correlations in the outputs of the beam splitters with a
thermal input, this produces a string of correlated bits for each person. At
this point, if the protocol has been successfully executed, a key may be
distilled from the bit strings, allowing Alice and Bob to communicate
securely. Comparing Alice and Bob’s results for a subset of measurements
allows them to calculate correlation coefficients, to verify that a thermal
source was used and correlated bits have been transmitted.

Figure 2: Heterodyne Detection. Heterodyne; or double homodyne, detection. As
initially shown in Figure 1, the measurer splits the incoming signal at a
50:50 beam splitter, and combines the outputs with local oscillators. The X
and P quadratures can then be measured separately using two pairs of
detectors.

We performed a Monte Carlo simulation of this protocol in Python with QuTiP
[16, 17]. The initial beam is created by randomly sampling Fock state values
from the thermal state distribution, with the beam splitters randomly
splitting an input beam into a pair of outputs. With a Fock state input, the
possible output Fock states of one arm of a beam splitter is described by a
binomial distribution. One of these possible outputs is selected at random.
This describes a portion of the incident photons being transmitted through the
beam splitter, with the remaining portion being reflected. Once all the beam
splitters are applied, each person receives a string of randomly distributed
Fock state measurements. Due to thermal states being a statistical mixture of
Fock states, this is an appropriate method of modelling the system.

Figure 3: Correlation Coefficients. Changes in the correlation coefficient as
a sample set of Alice and Bob’s measurement results ($n\approx 100000$) are
offset with respect to each other. Correlated measurements are observed in the
thermal protocol when time delays are taken into account. Also shown are
sample scatter graphs comparing Alice and Bob’s measurements, with and without
offsetting the data streams.

Using a sample measurement set produced through simulation, we can verify that
Alice and Bob are receiving correlated measurements by calculating the
correlation coefficient as Alice’s data is offset relative to Bob. We can see
from Figure 3 that the correlations survive beam splitters as expected of
thermal sources [10]. Offsetting Alice and Bob’s data streams shows a clear
difference between the correlation coefficients for synchronised measurements
and random noise, which would not be observed if a coherent source were used.

Given that we have observed correlations between Alice and Bob’s data strings,
we now derive bit strings from the Fock state measurements and proceed to
calculate Shannon mutual informations to test if a secure key can be produced.

##  4 Key Rates

After performing Python simulations, the Shannon mutual information;
$I_{S}\left(A;B\right)$, is calculated using the bit strings produced by each
person. This is a measurement of the information gained about one of the
involved systems from measurement of the other system.

We begin with the definition for the Shannon entropy for a single system,
$H\left(A\right)=-\sum_{i=0}^{n-1}p_{i}\log_{2}\left(p_{i}\right)$. This
describes the uncertainty in predicting the outcome should a measurement be
performed on the system where there are n possible measurement outcomes, with
outcome $i$ having a probability $p_{i}$ of occurring. For a binary bit string
with 0 and 1 being the only possible values, this can be simplified to:

$H\left(A\right)=-p_{0}\log_{2}\left(p_{0}\right)-\left(1-p_{0}\right)\log_{2}\left(1-p_{0}\right).$
(1)

Here, $p_{0}$ is the probability of measuring the 0 outcome. From this, the
mutual information
$I_{S}\left(A;B\right)=H\left(A\right)+H\left(B\right)-H\left(AB\right)$ can
be defined, where $H\left(AB\right)$ is calculated by iterating over the four
possible outcomes of two people measuring separate bit strings. Once the
Shannon entropy is calculated for each bit string, and the mutual information
values between the bit strings for each person is measured, we can see if key
distribution can be performed. Two classical options for producing usable keys
are considered, direct reconciliation and reverse reconciliation.

In direct reconciliation, Alice openly shares additional information in order
for corrections to be made to Bob’s bit string. For this to produce a secure
key, it is required [18] that entropy calculations from the produced bit
strings satisfy $K_{DR}=I_{S}\left(A;B\right)-I_{S}\left(A;E\right)>0$.

Alternatively, reverse reconciliation is the opposite method, where Bob
provides the information in order for Alice to make corrections. In this case,
successfully creating a secure key requires [18]
$K_{RR}=I_{S}\left(A;B\right)-I_{S}\left(B;E\right)>0$. Therefore, if one of
these inequalities are satisfied, a secret key can be produced. The secret key
rate K in this case is bounded such that [19] $\max\\{K_{DR},\,K_{RR}\\}\leq
K\left(A;B|E\right)\leq\min\\{I_{S}\left(A;B\right),\,I_{S}\left(A;B|E\right)\\}$.

Here, $H(X|Y)=H\left(XY\right)-H\left(Y\right)$ describes the conditional
mutual information, the uncertainty in a system, X, given a measurement
performed in a second system. So far, the Shannon mutual information values
have been used, which can verify security in the case of an individual attack
by Eve [20], in which they perform measurements on each pulse sent by Alice
before any error correction occurs between Alice and Bob.

If neither of the above reconciliation methods are available, advantage
distillation through protocols such as Cascade will allow keys to be produced
from the bit strings provided any secrecy is present [8]. As measurements have
already taken place by this point, this is purely classical error correction.
Sharing a random subset of the bit strings allows Alice and Bob to estimate
the error rate for use in such algorithms.

While the Shannon entropy calculated through simulation is interesting, it is
more useful to analyse the protocol through the von Neumann entropy. Here, we
will compare the results of two different methods of calculating von Neumann
entropy to sample Shannon entropies produced by the simulation.

##  5 Mutual Information and State Variance

Performing an analysis similar to that done by Qi et. al. (2017) [6] we can
calculate von Neumann mutual informations, $I_{N}\left(A;B\right)$. Using
Alice’s quadrature measurements, Bob’s corresponding measurements are
estimated, along with Alice’s uncertainty on Bob’s measurements. Eve’s
interception is done with a beam splitter of transmittance $\tau$ and
reflectance $\mu$. This gives:

$\frac{A_{X}}{n_{A}}=\frac{\hat{B}}{\tau n_{B}}$ (2)

where $A_{X}$ is one of Alice’s measured X quadrature values, and $\hat{B}$ is
Alice’s estimate of the corresponding quadrature values of the modes at Bob
detector. The detector efficiency for Alice, Bob and Eve are given by
$n_{A},\>n_{B},$ and $n_{E}$ respectively. By considering asymmetric beam
splitters, continuing the analysis shows the quadrature values of the modes
received by each person’s detectors, $X_{A},$ $X_{B}$, and $X_{E}$ are found:

$X_{A}=\frac{n_{A}}{2}x_{in}+\sqrt{1-\left(\frac{n_{A}}{2}\right)^{2}}v_{A}+N_{A},$
(3)

$X_{B}=\frac{\tau n_{B}}{2}x_{in}+\sqrt{1-\left(\frac{\tau
n_{B}}{2}\right)^{2}}v_{B}+N_{B},$ (4)

$X_{E}=\frac{\mu n_{E}}{2}x_{in}+\sqrt{1-\left(\frac{\mu
n_{E}}{2}\right)^{2}}v_{E}+N_{E},$ (5)

where $x_{in}$ is the quadrature output from the source, $v_{A},$ $v_{B},$ and
$v_{E}$ describe the noise introduced at the beam splitters between the source
and each person, and loss at their detector. $N_{A},$ $N_{B},$ and $N_{E}$
describes Gaussian noise added at each person’s detector.

Taking the introduced noise to be described by a Gaussian distribution with
mean zero and variance one, we can calculate the uncertainty Alice has on
Bob’s measurements, then we can perform a similar analysis for Bob and Eve:

$\Delta_{AB}=\left\langle\left(\hat{B}-X_{B}\right)^{2}\right\rangle=\left(\frac{\tau
n_{B}}{n_{A}}\right)^{2}\left(1-\frac{n_{A}}{2}+\left\langle
N_{A}^{2}\right\rangle\right)+1+\left\langle N_{B}^{2}\right\rangle,$ (6)

$\Delta_{BE}=\left(\frac{\mu n_{E}}{\tau
n_{B}}\right)^{2}\left(1-\frac{\left(\tau n_{B}\right)^{2}}{2}+\left\langle
N_{B}^{2}\right\rangle\right)+1+\left\langle N_{E}^{2}\right\rangle.$ (7)

The mutual information for Gaussian states can be shown to be [6]:

$I_{N}\left(A:B\right)=\frac{1}{2}\log_{2}\left(\frac{V+\chi}{1+\chi}\right),$
(8)

where V is the variance of the input thermal state and $\chi$ is the added
noise. If $\chi_{line}$ is the noise added in the channels, and $\chi_{hom}$
is the detection noise, the total added noise in a channel with transmittance
T is given by [6]:

$\chi=\chi_{line}+\frac{\chi_{hom}}{T},$ (9)

$\chi_{line}=\frac{1}{T}-2+\Delta,$ (10)

$\chi_{hom}=\frac{1+\left\langle N^{2}\right\rangle}{n_{B}}-1,$ (11)

where we have taken $T=1$ and assumed equal detector noise for each person,
such that $\left\langle N_{A}^{2}\right\rangle=\left\langle
N_{B}^{2}\right\rangle=\left\langle N_{E}^{2}\right\rangle=\left\langle
N^{2}\right\rangle=1$. This simplified setup gives $\chi=\Delta$. Figure 4
shows the plots of various mutual information values as variance is adjusted.
Eve’s beam splitter is assumed to be 50:50.

Figure 4: Mutual information calculations using uncertainty. The von Neumann
mutual informations $I_{N}\left(A;B\right)$, $I_{N}\left(B;E\right)$ as the
variance of the input thermal state, $\langle n\rangle\left(\langle
n\rangle+1\right)$ is changed. This is calculated through uncertainty in Alice
and Eve’s estimates of Bob’s measurements.

##  6 Covariance Matrix Description

To verify this behaviour, we can use a second method to calculate von Neumann
entropies, also extending Section 4 to allow for entropy calculations using
the quantum state of the system, rather than Shannon entropies of
measurements. As the states involved in this protocol are Gaussian, they can
be completely described with covariance matrices. For an N-mode state $\rho$,
the covariance matrix $\gamma$ is defined as [21]:

$\gamma_{ij}=\Tr\left[\rho\frac{1}{2}\left\\{\left(\hat{r}_{i}-d_{i}\right),\>\left(\hat{r}_{j}-d_{j}\right)\right\\}\right].$
(12)

Here, $r=\left(\hat{X}_{1},\hat{P}_{1},...,\hat{X}_{N},\hat{P}_{N}\right)$
consists of a pair of quadrature operators for each mode, $\hat{X}_{i}$ and
$\hat{P_{i}}$, with $d_{i}=\left\langle\hat{r}_{i}\right\rangle$ denoting
their expectation values. For the inputs into the initial splitter, the
covariance matrix $\gamma_{12}$ is given by:

$\gamma_{12}=\left[\begin{array}[]{cccc}V&0&0&0\\\ 0&V&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{array}\right]$ (13)

Where $V$ is the variance of the quadratures of the beam output by the thermal
source. This fully describes the thermal and vacuum modes input into the
initial beam splitter. Through applying the beam splitter transformations to
the relevant modes, the covariance matrix of the final state is calculated.
The transformation, S, for a beam splitter with transmittance $\tau$ and
reflectance $\mu$ is given by [21]:

$S\left(\tau,\mu\right)=\left[\begin{array}[]{cc}\tau&\mu\\\
-\mu&\tau\end{array}\right]\otimes I,$ (14)

By applying the beam splitters to the appropriate modes, the final covariance
matrix can be found:

$\gamma_{A_{1}A_{2}B_{1}B_{2}E_{1}E_{2}}=\left[\begin{array}[]{ccc}\gamma_{A_{1}A_{2}}&C_{AB}&C_{AE}\\\
C_{AB}^{T}&\gamma_{B_{1}B_{2}}&C_{BE}\\\
C_{AE}^{T}&C_{BE}^{T}&\gamma_{E_{1}E_{2}}\end{array}\right].$ (15)

The sub-matrices are given by:

$\gamma_{A_{1}A_{2}}=\left[\begin{array}[]{cc}\frac{1}{4}\left(V+3\right)&-\frac{1}{4}\left(V-1\right)\\\
-\frac{1}{4}\left(V-1\right)&\frac{1}{4}\left(V+3\right)\end{array}\right]\otimes
I,$ (16)

$\gamma_{B_{1}B_{2}}=\left[\begin{array}[]{cc}\frac{\tau^{2}}{4}\left(V+1\right)+\frac{1+\mu^{2}}{2}&-\frac{\tau^{2}}{4}\left(V+1\right)+\frac{1-\mu^{2}}{2}\\\
-\frac{\tau^{2}}{4}\left(V+1\right)+\frac{1-\mu^{2}}{2}&\frac{\tau^{2}}{4}\left(V+1\right)+\frac{1+\mu^{2}}{2}\end{array}\right]\otimes
I,$ (17)

$\gamma_{E_{1}E_{2}}=\left[\begin{array}[]{cc}\frac{\mu^{2}}{4}\left(V+1\right)+\frac{1+\tau^{2}}{2}&-\frac{\mu^{2}}{4}\left(V+1\right)+\frac{1-\tau^{2}}{2}\\\
-\frac{\mu^{2}}{4}\left(V+1\right)+\frac{1-\tau^{2}}{2}&\frac{\mu^{2}}{4}\left(V+1\right)+\frac{1+\tau^{2}}{2}\end{array}\right]\otimes
I,$ (18)

$C_{AB}=\left[\begin{array}[]{cc}\frac{\tau}{4}\left(1-V\right)&-\frac{\tau}{4}\left(1-V\right)\\\
-\frac{\tau}{4}\left(1-V\right)&\frac{\tau}{4}\left(1-V\right)\end{array}\right]\otimes
I,$ (19)

$C_{AE}=\left[\begin{array}[]{cc}-\frac{\mu}{4}\left(1-V\right)&\frac{\mu}{4}\left(1-V\right)\\\
\frac{\mu}{4}\left(1-V\right)&-\frac{\mu}{4}\left(1-V\right)\end{array}\right]\otimes
I,$ (20)

$C_{BE}=\left[\begin{array}[]{cc}-\frac{\tau\mu}{4}\left(V-1\right)&\frac{\tau\mu}{4}\left(V-1\right)\\\
\frac{\tau\mu}{4}\left(V-1\right)&-\frac{\tau\mu}{4}\left(V-1\right)\end{array}\right]\otimes
I.$ (21)

Here, $\gamma_{A_{1}A_{2}}$ is the covariance matrix describing the two modes
Alice receives at their pair of detectors, with $C_{AB}$ describing covariance
between Alice’s modes and Bob’s. The remaining sub-matrices are similarly
defined. From this, von Neumann entropy values are calculated using symplectic
eigenvalues. For a covariance matrix $\gamma$, the von Neumann entropy is
given by [21]:

$S_{N}\left(\gamma\right)=\sum_{i}G\left(\frac{\lambda_{i}-1}{2}\right)$ (22)

Where $G\left(x\right)=\left(x+1\right)\log_{2}\left(x+1\right)-x\log_{2}x$,
and $\lambda_{i}$ are the symplectic eigenvalues of $\gamma$. For the
covariance matrix for a single mode system, $\gamma_{1}$, this is given by
$\lambda^{2}=\left|\gamma_{1}\right|$. For a two-mode state with the
covariance matrix $\gamma_{12}$, taking
$\Delta=\left|\gamma_{1}\right|+\left|\gamma_{2}\right|-2\left|C\right|$
allows the two symplectic eigenvalues, $\lambda_{+}$ and $\lambda_{-}$ to be
calculated:

$\left(\lambda_{\pm}\right)^{2}=\frac{1}{2}\left(\Delta\pm\left[\Delta^{2}-4\left|\gamma_{12}\right|\right]^{\frac{1}{2}}\right).$
(23)

Mutual informations calculated in this way can be plotted against variance,
this is displayed in Figure 5. Upper and lower bounds can be placed on the
mutual information values in the same manner as when Shannon entropies were
used. However, requiring
$K_{RR}=I_{N}\left(A;B\right)-I_{N}\left(B;E\right)>0$, where
$I_{N}\left(A;B\right)=S_{N}\left(\gamma_{A}\right)+S_{N}\left(\gamma_{B}\right)-S_{N}\left(\gamma_{AB}\right)$,
allows for security against a stronger set of attacks. In the case of these
”collective attacks”, Eve does not perform measurements until after classical
communication between Alice and Bob has occurred. In the example shown in
Figure 5, it can be seen that as the variance of the thermal state is
increased, the protocol remains secure in the case Eve uses a 50:50 beam
splitter. In this case, reverse reconciliation is used as $K_{RR}$ is
positive. Additionally, analysing either the covariance matrix or measurement
uncertainty both produce mutual information graphs which follow similar
patterns as variance is increased.

Figure 5: Mutual information calculations using covariance. Von Neumann mutual
information calculations plotted against thermal state variance, found using
the covariance matrix of the final state. Here, Eve performs interception
using a 50:50 beam splitter.

With two methods of calculating von Neumann entropy displaying similar
behaviour, we may now compare the outputs to Shannon entropies calculated
through the simulation.

##  7 Results

We performed calculations of the Shannon entropy for the three bit strings and
the mutual information between each pair of strings. These strings were
produced through the Python simulation. Currently no loss or noise is
considered, this allows the simulation to be performed with a lossless Eve.
The transmittance of Eve’s splitter is varied to measure the effect of Eve’s
interception strength on the mutual information values. Also calculated was
the von Neumann entropy of each mode and pair of modes, using equation 22 and
the covariance matrix describing the final state of the system. The results of
these calculations and measurements are shown in Figure 6.

Meeting the restrictions placed on von Neumann mutual information which ensure
secrecy in the case of a collective attack,
$I_{N}\left(A;B\right)-I_{N}\left(B;E\right)>0$ or
$I_{N}\left(A;B\right)-I_{N}\left(A;E\right)>0$, allow the protocol to be
secure against a greater range of attacks than the restrictions based on
Shannon entropy. Both are included here so that it can be seen that changes in
von Neumann entropies are reflected in the Shannon entropies of the bit
strings derived by each person after the protocol has been carried out.

It can be seen from Figure 6 that $I\left(A;B\right)-I\left(A;E\right)$
crosses zero in both cases when Eve’s beam splitter reflects half of the beam
sent to Bob. This is expected as Bob and Eve’s positions in the protocol are
interchangeable in this special case, so
$I\left(A;B\right)=I\left(A;E\right)$. If over half of Bob’s beam is reflected
by Eve, a key cannot be produced via direct reconciliation. However, the
second possible requirement of $I\left(A;B\right)-I\left(B;E\right)>0$ is
always satisfied in the no-loss scenario provided Eve’s interception beam
splitter has nonzero transmittance. This means that reverse reconciliation may
be used to produce a secret key and establish secure communication during
collective or individual attacks. This would allow the thermal state central
broadcast protocol to be used as a method of quantum key distribution. It is
also clear that the von Neumann entropy has a higher magnitude than Shannon,
this is expected due to the presence of discord in the system, which the
Shannon entropy does not consider.

Figure 6: Information with varying interception strength. Calculations of von
Neumann entropy using the covariance matrix and measurements of Shannon
entropy taken from the simulation as the transmittance of Eve’s beam splitter
is varied. The notable result is that $I\left(A;B\right)-I\left(B;E\right)>0$
holds in both cases provided Bob receives a nonzero proportion of the signal
sent to him. This used an average photon number of 200. Error bars describe
one standard deviation. Due to discord in the system, von Neumann entropies
have larger magnitudes than Shannon entropies.

##  8 Conclusions

When considering a system without loss and noise, the lower bound placed on
the key rate when reverse reconciliation is considered is positive under a
beam splitter attack even when Eve has zero loss. This would allow for a
secret key to be produced between Alice and Bob, and therefore secure
communication could take place in the presence of an eavesdropper performing
collective or individual attacks. Additionally, two separate methods of von
Neumann mutual information analysis both showed that thermal sources with
higher variance than those that could be produced in the Monte Carlo
simulation allowed for a key to be produced with little change in the lower
bound of the key rate.

Future work in this area could focus on examining the effects of adding noise
and loss into different channels, checking if a positive key rate could be
maintained. This is especially relevant for thermal states due to added noise
being a large barrier to successful QKD. Additionally, a practical setup
following the diagram shown in Figure 1 would allow the protocol to be
performed experimentally. This allows real key rates to be measured and would
show if the protocol continues to be functional when using thermal sources
likely to be employed in modern communication.

## Acknowledgements

Work was undertaken on ARC4, part of the High Performance Computing facilities
at the University of Leeds, UK. DJ is supported by the Royal Society and also
a University Academic Fellowship. This work was supported by the Northern
Triangle Initiative Connecting capability fund as well as funding from the UK
Quantum Technology Hub for Quantum Communications Technologies EP/M013472.
Data used to plot the Shannon entropy graph in Figure 6 is available from the
Research Data Leeds Repository with the identifier https://doi.org/10.5518/944
[22].

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# Modular curves and their pseudo-analytic cover

B.Zilber111Supported by EPSRC Program grant “Symmetries and Correspondences”
and C.Daw222Supported by EPSRC New Investigator award EP/S029613/1

We find a natural $L_{\omega_{1},\omega}$-axiomatisation $\Sigma$ of a
structure on the upper half-plane $\mathbb{H}$ as the covering space of
modular curves. The main theorem states that $\Sigma$ has a unique model in
every uncountable cardinal.

The proof relies heavily on the theory of complex multiplication and the work
on Langland’s conjecture on the conjugation of Shimura varieties. We also use
the earlier work on a related problem by C.Daw and A.Harris. The essential
difference between the setting of this work and that of the current paper is
that the former was in the language which named the CM-points of the modular
curves while our results here are over ${\mathbb{Q}}.$

##  1 Introduction

###### 1.1

Motivation and origins. The original aim of the paper is the extension of the
project categoricity of non-elementary theories of analytic covers from
“abelian” cases, such as $\exp:{\mathbb{C}}\to\mathbb{G}_{m}$ or
$\exp_{\Lambda}:{\mathbb{C}}\to\mathrm{E}_{\Lambda}\ (\mbox{elliptic curve}),$
to hyberbolic curves and possibly wider, see e.g. the survey part in [1] for
some history and references.

The first obstruction for this project is that an adequate formalism (that is
the language) in which such an analytic cover can be properly presented is not
easy to determine. Some attempts in this direction were by M.Gavrilovich [2],
as well as later attempts by A.Harris [3]. The one which we found satisfactory
and applied here is based on the formalism close to the one applied in the
recent [4].

More specifically, we formalise e.g. the case
$\exp:{\mathbb{C}}\to\mathbb{G}_{m}$ as a structure with two sorts
${\mathbb{U}}$ and $\mathrm{F},$ where ${\mathbb{U}}$ is the complex numbers
with the ${\mathbb{Q}}$-module structure and the distinguished subgroup $2\pi
i{\mathbb{Z}},$ and $\mathrm{F}$ is the complex numbers as the field. Then
$\exp:{\mathbb{U}}\to\mathbb{G}_{m}(\mathrm{F})$ is the map between the two
sorts. But in fact along with $\exp$ we automatically get in abelian cases the
family

$\exp_{n}:{\mathbb{U}}\to\mathbb{G}_{m}(\mathrm{F}),\ \
\exp_{n}:x\mapsto\exp(\frac{x}{n})$

which of course agrees with the system of finite covers

${\rm pr}_{k,n}:\mathbb{G}_{m}(\mathrm{F})\to\mathbb{G}_{m}(\mathrm{F});\
y\mapsto y^{\frac{k}{n}},\mbox{ when }n|k.$

This we describe axiomatically by simple $L_{\omega_{1},\omega}$ sentences. In
terms of model-theoretic classification such a structure is a fusion between a
locally modular structure ${\mathbb{U}}$ and an algebraically closed field
$\mathrm{F}.$ The case $\exp_{\Lambda}:{\mathbb{C}}\to\mathrm{E}_{\Lambda}$
similarly represents a fusion between a locally modular structure
${\mathbb{U}}_{\Lambda},$ and an algebraically closed field $\mathrm{F},$
where ${\mathbb{U}}_{\Lambda}$ is a ${\mathbb{Q}}$-module with distinguished
${\mathbb{Z}}$-module $\Lambda$ with an alternating form on it (accounting for
the Weil pairing).

###### 1.2

It is not clear a priori why such $L_{\omega_{1},\omega}$-theories should be
categorical in uncountable cardinals and the fact that they are must be of
some significance.

The geometric value of the project is perhaps in the fact that the formulation
of the categorical theory of the universal cover of a variety ${\mathbb{X}}$
(essentially the description of ${\mathbb{U}}$) is essentially a formulation
of a complete formal invariant of ${\mathbb{X}}.$ Indeed, the most interesting
outcome of the earlier works was establishing an equivalence between
categoricity of the cover of $\mathrm{E}_{\Lambda}$ and the conjuction of the
two arithmetic facts:

(i) the complete classification of the Galois action on the torsion of
$\mathrm{E}_{\Lambda}$ (the open image theorem by J.-P.Serre) and

(ii) the Kummer theory for $\mathrm{E}_{\Lambda}$ (M.Bashmakov and K.Ribet)

For abelian varieties the success of the program depends on the extension of
analogues of (i) and (ii) to abelian varieties, and (ii) is known due to
K.Ribet and M.Larsen. However, an analogue of Serre’s theorem for abelian
varieties is an open problem and therefore the best categoricity result here
is under the assumption that the language names points of the kernel of the
exponential map. This autmomatically removes the problem of determining the
Galois action on the torsion points at the cost of weakening the formal
invariant of ${\mathbb{X}}/\mathrm{k},$ the theory of the cover of
${\mathbb{X}}/\mathrm{k},$ to the formal invariant of ${\mathbb{X}}/K$ where
$K$ is the field obtained from $\mathrm{k}$ by adjoining all torsion points of
${\mathbb{X}}.$

The case of hyperbolic curves ${\mathbb{X}}$ was first considered by A.Harris
[3] and C.Daw and A.Harris [13] in the context of modular curves
$\mathbb{Y}(\Gamma)$ in the formalism (the choice of the language) which has
names for each element of $\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ acting on the
upper half plane $\mathbb{H}$ by Möbius transformations. The proof of
categoricity in this setting required essentially Serre’s open image theorem
for products of non-CM elliptic curves. Interestingly, in the analysis of
$(\mathbb{H},j_{N},\mathbb{Y}(N))$ Serre’s theorem plays rather the role of
(ii), while (i) is not needed since naming elements of
$\mathrm{GL}_{2}({\mathbb{Q}})$ fixes the special points (CM-points). This has
a cost: one works out the formal invariant of $\mathbb{Y}(\Gamma)$ over the
extension of the natural field of definition by special points.

###### 1.3

Our setting. Our current interest is the case of the universal cover of
hyperbolic curves, e.g. the complex curve
${\mathbb{X}}={\mathbf{P}}^{1}\setminus\\{0,1,\infty\\}.$ However, before
approaching this case we set ourselves a simpler task of the cover of the
modular curve $\mathbb{Y}(N)$ universal in the class of modular curves, which
means that our structure incorporates the analytic covering maps

$j_{n}:\mathbb{H}\to\mathbb{Y}(n),\mbox{ for all $n$ such that }N|n$

agreeing with the algebraic finite covers

${\rm pr}_{n,m}:\mathbb{Y}(n)\to\mathbb{Y}(m),\mbox{ for all $n,m$ such that
}N|m,\ m|n.$

In fact it is enough to classify the case $N=1,$
$\mathbb{Y}(1)=\mathbb{Y}=\mathbf{A}^{1}$ the affine line (the fact that for
some $n$ the covers $j_{n}:\mathbb{H}\to\mathbb{Y}(n)$ are ramified does not
matter in our setting).

Note that $\mathbb{Y}(2)={\mathbf{P}}^{1}\setminus\\{0,1,\infty\\}$ when we
consider $\mathbb{Y}(2)$ as an algebraic curve (without the level structure).

The important difference with the case of the proper universal cover is that,
instead of the profinite completion $\hat{\Gamma}(N)$ of the respective
fundamental group, in the modular setting one has the group
$\tilde{\Gamma}(N),$ the completion in the topology based on congruence
subgroups, which for $N=1$ gives us

$\tilde{\Gamma}(1)=\tilde{\Gamma}=\mathrm{SL}(2,\hat{{\mathbb{Z}}}),$

where

$\hat{{\mathbb{Z}}}=\lim{\mathbb{Z}}/n{\mathbb{Z}},$

the projective limit of residue rings.

###### 1.4

The key problem, similar to earlier cases, is in classifying the saturated
version of $\mathbb{H}$ in the structure
$(\mathbb{H},j_{n},\mathbb{Y}(n))_{n\in{\mathbb{N}}}$ which is essentially
reducible to understanding the structure on the projective limit

$\tilde{\mathbb{H}}:=\lim_{\leftarrow}\Gamma(n)\backslash\mathbb{H}\cong\lim_{\leftarrow}\mathbb{Y}(n).$

This includes giving a detailed enough description of the action of the
automorphism group $\mathrm{Aut}\,{\mathbb{C}}$ on $\tilde{\mathbb{H}}$ and,
in particular, its action on $\tilde{\mathbb{H}}(\mathrm{CM}),$ the special
points of the structure (equivalently, the action of
$\mathrm{Gal}({\mathbb{Q}}(\mathrm{CM}):{\mathbb{Q}})$ on
$\tilde{\mathbb{H}}(\mathrm{CM})$), the analogue of (i) of 1.2.

It turns out that this problem is closely connected to the problem/conjecture
on conjugation of Shimura varieties posed by R.Langlands in [5] and solved in
various degrees of generality by G.Shimura, P.Deligne, M.Borovoi, J.Milne and
others; see [8] in a most general form, and [9] for the case of Shimura
curves. In the interpretation of Borovoi [7] Langlands’ conjecture amounts to
characterising $\tilde{\mathbb{H}}$ as a structure with $\mathbb{H},$ its
substructure, and describing the way in which $\mathbb{H}$ is conjugated
inside $\tilde{\mathbb{H}}$ by $\mathrm{Aut}\,{\mathbb{C}}$ (in the case of
general Shimura varieties). We believe that the categorical axiomatisation of
$(\mathbb{H},j_{n},\mathbb{Y}(n))_{n\in{\mathbb{N}}}$ established in the paper
implies Langlands’ conjecture (for modular curves) but leave the proof of this
statement for future work.

The theory resulting from Langlands’ Conjecture proves uniqueness of so called
canonical models of Shimura varieties. This involves an advanced theory of
complex multiplication and Artin’s reciprocity map. The results allow one to
identify the action of $\mathrm{Aut}\,{\mathbb{C}}$ on a single CM-point (that
is the Galois orbit of the point), see [8], 12.8. We need a stronger result:
our paper [10] goes further and identifies the action of
$\mathrm{Gal}({\mathbb{Q}}(\mathrm{CM}):{\mathbb{Q}})$ on
$\tilde{\mathbb{H}}(\mathrm{CM}),$ at the same time describing all the
relations between CM-points. One of the interesting model-theoretic forms of
the main result of [10] is that the field ${\mathbb{Q}}(\mathrm{CM})$ as a
structure is $L_{\omega_{1},\omega}$-bi-interpretable with a certain structure
formulated purely in terms of the ring $\mathbb{A}_{f}$ of finite adeles over
${\mathbb{Q}}.$

Theorem 4.1 in section 4 below formulates the main result of [10] in the form
suitable for the current purposes. In particular, for a certain locally
modular structure of trivial type on the set $\tilde{\mathbb{H}},$ which we
term $\tilde{\mathbb{H}}^{\mathrm{Pure}},$ the theorem states:

$\mathrm{Aut}\,\tilde{\mathbb{H}}^{\mathrm{Pure}}(\mathrm{CM})\cong\mathrm{Gal}({\mathbb{Q}}(\mathrm{CM}):{\mathbb{Q}}).$

The structure can be formulated in terms of the action of a large subgroup
$\tilde{\mathrm{G}}$ of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ on
$\tilde{\mathbb{H}}$ and has a “standard” version $\mathbb{H}^{\mathrm{Pure}}$
which is based on the actual upper half-plane $\mathbb{H}$ and is given in
terms of the action of $\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ and complex
multiplication.

###### 1.5

With Theorem 4.1 in hand we can apply the model-theoretic techniques on
categoricity developed in earlier works (see e.g. the survey in [1]) and in
particular the proof in [13] which we follow quite closely (and so use the
Serre open image theorem).

The axioms $\Sigma$ of the resulting $L_{\omega_{1},\omega}$-theory consist,
as in abelian examples, of three parts:

  * A

The axioms describing ${\mathbb{U}}^{\mathrm{Pure}}$ (the abstract version of
$\mathbb{H}^{\mathrm{Pure}}$), a locally modular structure of trivial type
with the action of the group $\mathrm{G}$ isomorphic to
$\mathrm{GL}_{2}^{+}({\mathbb{Q}}),$ and its subgroups corresponding to
$\Gamma(n).$

  * B

The axioms describing an algebraically closed field $\mathrm{F}$ of
characteristic 0 and curves
$\mathbb{Y}(n)\subset{\mathbf{P}}^{3}(\mathrm{F}).$

  * C

The axioms describing $j_{n}:{\mathbb{U}}^{\mathrm{Pure}}\to\mathbb{Y}(n)$
obtained via the translation of relations on ${\mathbb{U}}^{\mathrm{Pure}}$ to
special relations on the $\mathbb{Y}(n).$

Main Theorem. The above system of axioms is satisfied by the standard complex
structure $\mathbb{H}$ and has, up to isomorphism, a unique model in any
uncountable cardinal.

###### 1.6

The first author whishes to express his gratitude to J.Derakhshan for his
interest in the work and his help with some of the mathematical issues in the
paper.

The second author would like to thank the EPSRC for its support via a New
Investigator Award (EP/S029613/1). He would also like to thank the University
of Oxford for having him as a Visiting Research Fellow.

##  2 Groups acting on $\mathbb{H}.$

###### 2.1

Groups $\mathrm{G}$ and $\Gamma$ and their generators.

Generators of $\Gamma:=\mathrm{SL}_{2}({\mathbb{Z}})$ are represented by
matrices

$\mathbf{s}=\left(\begin{array}[]{ll}0-1\\\ 1\ \ \ 0\end{array}\right),\ \
\mathbf{t}=\left(\begin{array}[]{ll}1\ 1\\\ 0\ 1\end{array}\right):$
$\mathbf{s}^{2}=-\mathbf{I},\ \
(\mathbf{s}\mathbf{t})^{3}=\left(\begin{array}[]{ll}0-1\\\ 1\ \ \
1\end{array}\right)^{3}=-\mathbf{I}.$

Generators of $\mathrm{G}:=\mathrm{GL}_{2}^{+}({\mathbb{Q}}):$

$\mathbf{s},\mathbf{t}\mbox{ and }\mathbf{d}_{q}:=\left(\begin{array}[]{ll}q\
\ 0\\\ 0\ \ 1\end{array}\right),q\in{\mathbb{Q}}_{+}$

satisfying:

$\mathbf{d}_{q}\mathbf{d}_{r}=\mathbf{d}_{qr},$
$\mathbf{s}\mathbf{d}_{q}=q\mathbf{d}_{q}^{-1}\mathbf{s},$
$\mathbf{d}_{n}\mathbf{t}=\mathbf{t}^{n}\mathbf{d}_{n},\mbox{ for
}n\in{\mathbb{N}}.$

We will use

$\mathbf{d}^{\prime}_{q}:=\mathbf{s}\mathbf{d}_{q}\mathbf{s}^{-1}=\left(\begin{array}[]{ll}1\
\ 0\\\ 0\ \ q\end{array}\right),q\in{\mathbb{Q}}_{+}$
$\mathbf{t}_{-}=\left(\begin{array}[]{ll}\ 1\ \ \ 0\\\ -1\ \
1\end{array}\right),$

which satisfies

$\mathbf{s}\mathbf{t}\mathbf{s}^{-1}=\mathbf{t}_{-};\ \
\mathbf{d}_{n}\mathbf{t}_{-}^{n}=\mathbf{t}_{-}\mathbf{d}_{n},\mbox{ for
}n\in{\mathbb{N}},$

and subgroups

$\Delta({\mathbb{Q}}_{+})=\\{\mathbf{d}_{q}:\ q\in{\mathbb{Q}}_{+}\\}\mbox{
and }\Delta^{\prime}({\mathbb{Q}}_{+})=\\{\mathbf{d}^{\prime}_{q}:\
q\in{\mathbb{Q}}_{+}\\}.$ (1)

Similar notation also make sense for the multiplicative group of a commutative
ring $R,$

$\Delta(R^{\times})=\\{\mathbf{d}_{q}:\ q\in R^{\times}\\}\mbox{ and
}\Delta^{\prime}(R^{\times})=\\{\mathbf{d}^{\prime}_{q}:\ q\in R^{\times}\\}.$

###### 2.2

Remark. Note that all automorphisms of $\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ are
inner and the only non-identity automorphism which fixes $\Delta^{\prime}$
element-wise and preserves the subgroup $\Gamma$ is the involution

$g\mapsto\check{g}:=\mathbf{d}_{-1}\cdot g\cdot\mathbf{d}_{-1}.$

###### 2.3

Special points on $\mathbb{H}.$ Let

$\mathrm{E}=\\{\left(\begin{array}[]{ll}a\ b\\\ c\
d\end{array}\right)\in\mathrm{G}:(d-a)^{2}<-4bc\\}=\\{e\in\mathrm{G}:\mathrm{tr}^{2}\,e<4\det
e\\},$

the set of elliptic transformations.

These are exactly the elements for which there is a unique fixed point
$\tau_{e}\in\mathbb{H}$ which can be found by solving the equation

$cx^{2}+(d-a)x-b=0,\ \ \tau_{e}:=x,\ \ \Im x>0.$ (2)

Note that elements of the centre

$\mathrm{Z}=\\{\left(\begin{array}[]{ll}a\ 0\\\ 0\
a\end{array}\right)\in\mathrm{G}\\}$

act as identity on $\mathbb{H}.$

###### 2.4

Remark. (i) $\mathrm{E}$ is invariant under automorphisms of $\mathrm{G}$
since $g\cdot e\cdot g^{-1}$ fixes $g\tau_{e}$ and thus belongs to
$\mathrm{E}.$

(ii) The subgroup $\mathrm{St}_{\tau}\subset\mathrm{G}$ fixing point
$\tau=\tau_{e}$ is definable by the condition that $\mathrm{St}_{\tau}=C(e),$
the centraliser of $e$ in the group. This is a general fact for Mobius
transformations over any field of characteristic 0.

##  3 Projective limit

###### 3.1

We consider the structure $\tilde{\mathbb{H}},$ the projective limit of
structures

$\tilde{\mathbb{H}}=\lim_{\leftarrow}\Gamma(N)\backslash\mathbb{H},$ (3)

along the projective system of

$\Gamma(N)=\left\\{\left(\begin{array}[]{ll}a\ b\\\ c\
d\end{array}\right)\in\Gamma,\ \ a\equiv d\equiv 1\mod N,\ c\equiv b\equiv
0\mod N\right\\},$

normal subgroups of $\Gamma=\mathrm{SL}(2,{\mathbb{Z}}).$

The structures $\Gamma(N)\backslash\mathbb{H}$ are identified with the
classical complex modular curves $\mathbb{Y}(N)$ (without the full level $N$
structure). These are non-singular algebraic curves and therefore can be
realised as quasi-projective algebraic curves in
${\mathbf{P}}^{3}({\mathbb{C}}).$ Moreover, the realisation can be obtained
over ${\mathbb{Q}},$ see e.g. section 4 of [11] or the paper [12], section 2.
Another argument for this fact is given in [10], 3.2(8).

The classical modular functions

$j_{N}:\mathbb{H}\to\mathbb{Y}(N);\ \ \tau\mapsto\Gamma(N)\cdot\tau$

are holomorphic and the curves $\mathbb{Y}(N)$ are finite coverings of
$\mathbb{Y}(N/d),$ for $d|N,$ via the projection maps

${\rm pr}_{N,N/d}:j_{N}(\tau)\mapsto j_{N/d}(\tau),\ \ \
\mathbb{Y}(N)\to\mathbb{Y}(N/d).$ (4)

###### 3.2

The definition (3) implies the existence of maps

${j}_{N}:\tilde{\mathbb{H}}\to\Gamma(N)\backslash\mathbb{H}=\mathbb{Y}(N)$

which we are going to use below.

By definition, any $\tau\in\tilde{\mathbb{H}}$ is uniquely determined by the
sequence

${j}_{n}(\tau)\in\mathbb{Y}(n),\ n\in{\mathbb{N}}$

and the sequence has the property

${\rm pr}_{n,n/d}({j}_{n}(\tau))={j}_{n/d}(\tau),\mbox{ for each }d|n.$

###### 3.3

Lemma. Any $\tau\in\mathbb{H}$ is uniquely determined by the sequence

${j}_{n}(\tau)\in\mathbb{Y}(n):\ n\in{\mathbb{N}}.$

This gives the canonical embedding

$\mathbb{H}\hookrightarrow\tilde{\mathbb{H}}.$

Proof. The first statement follows from the fact that
$\bigcap_{N}\Gamma(N)=\\{1\\}.$ The second statement is the consequence of the
fact that the sequence satisfies

${\rm pr}_{n,n/d}(j_{n}(\tau))=j_{n/d}(\tau),\mbox{ for each }d|n.$

$\Box$

###### 3.4

Remark. The system of covers (4) is not étale. However, by removing finitely
many points on $\mathbb{Y}(1)$ and all the points on $\mathbb{Y}(n)$ over
these one gets smooth curves $\mathbb{Y}^{-}(n)$ and a projective system of
covers

${\rm pr}_{n,n/d}:\mathbb{Y}^{-}(n)\to\mathbb{Y}^{-}(n/d)$

which is étale.

Since the construction of $\tilde{\Gamma}$ depends on generic fibres we have
the same $\tilde{\Gamma}$ for the construction corresponding to the system of
étale covers.

###### 3.5

The projective limit as the structure $\tilde{\mathbb{H}}^{\mathrm{Full}}.$

The analysis and study of the projective limit of modular curves
$\mathbb{Y}(n)$ with the $n$-level structure defined over
${\mathbb{Q}}(\zeta_{n})$ is readily reducible to the study of canonical
models of Shimura varieties, see [8], and more specifically canonical models
of modular curves, [9]. The case of curves $\mathbb{Y}(n)$ over ${\mathbb{Q}}$
as above in the setting appropriate for our purposes is studied in [10]. The
structure $\tilde{\mathbb{H}}^{\mathrm{Full}}$ is described therein as the
quotient
$\Delta(\hat{{\mathbb{Z}}}^{\times})\backslash\mathbf{S}^{\mathrm{Full}}$ of a
more fundamental structure $\mathbf{S}^{\mathrm{Full}}$ associated with the
Shimura datum $(\mathbb{H}\cup-\mathbb{H},\mathrm{GL}_{2}).$

The main conclusions, see [10], 3.22, are as follows:

(a) There is a group $\tilde{\mathrm{G}}$ acting on $\tilde{\mathbb{H}}.$

$\tilde{\mathrm{G}}\cong\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})\subset\mathrm{GL}_{2}(\mathbb{A}_{f}),$

where $\mathbb{A}_{f}$ is the ring of finite adeles.

One can speak about an action of
$\mathbf{g}\in\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})$ on
$\tilde{\mathbb{H}}$ once an isomorphism
$\varphi:\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})\to\tilde{\mathrm{G}}$
is provided. Call $\varphi$ a naming isomorphism. The naming isomorphisms form
the family

$\\{\varphi_{\mu}:\
\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})\to\tilde{\mathrm{G}}\
|\ \mu\in\hat{{\mathbb{Z}}}^{\times}\\};\ \mbox{ set
}\mathbf{g}^{\mu}:=\varphi_{\mu}(\mathbf{g}),$

which satisfies

$\mathbf{g}^{\mu\cdot\lambda}=(\mathbf{d}_{\lambda}^{-1}\cdot\mathbf{g}\cdot\mathbf{d}_{\lambda})^{\mu}$

where the conjugation by $\mathbf{d}_{\lambda}$ is in the ambient group
$\mathrm{GL}_{2}(\mathbb{A}_{f}).$

(b) Any
$\mathbf{g}\in\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})$
determines the 0-definable subset of $\tilde{\mathrm{G}}:$

$\mathbf{g}^{\Delta}=\\{\mathbf{g}^{\mu}:\mu\in\hat{{\mathbb{Z}}}^{\times}\\}.$

Each $\mathbf{g}^{\mu}$ gives rise to the sequence of algebraic curves defined
over $\bar{{\mathbb{Q}}},$

$C^{\mu}_{\mathbf{g},N}\subset\mathbb{Y}(N)\times\mathbb{Y}(N);\ \
C^{\mu}_{\mathbf{g},N}:=\\{\langle{j}_{N}(u),{j}_{N}(\mathbf{g}^{\mu}\cdot
u)\rangle:u\in\tilde{\mathbb{H}}\\}$

(here $C^{\mu}_{\mathbf{g},N}$ corresponds to $C^{\mu}_{\mathbf{g},\approx K}$
of [10]) and the limit curve

$C^{\mu}_{\mathbf{g}}\subset\tilde{\mathbb{H}}\times\tilde{\mathbb{H}};\ \
C^{\mu}_{\mathbf{g}}:=\\{\langle
u,\mathbf{g}^{\mu}u\rangle:u\in\tilde{\mathbb{H}}\\}.$

(c) For a fixed
$\mathbf{g}\in\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})$ we
obtain the finite family of curves on $\mathbb{Y}(N)\times\mathbb{Y}(N)$

$\\{C^{\mu}_{\mathbf{g},N},\ \mu\in\hat{{\mathbb{Z}}}^{\times}\\}$

where

$\mu-\lambda\in N\hat{{\mathbb{Z}}}\Longrightarrow
C^{\mu}_{\mathbf{g},N}=C^{\lambda}_{\mathbf{g},N}.$

These curves are irreducible components of the algebraic curve
$C_{\mathbf{g},N}$ defined over ${\mathbb{Q}}:$

$C_{\mathbf{g},N}=\bigcup_{\mu\in\hat{{\mathbb{Z}}}^{\times}}C^{\mu}_{\mathbf{g},N}\subset\mathbb{Y}(N)\times\mathbb{Y}(N).$

In the limit one obtains the the infinite-component limit curve on
$\tilde{\mathbb{H}}\times\tilde{\mathbb{H}}:$

$C_{\mathbf{g}}=\bigcup_{\mu\in\hat{{\mathbb{Z}}}^{\times}}C^{\mu}_{\mathbf{g}}.$

$C_{\mathbf{g}}$ is defined over ${\mathbb{Q}}$ too.

(d) The irreducible components $C^{\mu}_{\mathbf{g},N}$ of $C_{\mathbf{g},N}$
are Galois conjugated over ${\mathbb{Q}}.$

Clearly $C_{\mathbf{g}}=C_{\mathbf{g}^{\prime}}$ for
$\mathbf{g}^{\prime}=\mathbf{g}^{\mu}.$ The image of $C_{\mathbf{g}}$ under
$j_{N}\times j_{N}$ is $C_{\mathbf{g},N}.$

(e) For any
$\mathbf{g}\in\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f}),$
the definable 4-ary relation on $\tilde{\mathbb{H}}$

$\mathrm{Comp}_{\mathbf{g}}(s_{1},s_{2},t_{1},t_{2}):\equiv\exists
h\in\tilde{\mathrm{G}}\ s_{2}=h\cdot s_{1}\ \&\ t_{2}=h\cdot t_{1}\ \&\
C_{\mathbf{g}}(s_{1},s_{2})$

determines the condition that $\langle s_{1},s_{2}\rangle$ and $\langle
t_{1},t_{2}\rangle$ belong to the same irreducible component of a
$C_{\mathbf{g}}.$

For each $N$ the relation $\mathrm{Comp}_{\mathbf{g},N}$ on $\mathbb{Y}(N)$ is
the image of $\mathrm{Comp}_{\mathbf{g}}$ under $j_{N},$ the relation which
determines the decomposition of the algebraic curve $C_{\mathbf{g},N}$ into
irreducible components. This relation is invariant under
$\mathrm{Gal}_{\mathbb{Q}}$ and so definable over ${\mathbb{Q}}.$

We also have on $\tilde{\mathbb{H}}$ the definable equivalence

$j_{N}(s_{1})=j_{N}(s_{2}),$

which can be equivalently given by:

$\exists\gamma\in\tilde{\Gamma}(n)\ s_{2}=\gamma\cdot s_{1}.$

(f) The points $s$ in $\tilde{\mathbb{H}}$ which are fixed by a
$g\in\tilde{\mathrm{G}}\setminus\varphi_{\mu}(\mathrm{Z})$ ($\mathrm{Z}$ is
the centre of $\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ will be called special, or
CM-points of $\tilde{\mathbb{H}}.$ $j_{N}(s)$ is a CM-point in $\mathbb{Y}(N)$
and, in particular, is algebraic.

The paper [10] describes the binary relations on $\tilde{\mathbb{H}}$ written
as

$\langle t_{1},t_{2}\rangle\in\mathrm{tp}(s_{1},s_{2})$

defined for each pair $s_{1},s_{2}$ of CM-points. The reation is valid if and
only if there is an automorphism $\sigma$ of the projective system (4) such
that $\sigma:\langle s_{1},s_{2}\rangle\mapsto\langle t_{1},t_{2}\rangle.$ The
relations are invariant under automorphisms of the projective system (i.e.
defined over ${\mathbb{Q}}$) for each choice of $s_{1},s_{2}.$

The image of the relation under $j_{N}$ is the relation on $\mathbb{Y}(N)$

$\langle y_{1},y_{2}\rangle\in\mathrm{tp}_{N}(x_{1},x_{2})$

which holds if and only if $\langle y_{1},y_{2}\rangle$ is Galois conjugated
to $\langle x_{1},x_{2}\rangle$ over ${\mathbb{Q}}.$

###### 3.6

Definition. $\tilde{\mathbb{H}}^{\mathrm{Pure}}$ is the structure with the
universe $\tilde{\mathbb{H}}$ and relations $C_{\mathbf{g}}(s_{1},s_{2}),$
$\mathrm{Comp}_{\mathbf{g}}(s_{1},s_{2},t_{1},t_{2})$
($\mathbf{g}\in\Delta({\mathbb{Q}}_{+})\cdot\mathrm{SL}_{2}(\mathbb{A}_{f})$),
$j_{N}(s_{1})=j_{N}(s_{2})$ and $\langle
t_{1},t_{2}\rangle\in\mathrm{tp}(s_{1},s_{2}).$

$\tilde{\mathbb{H}}^{\mathrm{Full}}$ is the multisorted structure with sorts
$\tilde{\mathbb{H}}$ and $\mathbb{Y}(N),$ $N\in{\mathbb{N}},$ and relations:

\- on $\tilde{\mathbb{H}}$ the relations of
$\tilde{\mathbb{H}}^{\mathrm{Pure}};$

\- on the $\mathbb{Y}(N)$ the Zariski closed relations defined over
${\mathbb{Q}};$

\- the maps $j_{N}:\tilde{\mathbb{H}}\to\mathbb{Y}(N)$ and ${\rm
pr}_{N,N/d}:\mathbb{Y}(N)\to\mathbb{Y}(N/d).$

$\tilde{\mathbb{H}}^{\mathrm{Pure}}(\mathrm{CM})$ and
$\tilde{\mathbb{H}}^{\mathrm{Full}}(\mathrm{CM})$ are substructures of the
structures with universes restricted to their special points.

${\mathbb{H}}^{\mathrm{Full}}$ and ${\mathbb{H}}^{\mathrm{Pure}}$ are
“standard” substructures of $\tilde{\mathbb{H}}^{\mathrm{Full}}$ and
$\tilde{\mathbb{H}}^{\mathrm{Pure}},$ respectively, with the sort $\mathbb{H}$
in place of $\tilde{\mathbb{H}}.$

In particular, relations $C_{\mathbf{g}}$ and $\mathrm{Comp}_{\mathbf{g}}$ are
empty in ${\mathbb{H}}^{\mathrm{Full}}$ unless
$\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}}).$ Also, by [10], 3.24.3, the
relation $\langle t_{1},t_{2}\rangle\in\mathrm{tp}(s_{1},s_{2})$ on
$\mathbb{H}$ can be expressed in terms of actions by elements of
$\mathrm{GL}_{2}^{+}({\mathbb{Q}}):$ if $s_{1},s_{2}$ are fixed points of
respective $g_{1},g_{2}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ then either
$t_{1}=s_{1},$ $t_{2}=s_{2},$ or $t_{1},t_{2}$ are fixed points of the
cojugates $\check{g}_{1}:=\mathbf{d}_{-1}g_{1}\mathbf{d}_{-1}$ and
$\check{g}_{2}:=\mathbf{d}_{-1}g_{2}\mathbf{d}_{-1}$ respectively.

Remarks. (i) Since components of the $C_{\mathbf{g}}(s_{1},s_{2})$ are graphs
of actions of elements of the group $\tilde{\mathrm{G}},$ we often look at
$\tilde{\mathbb{H}}$ as a $\tilde{\mathrm{G}}$-set.

(ii) A corollary of definitions is that the relations $C_{\mathbf{g}},$
$\mathrm{Comp}_{\mathbf{g}},$ and $\langle
t_{1},t_{2}\rangle\in\mathrm{tp}(s_{1},s_{2})$ are projective limits of the
relations $C_{\mathbf{g},N},$ $\mathrm{Comp}_{\mathbf{g},N},$ and $\langle
t_{1},t_{2}\rangle\in\mathrm{tp}_{N}(s_{1},s_{2})$ on the $\mathbb{Y}(N).$
That is the relations on the sort $\tilde{\mathbb{H}}$ are positive-type-
definable in terms of pull-backs of the respective relations on the
$\mathbb{Y}(N)$ along with the pull-back of equality.

We thus can, up to $L_{\omega_{1},\omega}$-bi-interpretability, identify
$\tilde{\mathbb{H}}^{\mathrm{Pure}}$ as the structure given by the pull-backs
of $C_{\mathbf{g},N},$ $\mathrm{Comp}_{\mathbf{g},N},$ $\langle
t_{1},t_{2}\rangle\in\mathrm{tp}_{N}(s_{1},s_{2})$ and equality. Call it the
pull-backs version of $\tilde{\mathbb{H}}^{\mathrm{Pure}}.$ In the same sense
we speak about the pull-backs version of ${\mathbb{H}}^{\mathrm{Pure}},$ a
substructure of the pull-backs version of
$\tilde{\mathbb{H}}^{\mathrm{Pure}}.$

###### 3.7

Lemma. The pull-backs versions of $\tilde{\mathbb{H}}^{\mathrm{Pure}}$ and of
${\mathbb{H}}^{\mathrm{Pure}}$ are locally modular of trivial type.

The structures satisfy

${\mathbb{H}}^{\mathrm{Pure}}\prec\tilde{\mathbb{H}}^{\mathrm{Pure}}.$

Proof. The structure ${\mathbb{H}}^{\mathrm{Pure}}$ satisfies the assumptions
of Theorems 4.11 and 4.14 of [1]. It follows that its theory has quantifier
elimination and is of trivial type. $\tilde{\mathbb{H}}^{\mathrm{Pure}}$ is
then its elementary extension in the obvious way. $\Box$

###### 3.8

Remarks. It is easy to see that the centre $\mathrm{Z}$ of
$\mathrm{GL}^{+}_{2}({\mathbb{Q}})$ acts on $\tilde{\mathbb{H}}$ trivially.

Note that in the definitions above the curves $\mathbb{Y}(N)$ are curves over
${\mathbb{C}}$ (defined over ${\mathbb{Q}}$) and so the points of
$\tilde{\mathbb{H}}$ are limits of ${\mathbb{C}}$-points.

However the definitions and results are valid in the context of curves
$\mathbb{Y}(N)$ over $\mathrm{F},$ an abtract algebraically closed field of
characteristic $0.$ In this case $\mathbb{Y}_{\mathrm{F}}(N)$ are curves over
$\mathrm{F}.$

###### 3.9

Define $\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}$ and
$\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}$ to be the respective
structures obtained as the projective limit of the
$\mathbb{Y}_{\mathrm{F}}(N),$ for an arbitrary algebraically closed field of
characteristic $0.$

###### 3.10

Remark. Note that

$\tilde{\mathbb{H}}^{\mathrm{Pure}}(\mathrm{CM})=\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}(\mathrm{CM})$

since both sides are the structures obtained by taking the projective limit of
the respective substructures $\mathbb{Y}(n)^{\mathrm{Pure}}(\mathrm{CM})$ on
the curves. The same is true for the full structures:

$\tilde{\mathbb{H}}^{\mathrm{Full}}(\mathrm{CM})=\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}(\mathrm{CM}).$

###### 3.11

Define

$\tilde{\mathrm{E}}=\\{e\in\tilde{\mathrm{G}}\setminus\mathrm{Z}\ \exists
u\in\tilde{{\mathbb{U}}}\ e\cdot u=u\\}.$

Set

$\tilde{\mathrm{G}}_{*}=(\tilde{\mathrm{G}},\tilde{\Gamma},\tilde{\mathrm{E}},\\{\mathbf{d}_{q},\
\mathbf{d}^{\prime}_{q}:q\in{\mathbb{Q}}_{+})$

the structure on the group $\tilde{\mathrm{G}}$ with distinguished subgroup
$\tilde{\Gamma},$ distinguished subset $\tilde{\mathrm{E}}$ and distinguished
elements $\mathbf{d}_{q}$ and $\mathbf{d}^{\prime}_{q}.$

Analogously,

${\mathrm{G}}_{*}=({\mathrm{G}},{\Gamma},{\mathrm{E}},\\{\mathbf{d}_{q},\
\mathbf{d}^{\prime}_{q}:q\in{\mathbb{Q}}_{+}\\}).$

Clearly,

$\mathrm{G}_{*}\subset\tilde{\mathrm{G}}_{*}$

as structures.

###### 3.12

Remark. There is an embeding

$\tilde{\mathrm{G}}\subset\mathrm{GL}_{2}(\mathbb{A}_{f})$

which is an identity on the diagonal elements $\mathbf{d}_{q},\
\mathbf{d}^{\prime}_{q}:q\in{\mathbb{Q}}_{+}.$ It is easy to see that such an
embedding is determined uniquely, up to the conjugation by $\mathbf{d}_{\mu},$
$\mu\in\hat{{\mathbb{Z}}}^{\times}.$

In particular, we may assume that elements $g$ of $\tilde{\mathrm{G}}$ are
also elements of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ and thus the conjugation by
an element $\mathbf{d}_{\lambda}\in\Delta,$

$g\mapsto\mathbf{d}_{\lambda}\cdot g\cdot\mathbf{d}_{\lambda}^{-1}$

is well defined.

###### 3.13

Lemma. Consider the natural embedding
$\mathrm{G}_{*}\subset\tilde{\mathrm{G}}_{*}$ and let

$\psi:\mathrm{G}_{*}\twoheadrightarrow\mathrm{G}^{\prime}_{*}\subset\tilde{\mathrm{G}}_{*}$

be a partial isomorphism of $\tilde{\mathrm{G}}_{*}.$

$\psi$ can be extended to an automorphism
$\tilde{\psi}:\tilde{\mathrm{G}}_{*}\to\tilde{\mathrm{G}}_{*}.$ Moreover,
there is $\lambda\in\hat{{\mathbb{Z}}}^{\times}$ such that

$\tilde{\psi}:g\mapsto\mathbf{d}_{\lambda}\cdot
g\cdot\mathbf{d}_{\lambda}^{-1},\mbox{ for all }g\in\tilde{\mathrm{G}}.$

Proof. Recall that the subgroup $\Gamma\subset\mathrm{G}$ is generated by
elements $\mathbf{s}$ and $\mathbf{t},$ and $\mathrm{G}$ is generated over
$\Gamma$ by elements $\mathbf{d}_{q},$ $q\in{\mathbb{Q}}_{+}.$

Let
$\mathbf{s}^{\prime}=\psi(\mathbf{s})\in\Gamma^{\prime}\subset\mathrm{G}^{\prime}.$
Then

$\mathbf{s}^{\prime}\in\tilde{\mathrm{G}},\ \
\mathbf{s}^{\prime}\cdot\mathbf{d}_{-1}=\mathbf{d}^{\prime}_{-1}\mathbf{s}^{\prime}\mbox{
and }\ \
{\mathbf{s}^{\prime}}^{2}=-\mathbf{I}=\mathbf{d}_{-1}\cdot\mathbf{d}^{\prime}_{-1}.$

It is easy to see that the three equations imply that, for some
$\lambda\in\hat{{\mathbb{Z}}}^{\times},$

$\mathbf{s}^{\prime}=\left(\begin{array}[]{ll}\ 0\ \ \ \ \lambda\\\
-\lambda^{-1}\
0\end{array}\right)=\mathbf{d}_{\lambda}\cdot\mathbf{s}\cdot\mathbf{d}_{\lambda}^{-1}.$

Let $\mathbf{t}^{\prime}=\psi(\mathbf{t})\in\mathrm{G}^{\prime}.$ Then for
each $n\in{\mathbb{N}}_{>0},$

$\mathbf{t}^{\prime}\in\tilde{\Gamma},\ \
\mathbf{d}_{n}\cdot\mathbf{t}^{\prime}\cdot\mathbf{d}_{n}^{-1}={\mathbf{t}^{\prime}}^{n}\mbox{
and }(\mathbf{s}^{\prime}\mathbf{t}^{\prime})^{3}=\mathbf{I}.$

It follows

$\mathbf{t}^{\prime}=\left(\begin{array}[]{ll}1\ \lambda\\\ 0\
1\end{array}\right)=\mathbf{d}_{\lambda}\cdot\mathbf{t}\cdot\mathbf{d}_{\lambda}^{-1}.$

Thus, $\mathrm{G}^{\prime}$ is the group generated by $\mathbf{s}^{\prime},$
$\mathbf{t}^{\prime}$ and $\mathbf{d}_{q},$ $q\in{\mathbb{Q}}_{+},$ and

$\psi:g\mapsto\mathbf{d}_{\lambda}\cdot g\cdot\mathbf{d}_{\lambda}^{-1}\mbox{
for all }g\in\mathrm{G}.$

Take

$\tilde{\psi}:g\mapsto\mathbf{d}_{\lambda}\cdot
g\cdot\mathbf{d}_{\lambda}^{-1}\mbox{ for all }g\in\tilde{\mathrm{G}}.$

It is clear that $\tilde{\psi}$ preserves $\tilde{\Gamma}$, $\mathbf{d}_{q}$
and $\mathbf{d}^{\prime}_{q},$ $q\in{\mathbb{Q}}_{+}.$

Finally note that $g\in\tilde{\mathrm{E}}$ if and only if
$\tilde{\psi}(g)\in\tilde{\mathrm{E}}.$ This follows from the description of
fixed points in 3.22D(b) of [10].

$\Box$

##  4 The action of $\mathrm{Aut}\,\mathrm{F}$ on
$\tilde{{\mathbb{U}}}^{\mathrm{Full}}_{\mathrm{F}}.$

The following is essentially a corollary of the main results of [10].

###### 4.1

Theorem.

(i) $\mathrm{Gal}({\mathbb{Q}}(\mathrm{CM}):{\mathbb{Q}})$ acts naturally on
$\tilde{{\mathbb{U}}}^{\mathrm{Pure}}_{\mathrm{F}}(\mathrm{CM}),$ and

$\mathrm{Gal}({\mathbb{Q}}(\mathrm{CM}):{\mathbb{Q}})=\mathrm{Aut}\,\tilde{{\mathbb{U}}}^{\mathrm{Pure}}_{\mathrm{F}}(\mathrm{CM}).$

(ii) Any autmorphism of
$\tilde{{\mathbb{U}}}^{\mathrm{Pure}}_{\mathrm{F}}(\mathrm{CM})$ extends to an
automorphism of $\tilde{{\mathbb{U}}}^{\mathrm{Full}}_{\mathrm{F}}.$

(iii) For
$s_{1},\ldots,s_{n},t_{1},\ldots,t_{n}\in\tilde{{\mathbb{U}}}^{\mathrm{Pure}}_{\mathrm{F}}(\mathrm{CM}):$

$\exists\,\sigma\in\mathrm{Aut}\,\tilde{{\mathbb{U}}}^{\mathrm{pure}}(\mathrm{CM})_{\mathrm{F}}\
\bigwedge_{i=1}^{n}\sigma:s_{i}\mapsto t_{i}\ \ \ \Leftrightarrow\ \ \ \ \
\vDash\bigwedge_{i=1}^{n}\langle
t_{1},t_{i}\rangle\in\mathrm{tp}(s_{1},s_{i}).$

Proof. We rely heavily on [10].

We use the fact 3.10:

$\tilde{\mathbb{H}}^{\mathrm{Pure}}(\mathrm{CM})=\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}(\mathrm{CM}).$

Given that by definition
$\tilde{\mathbb{H}}^{\mathrm{Pure}}=\mathbf{S}^{\mathrm{pure}}_{\approx},$
3.21 gives us (i) of the theorem. In particular, any automorphism $\sigma$ of
$\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}(\mathrm{CM})$ is induced by
an automorphism $\rho\in\mathrm{Aut}\,\mathrm{F}$ acting on the CM-points,
that is $\sigma=\rho_{|\mathrm{CM}}.$ But

$\mathrm{Aut}\,\tilde{{\mathbb{U}}}^{\mathrm{Full}}_{\mathrm{F}}=\mathrm{Aut}\,\mathrm{F}$

by construction. In other words $\sigma$ lifts to an automorphism $\rho$ of
$\tilde{{\mathbb{U}}}^{\mathrm{Full}}_{\mathrm{F}}.$ This proves (ii).

Finally, Theorem 3.23 implies (iii). $\Box$

###### 4.2

Lemma. Any $\psi\in\mathrm{Aut}\,\tilde{\mathrm{G}}_{*}$ can be extended to
$\tilde{\psi}\in\mathrm{Aut}\,\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}(\mathrm{CM}).$

Proof As noted above
$\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}(\mathrm{CM})=\tilde{\mathbb{H}}^{\mathrm{Pure}}(\mathrm{CM})$
so we may argue in the setting of $\tilde{\mathbb{H}}.$

By 3.13 $\psi$ has the form
$\mathbf{g}^{\mu}\mapsto\mathbf{g}^{\mu\cdot\lambda}=\mathbf{d}_{\lambda}\cdot\mathbf{g}^{\mu}\cdot\mathbf{d}_{\lambda}^{-1},$
for some $\lambda\in\hat{{\mathbb{Z}}}^{\times}.$ And by 3.5(d) $\psi$ is
induced by a $\sigma\in\mathrm{Gal}_{\mathbb{Q}}.$ In its turn $\sigma$ acts
on the CM-points of $\tilde{{\mathbb{U}}}_{F}$ and induces a
$\tilde{\psi}\in\mathrm{Aut}\,\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Pure}}(\mathrm{CM}),$
as required. $\Box$

##  5 Axiomatisation of $\mathbb{H}_{\mathrm{F}}^{\mathrm{Full}}$ and
${\mathbb{U}}_{\mathrm{F}}^{\mathrm{Full}}.$

###### 5.1

The language and axioms $\Sigma$.

The language $\mathcal{L}(j_{n\in{\mathbb{N}}})$ is 3-sorted, with sorts
${\mathbb{U}},\mathrm{G}$ and $\mathrm{F}.$ The structure on $\mathrm{F}$ is
that of a field given in a standard ring language, the structure on
$\mathrm{G}$ and ${\mathbb{U}}$ is that of a group acting on ${\mathbb{U}}$
with distinguished subsets
$\mathrm{E},\Gamma,\\{\mathbf{d}_{q},q\in{\mathbb{Q}}^{\times}\\},\\{\mathbf{d}^{\prime}_{q},q\in{\mathbb{Q}}^{\times}\\}.$
Note that
$\\{\mathbf{d}_{q},q\in{\mathbb{Q}}^{\times}\\}=\Delta({\mathbb{Q}})$ with all
its elements named and the same for
$\\{\mathbf{d}^{\prime}_{q},q\in{\mathbb{Q}}^{\times}\\}=\Delta^{\prime}({\mathbb{Q}}).$
Note that

$\mathrm{GL}_{2}^{+}({\mathbb{Q}})=\Delta({\mathbb{Q}}_{>0})\cdot\mathrm{SL}_{2}({\mathbb{Z}})=\Delta^{\prime}({\mathbb{Q}}_{>0})\cdot\mathrm{SL}_{2}({\mathbb{Z}})$

and so group $\mathrm{G}$ is isomorphic to $\mathrm{GL}_{2}^{+}({\mathbb{Q}})$
with distinguished elements of $\Delta,$ $\Delta^{\prime}$ and subgroup
$\Gamma\cong\mathrm{SL}_{2}({\mathbb{Z}})$ will have the same structure.

Note that $\mathrm{GL}^{+}_{2}({\mathbb{Q}})$ is invariant under the
involutive transformation

$g\mapsto\check{g}:=\mathbf{d}_{-1}\cdot g\cdot\mathbf{d}_{-1}$

where
$\mathbf{d}_{-1}\in\mathrm{GL}_{2}({\mathbb{Q}})\subset\mathrm{GL}_{2}(\mathbb{A}_{f}).$

The maps $j_{n}$ have ${\mathbb{U}}$ as their domain and have quasi-projective
curves $\mathbb{Y}(n)$ as their range.

We consider the following system of axioms $\Sigma$ which is split into five
groups of axioms :

Group axioms:

$(\mathrm{G},\Gamma,\mathrm{E},\\{\mathbf{d}_{q}:q\in{\mathbb{Q}}_{>0}\\})\cong(\mathrm{GL}^{+}_{2}({\mathbb{Q}}),\mathrm{SL}_{2}({\mathbb{Z}}),\mathrm{E}({\mathbb{Q}}),\\{\mathbf{d}_{q}:q\in{\mathbb{Q}}_{>0}\\}).$
(5)

Note that $\Gamma(n)$ is definable from the data, namely the subgroup
$\Gamma_{0}(n)$ of matrices of the form $\mathbf{g}=\mathbf{t}^{nm}$ are
definable by condition

$\exists\gamma\in\Gamma\
\mathbf{g}=\mathbf{d}_{n}\cdot\gamma\cdot\mathbf{d}_{n}^{-1},$

and $\Gamma(n)$ can be defined as the normal closure of $\Gamma_{0}(n).$

Let $\mathrm{Z}=\mathrm{Centre}(\mathrm{G}).$

Action axiom: $\mathrm{G}\mbox{ acts on }{\mathbb{U}};$

$\begin{array}[]{lll}\forall
g\in\mathrm{G}\setminus(\mathrm{E}\cup\mathrm{Z})\ \forall u\in{\mathbb{U}}\
g\cdot u\neq u,\\\ \forall g\in\mathrm{Z}\ \forall u\in{\mathbb{U}}\ g\cdot
u=u,\\\ \forall e\in\mathrm{E}\ \exists!u_{e}\in{\mathbb{U}}\ e\cdot
u_{e}=u_{e}.\end{array}$ (6)

Fibre formula:

$\forall u,v\in{\mathbb{U}}\
j_{n}(u)=j_{n}(v)\leftrightarrow\exists\gamma\in\Gamma(n)\ v=\gamma\cdot u$
(7)

ACF0 axioms and sorts $\mathbb{Y}(n)$:

$\mathrm{F}\vDash\mathrm{ACF}_{0}$ (8)

and

$\mathbb{Y}(n)\subset{\mathbf{P}}^{3}(\mathrm{F});\ \ {\rm
pr}_{n.m}:\mathbb{Y}(n)\to\mathbb{Y}(m),\mbox{ for }m|n$

are given by specific equations over ${\mathbb{Q}}.$

Axioms - Functional equations:

$j_{n}:{\mathbb{U}}\twoheadrightarrow\mathbb{Y}(n);\ \ {\rm pr}_{n,m}\circ
j_{n}=j_{m}\mbox{ for each }m|n;$ (9) $\begin{array}[]{ll}\ \ \ \ \ \
\mbox{For each
}\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}}),\,n\in{\mathbb{N}},\,\mbox{
and }\mu\in\hat{{\mathbb{Z}}}^{\times},\ \exists g\in\mathbf{g}^{\Delta}\
\forall u,v\in{\mathbb{U}}:\\\ \langle j_{n}(u),j_{n}(v)\rangle\in
C_{\mathbf{g},n}^{\mu}\leftrightarrow\exists
v^{\prime},u^{\prime}\in{\mathbb{U}}\ v^{\prime}=g\cdot u^{\prime}\ \&\
j_{n}(u^{\prime})=j_{n}(u)\&j_{n}(v^{\prime})=j_{n}(v)\end{array}$ (10)

For each $n$ and any
$g_{1}\in\mathbf{g}_{1}^{\Delta},\ldots,g_{m}\in\mathbf{g}_{m}^{\Delta}$:

$\begin{array}[]{ll}\\{\langle
j_{n}(u),j_{n}(g_{1}u),\ldots,j_{n}(g_{m}u)\rangle:\ u\in{\mathbb{U}}\\}=\\\
=\mbox{ an irreducible component of }\\\ \\{\langle
y,y_{1},\ldots,y_{m}\rangle\in\mathbb{Y}(n)^{m+1}:\ \bigwedge_{i=1}^{m}\langle
y,y_{i}\rangle\in C_{\mathbf{g}_{i},n}\\}\end{array}$ (11)

###### 5.2

Commentary. Axioms (5), (6) and (7) describe the structure
${\mathbb{U}}_{\mathrm{F}}^{\mathrm{Pure}}.$

Axiom (8) describes the algebro-geometric part of
${\mathbb{U}}_{\mathrm{F}}^{\mathrm{Full}},$ the structure on the
$\mathrm{F}$-curves $\mathbb{Y}(n).$

Axioms (9)-(11) describes the connection between
${\mathbb{U}}_{\mathrm{F}}^{\mathrm{Pure}}$ and the $\mathbb{Y}(n)$ in terms
of the $j_{n}.$ More precisely, (10) states, in accordance with 3.5, that an
algebraic curve of the form $C^{\mu}_{\mathbf{g},n}$ is the image of the graph
of $g$ under $j_{n}$ for some $g\in\mathbf{g}^{\Delta}.$ Combining this with
(11) for $m=1$ and using the fact that curves $C^{\mu}_{\mathbf{g},n}$ are
irreducible the axioms imply that for all $g\in\mathbf{g}^{\Delta}$ the image
of the graph of $g$ under $j_{n}$ is of the form $C^{\mu}_{\mathbf{g},n}$ for
some $\mu\in\hat{{\mathbb{Z}}}^{\times}.$

Note that for a given $\mathbf{g}_{1},\ldots,\mathbf{g}_{m}$ the algebraic
subvariety $\\{\langle y,y_{1},\ldots,y_{m}\rangle:\
\bigwedge_{i=1}^{m}\langle y,y_{i}\rangle\in C_{\mathbf{g}_{i},n}\\}$ of
$\mathbb{Y}(n)^{m+1}$ of (11) is well-defined (although its explicit
representation in terms of equations could be hard to obtain) and thus there
is a well-defined finite list of its irreducible components.

###### 5.3

Lemma. $\mathbb{H}_{\mathbb{C}}^{\mathrm{Full}}$ is a model of axioms
$\Sigma.$

Proof. The axioms just describe formally the structure
$\mathbb{H}_{\mathbb{C}}^{\mathrm{Full}}$ following 3.5. $\Box$

###### 5.4

Lemma. For any model ${\mathbb{U}}_{\mathrm{F}}^{\mathrm{Full}}$ of $\Sigma$
there is an embedding

$i:{\mathbb{U}}_{\mathrm{F}}^{\mathrm{Full}}\hookrightarrow\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}$
(12)

which is the identity on $\mathrm{F}.$

Proof. $\tilde{{\mathbb{U}}}$ in
$\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}$ is given as the universal
cover of $\mathbb{Y}(\mathrm{F}),$ with respect to the class of modular
curves. By axiom (9) ${\mathbb{U}}$ covers every principal modular curve
$\mathbb{Y}(n)$ by $j_{n}:{\mathbb{U}}\to\mathbb{Y}(n).$ Hence there is a
unique map $i:{\mathbb{U}}\to\tilde{{\mathbb{U}}}$ such that

$j_{n}(i(u))=j_{n}(u).$ (13)

Now we constuct an emebdding of the groups $\mathrm{G}\to\tilde{\mathrm{G}}$
which agrees with the action and takes $\Gamma(n)$ to $\tilde{\Gamma}(n).$ In
order to do this recall that (10)-(11) imply that there is a bijective
correspondence between elements of $g\in\mathrm{G}$ and the sequences
$C_{\mathbf{g},n}^{\mu},$ $n\in{\mathbb{N}},$ where $\mu$ depends on $g$ (see
5.2 above). On the other hand the sequence $C_{\mathbf{g},n}^{\mu},$
$n\in{\mathbb{N}},$ determines a limit curve
$C_{\mathbf{g}}^{\mu}\subset\tilde{\mathbb{H}}^{2}$ (see 3.5(b)) and thus an
element, call it $i(g),$ of $\tilde{\mathrm{G}}$ together with its action on
$\tilde{\mathbb{H}}.$

Note that (13) together with the correspondnce of group actions imply that

$j_{n}(\gamma\cdot u)=j_{n}(u)\Leftrightarrow j_{n}(i(\gamma)\cdot
i(u))=j_{n}(i(u))$

and hence by (6) $\gamma\in\Gamma(n)\Leftrightarrow
i(\gamma)\in\tilde{\Gamma}(n).$

Finally the structure induced on the $\mathbb{Y}(n)_{\mathrm{F}}$ from
$\mathrm{F}$ is the same by virtue of (13) and axioms (8) and (9).

$\Box$

Given a model ${\mathbb{U}}_{\mathrm{F}}^{\mathrm{Full}}$ of $\Sigma$ and
$g\in\mathrm{E}$ we denote $u_{g}$ the unique fixed point of $g$ in
${\mathbb{U}}.$

Given $g_{1},\ldots,g_{k}\in\mathrm{G},$we denote
$\mathrm{Type}(g_{1},\ldots,g_{k}),$ the type of the tuple in $\mathrm{G},$ to
be the formula in terms of the structure $\mathrm{G}_{*}$ such that: if
$\mathrm{G}_{*}^{\prime}\cong\mathrm{G}_{*}$ and
$g^{\prime}_{1},\ldots,g^{\prime}_{k}\in\mathrm{G}^{\prime}$ then

$\langle g^{\prime}_{1},\ldots,g^{\prime}_{k}\rangle$ satisfies
$\mathrm{Type}(g_{1},\ldots,g_{k})$ if and only if there exists an isomorphism
$\psi:\mathrm{G}_{*}\to\mathrm{G}_{*}^{\prime},\ \langle
g_{1},\ldots,g_{k}\rangle\mapsto\langle
g^{\prime}_{1},\ldots,g^{\prime}_{k}\rangle.$

Since elements $g_{i}$ of $\mathrm{G}=\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ can
be written as the group words
$g_{i}=w_{i}(\mathbf{s},\mathbf{t},\mathbf{d}_{q}),$ it is immediate from 3.13
that

$\mathrm{Type}(g_{1},\ldots,g_{k})\equiv\exists s,t\in\Gamma\
s^{2}=-\mathbf{I}\ \&\ (st)^{3}=\mathbf{I}\ \&\
g_{i}=w_{i}(s,t,\mathbf{d}_{q}).$

We will write ${\mathbb{U}}_{\mathrm{F}}$ without superscript when speaking on
models of (5)-(11).

###### 5.5

Proposition. Let ${\mathbb{U}}_{\mathrm{F}}$ and
${\mathbb{U}}^{\prime}_{\mathrm{F}}$ be two models of $\Sigma.$

Let $\psi:\mathrm{G}_{*}\to\mathrm{G}^{\prime}_{*}$ be an isomorphism of
respective groups with extra structure and $g_{1},\ldots,g_{k}\in\mathrm{G},$
$g^{\prime}_{1},\ldots,g^{\prime}_{k}\in\mathrm{G}^{\prime}$ such that
$\psi:g_{i}\mapsto g^{\prime}_{i},$ $i=1,\ldots,k.$

Then, the $k$-tuples $\langle j_{N}(u_{g_{1}}),\ldots,j_{N}(u_{g_{k}})\rangle$
and $\langle
j_{N}(u_{g^{\prime}_{1}}),\ldots,j_{N}(u_{g^{\prime}_{k}})\rangle$ are Galois
conjugated.

Moreover, the algebraic type of $\langle
j_{N}(u_{g_{1}}),\ldots,j_{N}(u_{g_{k}})\rangle$ over ${\mathbb{Q}}$ is
determined by $\mathrm{Type}(g_{1},\ldots,g_{k}),$ the type of $\langle
g_{1},\ldots,g_{k}\rangle$ in $\mathrm{G}_{*}.$

Proof. By 5.4 we have two embeddings

${\mathbb{U}}_{\mathrm{F}}\subset\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}\mbox{
and
}{\mathbb{U}}^{\prime}_{\mathrm{F}}\subset\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}$

into the projective limit of curves $\mathbb{Y}(n)$ over $\mathrm{F}.$

By 3.13 $\psi$ can be extended to
$\tilde{\psi}\in\mathrm{Aut}\,\tilde{\mathrm{G}}_{*}.$ By 4.2 and 4.1(i) this
extends to an automorphism of
$\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}(\mathrm{CM}),$ that is an
automorphism
$\tilde{\psi}\in\mathrm{Gal}({\mathbb{Q}}(\mathrm{CM}):{\mathbb{Q}})$ acting
naturally on the CM-points of the curves $\mathbb{Y}(n)$ and respectively on
$\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}}(\mathrm{CM}).$ Thus the
special points $j_{N}(u_{g_{i}})$ are Galois conjugated to the special points
$j_{N}(u_{g^{\prime}_{i}}).$

Moreover, since the algebraic type of
$j_{N}(u_{g_{1}}),\ldots,j_{N}(u_{g_{k}})$ is determined in the substructure
$\tilde{{\mathbb{U}}}^{\mathrm{Full}}_{\mathrm{F}}(\mathrm{CM})$ of
$\tilde{{\mathbb{U}}}^{\mathrm{Full}}_{\mathrm{F}}$ and the substructure
coincides with $\tilde{\mathbb{H}}(\mathrm{CM}),$ the type is independent on
the choice of the model ${\mathbb{U}}_{\mathrm{F}}.$ $\Box$

###### 5.6

Corollary. Let $\bar{g}=\langle g_{1},\ldots,g_{m}\rangle\in\mathrm{G}^{m}.$
Then, in a model ${\mathbb{U}}_{\mathrm{F}}$ of $\Sigma,$ the set

$Z_{\bar{g},N}=\\{\langle
j_{N}(v),j_{N}(g_{1}v),\ldots,j_{N}(g_{n}v)\rangle:v\in{\mathbb{U}}\\}$

is an irreducible algebraic curve defined over the extension of ${\mathbb{Q}}$
by some special points $j_{N}(u_{e_{1}}),\ldots,j_{N}(u_{e_{k}}).$

If $\psi\in\mathrm{Aut}\,\tilde{{\mathbb{U}}}_{\mathrm{F}}^{\mathrm{Full}},$
$\psi:g_{i}\mapsto g^{\prime}_{i},$ $e_{l}\mapsto e^{\prime}_{l}$
$i=1,\ldots,m,$ $l=1,\ldots,k$ then $Z_{\bar{g},N}$ is Galois conjugated to
$Z_{\bar{g}^{\prime},N}.$

Moreover, the algebraic type of $Z_{\bar{g},N}$ over ${\mathbb{Q}}$ is
determined by the type of $\langle g_{1},\ldots,g_{k}\rangle$ in
$\mathrm{G}_{*}$ only in all models of $\Sigma.$

Proof. $Z_{\bar{g},N}$ is an irreducible algebraic curve by axiom (11). Curves
of this form were studied in [3] and [13]; they are special subvarieties
containing infinitely many special points and thus are defined over an
extension of ${\mathbb{Q}}$ by special points.

Since $\psi$ induces a Galois automorphism on CM-points, the statement of the
corollary follows. $\Box$

Given a model ${\mathbb{U}}_{\mathrm{F}}$ of $\Sigma$ and a naming isomorphism

$\varphi:(\mathrm{GL}^{+}_{2}({\mathbb{Q}}),\mathrm{SL}_{2}({\mathbb{Z}}),\mathrm{E}({\mathbb{Q}}),\\{\mathbf{d}_{q}:q\in{\mathbb{Q}}_{>0}\\})\to(\mathrm{G},\Gamma,\mathrm{E},\\{\mathbf{d}_{q}:q\in{\mathbb{Q}}_{>0}\\})$

define $\mathbf{U}_{\mathrm{F},\varphi}$ to be the expansion of the structure
${\mathbb{U}}_{\mathrm{F}}$ by names
$\mathbf{g}\in\mathrm{GL}^{+}_{2}({\mathbb{Q}})$ for each element
$\varphi(\mathbf{g})\in\mathrm{G}.$ We call the later structure an inessential
$\mathrm{G}$-expansions of the former. The language of
$\mathbf{U}_{\mathrm{F},\varphi}$ is the extension of the language
$\mathcal{L}(j_{n\in{\mathbb{N}}})$ by the names
$\mathbf{g}\in\mathrm{GL}^{+}_{2}({\mathbb{Q}})$ and the names for
$j_{N}(u_{\mathbf{g}}),$ for $\mathbf{g}\in E.$ The latter are names of
certain elements of $\mathbb{Y}(N),$ tuples in
$\bar{{\mathbb{Q}}}\subseteq\mathrm{F},$ so essentialy name points of the
field which are special. We refer to this language as
$\mathcal{L}(j_{n\in{\mathbb{N}}},\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}})).$
Below we are interested in reducts $\mathbf{U}^{(N)}_{\mathrm{F},\varphi}$ of
$\mathbf{U}_{\mathrm{F},\varphi}$ which ignore all the $j_{n}$ and
$\mathbb{Y}(n)$ except when $n=N.$ In this case we use the sublanguage
$\mathcal{L}(j_{N},\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}}))$ of
$\mathcal{L}(j_{n\in{\mathbb{N}}},\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}})).$

###### 5.7

Lemma. Two models ${\mathbb{U}}_{\mathrm{F}}$ and
${{\mathbb{U}}^{\prime}}_{\mathrm{F}}$ of $\Sigma$ are isomorphic if and only
if, for some $\varphi,\varphi^{\prime}$, $\mathbf{U}_{\mathrm{F},\varphi}$ is
isomorphic to ${\mathbf{U}^{\prime}}_{\mathrm{F},\varphi^{\prime}}.$

Proof. An isomorphism
$\alpha:{\mathbb{U}}_{\mathrm{F}}\to{{\mathbb{U}}^{\prime}}$ induces by the
group axiom (5) the isomorphism of groups
$\varphi:\mathrm{GL}^{+}_{2}({\mathbb{Q}})\to\mathrm{G}$ and
$\varphi^{\prime}:\mathrm{GL}^{+}_{2}({\mathbb{Q}})\to\mathrm{G}$ such that
$\varphi(g)\cdot u=\varphi^{\prime}(g)\cdot\alpha(u)$ and the $\mathrm{G}_{*}$
structures respected. This gives us the isomorphism of named structures
$\mathbf{U}_{\mathrm{F},\varphi}$ and
${\mathbf{U}^{\prime}}_{\mathrm{F},\varphi^{\prime}}$ which coincides with
$\alpha$ on ${\mathbb{U}}$ and $\mathrm{F}.$

Conversely, an isomorphism
$\alpha:\mathbf{U}_{\mathrm{F}}\to{\mathbf{U}^{\prime}}_{\mathrm{F}}$ becomes
the isomorphism
${\mathbb{U}}_{\mathrm{F}}\to{{\mathbb{U}}^{\prime}}_{\mathrm{F}}$ if we
ignore the naming of elements of $\mathrm{G}$ and $\mathrm{G}^{\prime}.$
$\Box$

##  6 The structure over the modular curve $\mathbb{Y}(N).$

###### 6.1

The structure $\mathbf{U}_{\mathrm{F}}$ and the A.Harris axioms.

[3] and [13] study substructures $\mathbf{U}_{\mathrm{F}}^{(N)}$ of
$\mathbf{U}_{\mathrm{F}}$ which are restrictions of the latter to sorts
${\mathbb{U}},$ $\mathrm{F}$ and $\mathbb{Y}(N),$ for a given $N.$ Since
elements $g$ of $\mathrm{G}$ are named, the special points of ${\mathbb{U}}$
and $\mathbb{Y}(N)$ get names $u_{g}$ and $j_{N}(u_{g}).$ Now the field
$\mathrm{F}$ has to be considered a field with constant symbols for CM-points
(and hence the automorphisms of the structure fix the subfield generated by
the CM-points).

Below we present axioms $\mathbf{\Sigma}(N)$ for
$\mathbf{U}_{\mathrm{F}}^{(N)}$ in language
$\mathcal{L}(j_{N},\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}}))$ in a form
which essentially copies axioms of [3] for $\mathbf{U}_{\mathrm{F}}^{(1)}.$
One of the differences with [3] and [13] is that we use
$\mathrm{GL}_{2}^{+}({\mathbb{Q}})$ where they used
$\mathrm{GL}_{2}^{+}({\mathbb{Q}})/\mathrm{Z}.$

Note that, for a finite tuple
$\bar{\mathbf{g}}=\langle\mathbf{g}_{1},\ldots,\mathbf{g}_{n}\rangle\in\mathrm{GL}_{2}^{+}({\mathbb{Q}})^{n},$
the algebraic curve $Z_{\bar{\mathbf{g}}}$ is uniquely defined by 5.6 in the
standard model $\mathbb{H}^{\mathrm{Full}}.$ We use these below in axiom (18).

Axioms $\mathbf{\Sigma}(N):$

Action axioms: For each
$\mathbf{g}\in\mathrm{GL}_{2}^{+}({\mathbb{Q}})\setminus(\mathrm{E}\cup\mathrm{Z}),$

$\forall u\in{\mathbb{U}}\ \mathbf{g}\cdot u\neq u$ (14)

For each $\mathbf{g}\in\mathrm{Z},$

$\forall u\in{\mathbb{U}}\ \mathbf{g}\cdot u=u$

For each $\mathbf{e}\in\mathrm{E},$

$\exists!u_{e}\in{\mathbb{U}}\ \mathbf{e}\cdot u_{e}=u_{e};$ (15)

The field axioms:

$\mathrm{F}\vDash\mathrm{ACF}_{0}$ (16)

and

$\mathbb{Y}(N)\subset{\mathbf{P}}^{3}(\mathrm{F})$

is given by specific equations over ${\mathbb{Q}}.$

Functional equations:

For any $\mathbf{e}_{1},\ldots,\mathbf{e}_{k}\in\mathrm{E},$ the special
points $j(u_{e_{1}}),\ldots,j(u_{e_{n}})$ satisfy

$p_{\bar{\mathbf{e}}}(j_{N}(u_{e_{1}}),\ldots,j_{N}(u_{e_{k}}))=0,$ (17)

for a polynomial $p_{\bar{e}}(X_{1},\ldots,X_{k})$ over ${\mathbb{Q}}$ if this
holds in the standard model.

For every finite tuple
$\bar{\mathbf{g}}=\langle\mathbf{g}_{1},\ldots,\mathbf{g}_{m}\rangle\in\mathrm{GL}_{2}^{+}({\mathbb{Q}})^{m},$
and the algebraic curve $Z_{\bar{\mathbf{g}}}$ defined over
${\mathbb{Q}}(j(u_{e_{1}}),\ldots,j(u_{e_{k}}))$

$\begin{array}[]{ll}\mbox{ the map }\ \ u\mapsto\langle
j_{N}(u),j_{N}(\mathbf{g}_{1}u),\ldots,j_{N}(\mathbf{g}_{m}u)\rangle\\\ \mbox{
is a surjection onto the the algebraic curve }Z_{\bar{g},N}.\end{array}$ (18)

The fibre formula:

$\forall u_{1},u_{2}\in{\mathbb{U}}\
j_{N}(u_{1})=j_{N}(u_{2})\leftrightarrow\bigvee_{\gamma\in\Gamma(N)}u_{2}=\gamma\cdot
u_{1}.$ (19)

###### 6.2

Theorem ([3], [13]) The set of axioms $\mathbf{\Sigma}(N)$ has a unique model
in every uncountable cardinal.

S. Eterovich in [15] extended the approach further to Shimura varieties
$\mathcal{A}_{2}$ and $\mathcal{A}_{3},$ moduli spaces of abelian varieties of
dimensions 2 and 3 respectively.

###### 6.3

Lemma. Suppose ${\mathbb{U}}_{\mathrm{F}}$ is a model of $\Sigma.$ Then
$\mathbf{U}_{\mathrm{F}}^{(N)}$ is a model of $\mathbf{\Sigma}(N).$

Proof. (14), (15) and (19) are immediate.

(17) follows from 5.5 and (18) from 5.6 since the tuples $\langle
j_{N}(e_{1}),\ldots,j_{N}(e_{k})\rangle$ and the curve
${\mathbb{Z}}_{\bar{\mathbf{g}},N}$ are determined by
$\mathbf{e}_{1},\ldots,\mathbf{e}_{n}$ and
$\mathbf{g}_{1},\ldots,\mathbf{g}_{m}$ in the standard model. $\Box$

###### 6.4

6.2 together with 6.3 fails to imply categoricity of $\Sigma.$

Nevertheless, the arguments from section 4 and 5 of [13] are sufficient to
prove categoricity of $\Sigma.$ We proceed with the arguments below.

We fix notation for the subfield

$L_{0}={\mathbb{Q}}(\mathrm{CM})\subset\mathrm{F}.$

###### 6.5

Lemma. Let $\bar{a}$ be an $m$-element subset of points of
$\mathbb{Y}(1)=\mathbf{A}^{1}(\mathrm{F})=\mathrm{F}$ and $L=L_{0}(\bar{a}).$
There exists a homomorphism

$\mathfrak{a}:\mathrm{Aut}\,(\mathrm{F}/L)\to\tilde{\Gamma}^{m}$

such that the number of $\mathrm{Aut}\,(\mathrm{F}/L)$-orbits in
$j^{-1}(\bar{a})$ is equal to the index of
$\mathfrak{a}(\mathrm{Aut}\,(\mathrm{F}/L))$ in $\tilde{\Gamma}^{m}.$

Proof. Consider the canonical action of $\mathrm{Aut}\,\mathrm{F}$ on
$\tilde{{\mathbb{U}}}$ and $\tilde{\Gamma}$.

Note that $\mathrm{Aut}\,(\mathrm{F}/L)$ fixes $\tilde{\Gamma}$ point-wise
since it fixes all the CM-points.

Let $\bar{u}\in\tilde{{\mathbb{U}}}^{m},$ $\bar{u}\in j^{-1}(\bar{a}),$
$\sigma\in\mathrm{Aut}\,(\mathrm{F}/L).$ Since $\bar{a}\subset L,$
$\sigma(\bar{u})$ belongs to the same fibre $j^{-1}(\bar{a})$ and hence

$\sigma(\bar{u})=\gamma_{\sigma,\bar{u}}\cdot\bar{u},\mbox{ for some
}\gamma_{\sigma,\bar{u}}\in\tilde{\Gamma}^{m}.$

Claim. $\gamma_{\sigma,\bar{u}}$ does not depend on the choice of $\bar{u}$ in
$j^{-1}(\bar{a}).$ The map

$\sigma\mapsto\gamma_{\sigma};\ \
\mathrm{Aut}\,(\mathrm{F}/L)\to\tilde{\Gamma}^{m}$

is a group homomorphism.

This follows by group-theoretic calculations.

By the same calculation, for a $\bar{v}\in j^{-1}(\bar{a}),$

$\exists\sigma\in\mathrm{Aut}\,(\mathrm{F}/L)\
\sigma\bar{u}=\bar{v}\Leftrightarrow\exists\gamma\in\mathfrak{a}(\mathrm{Aut}\,(\mathrm{F}/L))\
\gamma\bar{u}=\bar{v}.$

Hence $\mathrm{Aut}\,(\mathrm{F}/L)$-orbits are in a bijective correspondence
with $\mathfrak{a}(\mathrm{Aut}\,(\mathrm{F}/L))$-cosets in
$\tilde{\Gamma}^{m}.$

$\Box$

###### 6.6

Proposition. $\Sigma$ is $\kappa$-categorical for all uncounable $\kappa$ if
and only if the following two conditions hold:

  * (i)

given $\bar{a}$ such that its co-ordinates $a_{i}\notin L_{0}$ and in distinct
Hecke orbits, the image of $\mathrm{Aut}\,(\mathrm{F}/L)$ in
$\tilde{\Gamma}^{m}$ is of finite index;

  * (ii)

given an algebraically closed subfield $\mathrm{K}\subset\mathrm{F}$ and
$\bar{a}$ such that its co-ordinates $a_{i}\notin\mathrm{K}$ and in distinct
Hecke orbits, the image of $\mathrm{Aut}\,(\mathrm{F}/\mathrm{K}(\bar{a}))$ in
$\tilde{\Gamma}^{m}$ is of finite index.

Remark. The necessary condition (ii) is equivalent to a weaker condition which
assumes that $\mathrm{tr.deg}K(\bar{a})/\mathrm{K}=1.$

(i) and (ii) are effectively the conditions 4.7 and 4.8 of [13].

Proof of the Proposition. The same as the combined proofs of Theorems 4.1 and
4.9 in [13]. $\Box$

###### 6.7

Proposition. The conditions (i) and (ii) of 6.6 are satisfied.

Proof. See 5.1 and 5.2 of [13]. $\Box$

Remark. The key reference in the proof is the open image theorem of Serre,
[14].

As a corollary we get the main theorem.

###### 6.8

Theorem. $\Sigma$ is $\kappa$-categorical for all uncounable $\kappa.$

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