# CDDIP: Constrained Diffusion-Driven Deep Image Prior for Seismic Image
Reconstruction

Paul Goyes-Peñafiel*  —, Ulugbek S. Kamilov*  —, Henry Arguello*  — Manuscript
received July 24, 2024; revised Month 00, 000.This work was funded by the
Vicerrectoría de Investigación y Extensión from Universidad Industrial de
Santander under Project 3925.Paul Goyes-Peñafiel and Henry Arguello are with
the Department of Systems Engineering, Universidad Industrial de Santander,
Bucaramanga 680002, Colombia (e-mail<EMAIL_ADDRESS>henarfu@uis.edu.co)Ulugbek S. Kamilov is with the Department of Computer
Science and Engineering and Department of Electrical and Systems Engineering,
Washington University in St. Louis, St. Louis, MO 63130, USA (e-mail:
<EMAIL_ADDRESS>
###### Abstract

Seismic data frequently exhibits missing traces, substantially affecting
subsequent seismic processing and interpretation. Deep learning-based
approaches have demonstrated significant advancements in reconstructing
irregularly missing seismic data through supervised and unsupervised methods.
Nonetheless, substantial challenges remain, such as generalization capacity
and computation time cost during the inference. Our work introduces a
reconstruction method that uses a pre-trained generative diffusion model for
image synthesis and incorporates Deep Image Prior (DIP) to maintain data
consistency when reconstructing missing traces in seismic images. The proposed
method has demonstrated strong robustness and high reconstruction capability
for field and synthetic seismic images with different levels of structural
complexity, even in scenarios where test images were outside the training
domain. This indicates that our method can handle the high geological
variability of different exploration targets. Additionally, compared to other
state-of-the-art seismic reconstruction methods using diffusion models, our
approach reduces the number of neural function evaluations by up to 4x. Our
implementation is available at https://github.com/PAULGOYES/CDDIP.git

###### Index Terms:

Seismic data reconstruction, diffusion models, deep image prior, seismic
enhancement, consistent diffusion

##  I Introduction

EEnhancing post-stack seismic images is crucial for reducing exploration
risks, especially in conventional and unconventional energy sources such as
geothermal energy, shale gas, and rock-based hydrogen. In seismic processing,
post-stack images may exhibit missing or corrupted traces due to various
factors such as equipment malfunction, data acquisition errors, incomplete
coverage during data acquisition, signal attenuation, or interference from
surface or subsurface features [1].

The reconstruction traces problem in the seismic context has been mainly
addressed by methods based on deep learning, which can be primarily
categorized into supervised and unsupervised learning paradigms [2].
Supervised learning methods predominantly rely on end-to-end models that
necessitate large-scale datasets comprising pairs of corrupted images and
their corresponding labels. These models’ performance decreases when corrupt
images are outside the training domain, common in subsurface exploration due
to significant changes in geological formations and structural complexities
based on geographic location. To enhance the generalization capabilities of
supervised learning models, efforts have been made to harness the strengths of
generative models and integrate them into reconstruction learning frameworks
[3]. On the other hand, schemes based on denoisers have been proposed,
enabling reconstruction to be performed in a probabilistic fashion [4, 5].
Recently, seismic reconstruction has utilized unsupervised deep learning
through Deep Image Prior (DIP) [6, 7, 8]. This methodology enables training a
network solely with the measurements and inputting random noise. The approach
capitalizes on the convolutional neural networks’ capabilities to extract
features from the seismic image and adapts the network to the measurements
[7]. Thus, the need for extensive databases is alleviated, simplifying the
reconstruction of traces in diverse geological settings independent of data
availability. While DIP has demonstrated promising results in seismic data
reconstruction, one of its main drawbacks is its reliance on overfitting noise
towards the measurements. Therefore, DIP requires strict control over the
number of epochs and the implementation of proper early-stopping techniques
[9].

In the image restoration state-of-the-art, the probabilistic diffusion models
[10] have played a key role [11]. For instance, [12] proposed denoising
diffusion models for Plug-and-Play image restoration (DiffDPR) and solved the
inverse problem with a closed-form solution given by [13]. Also, applications
for Phase Retrieval [14] enforce the consistency of image generation with the
measurements using a subgradient of the least-squares data-fidelity term.
Specifically for seismic reconstruction, schemes that leverage diffusion
models and closed-form solutions have been proposed to condition seismic data
reconstruction and exploit generative models’ capabilities [15, 16, 17].
However, in those studies, the reconstruction task was tested on data within
the same training domain, and specifically in [16], the diffusion model was
retrained for different datasets. This is a disadvantage due to the limited
generalization of the reconstruction task.

This work presents a novel seismic reconstruction method using diffusion
models with consistent sampling. The diffusion model is trained to generate
samples from the posterior distribution by capturing the underlying structure
of the seismic images. These diffusion samples represent a possible
reconstruction of the missing traces. We employ a DIP solver instead of a
closed-form solution to generate images constrained by partial measurements.
Unlike closed-form solutions, DIP leverages convolutional neural networks to
extract image features, enhancing the generalization capabilities of our
approach. In addition, DIP acts as a prior distribution over the generated
images from the diffusion model, enforcing consistency based on the observed
traces. Consequently, we can reduce the number of diffusion steps needed to
accurately reconstruct missing traces by using scheduled steps to be denoised
by the DIP.

##  II Proposed method

The reconstruction of seismic traces is an ill-conditioned problem that
involves removing $n$ traces (i.e., columns) from a post-stack seismic image
$\boldsymbol{x}\in\mathbb{R}^{m\times n}$ with $m$ time (or depth for Prestack
Depth Migration) samples. In the view of computational imaging, the subsampled
seismic image $\boldsymbol{y}\in\mathbb{R}^{m\times n}$ due to missing traces
at positions $\mathbf{j}$ can be modeled by a degradation model given by the
following expression:

$\boldsymbol{y}=\mathbf{M}\odot\boldsymbol{x}+\boldsymbol{\omega},$ (1)

where $\mathbf{M}=[\ \mathbf{m}_{1},\mathbf{m}_{2},\cdots,\mathbf{m}_{n}]\
\in\\{0,1\\}^{m\times n}$ is the subsampling/masking operator with
$\mathbf{m}_{j}$ denoting the $j$-th column of $\mathbf{M}$, and $\odot$ is
the element-wise multiplication, and $\boldsymbol{\omega}$ is the measurement
noise. The structure of the non-zeros columns of $\mathbf{M}$ determines the
sampling scheme of the traces, and it can be uniform, irregular, or even
custom-designed [18]. In general, $\mathbf{m}_{j}$ can be expressed as

$\mathbf{m}_{j}=\begin{cases}0&\text{if }j\in\mathbf{j}\\\
1&\text{otherwise}.\end{cases}$ (2)

for $j=1,2,\cdots,n$

We propose to estimate $\boldsymbol{x}$ using the denoising diffusion
probabilistic model (DDPM) [10] and enforce consistency with the measurements
$\boldsymbol{y}$ through a Deep Image Prior (DIP) to improve the convergence
of the reconstruction problem.

The detailed steps are shown in Algorithm (1), where the inputs are the pre-
trained generative diffusion model $\boldsymbol{\epsilon}_{\theta}$,
measurements $\boldsymbol{y}$, the masking operator $\mathbf{M}$ and some
desired timestep schedule $\mathbf{t}=\\{t_{1},t_{2},t_{3},\cdots,t_{k}\\}$
where $k$ is the number of diffusion steps for the reconstruction, note that
$t_{1}=T$ is the number of total diffusion steps during training, and
$t_{k}=1$ is always the last diffusion step. The timestep sequence
$\mathbf{t}$ can be scheduled in a linear, quadratic, or exponential sampling
[12]. In step 1, the DIP solver’s trainable parameters $\Theta$ are randomly
initialized. In step 2, the first step is given by a random noise image with
normal standard distribution $\mathcal{N}(0,\mathbf{I})$. From steps 3 to 8,
the isotropic Gaussian noise is prepared to be used in step 9 to perform the
unconditional diffusion sampling $\tilde{\boldsymbol{x}}^{(t_{i})}_{0}$.

Algorithm 1 Deep Consistent Diffusion Sampling

1:$\boldsymbol{\epsilon}_{\theta}$, $\boldsymbol{y}$, $\mathbf{M}$,
$\mathbf{t}$ timestep schedule

2:Restored seismic image $\boldsymbol{x}_{1}$

3:Initialize $\mathcal{G}_{\Theta}$ with $\Theta$ randomly uniform

4:Sample $\boldsymbol{x}_{T}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$

5:for $i\;\textbf{in}\;1,2,3,\cdots,N$ do

6: if $t_{i}>1$ then

7: $\boldsymbol{z}\sim\mathcal{N}(0,\mathbf{I})$ $\triangleright$ Isotropic
gaussian noise

8: else if $t_{i}=1$ then

9: $\boldsymbol{z}=0$ $\triangleright$ Last diffusion step

10: end if

11: Unconditional diffusion sampling
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{\boldsymbol{x}}^{({t_{i}})}_{0}}=\frac{1}{\sqrt{\alpha}_{t_{i}}}\left(\boldsymbol{x}_{t_{i}}-\frac{1-\alpha_{{t_{i}}}}{\sqrt{1-\bar{\alpha}}_{t_{i}}}\boldsymbol{\epsilon}_{\theta}(\boldsymbol{x}_{t_{i}},{t_{i}})+\sqrt{1-\alpha_{t_{i}}}\cdot\boldsymbol{z}\right)$

12: DIP subproblem given $\boldsymbol{y}$
$\boldsymbol{\Theta}^{*}=\operatorname*{arg\,min}_{\Theta}\left\|\mathbf{M}\odot\mathcal{G}_{\boldsymbol{\Theta}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\tilde{\boldsymbol{x}}^{({t_{i}})}_{0}})-\boldsymbol{y}\right\|_{2}^{2}$

13: Data consistency regularized by DIP
$\boldsymbol{x}_{{t_{i+1}}}=\sqrt{\bar{\alpha}_{{t_{i+1}}}}\mathcal{G}_{\boldsymbol{\Theta}^{*}}(\tilde{\boldsymbol{x}}^{(t_{i})}_{0})+\sqrt{1-\bar{\alpha}_{{t_{i+1}}}}\cdot\boldsymbol{z}$

14:end for

15:return: $\boldsymbol{x}_{1}$

Although early diffusion steps $\tilde{\boldsymbol{x}}^{(t_{i})}_{0}$ remain
noisy, we leverage the capability of DIP to handle noise and estimate a clean
version solely based on the measurements $\boldsymbol{y}$. Thus, the role of
DIP in step 10 is to condition the generation of an image from DDPM via
$\mathcal{G}_{\boldsymbol{\Theta}}(\tilde{\boldsymbol{x}}^{(t_{i})}_{0})$ in
several sampling timesteps to ensure consistency with the measurements
$\boldsymbol{y}$. Therefore, it is worth mentioning that our approach does not
require closed-form solutions for the inverse problem. Conversely, our method
uses the DIP solver, which requires a few epochs. The DIP solution gradually
improves the image during each timestep of the denoising diffusion process.
Moreover, DIP is designed as a warm-starting model that leverages parameters
from the previous state solution.

In step 11, the regularized consistency
$\mathcal{G}_{\boldsymbol{\Theta}}(\tilde{\boldsymbol{x}}^{(t_{i})}_{0})$ is
back to the diffusion process, keeping the noise scheduled level for a desired
diffusion timestep using the traditional formulation of the forward process
with a distribution given by

$q(\boldsymbol{x}_{t_{i+1}}|\mathcal{G}_{\boldsymbol{\Theta}}(\tilde{\boldsymbol{x}}^{(t_{i})}_{0}))=\mathcal{N}(\boldsymbol{x}_{t_{i+1}};\sqrt{\bar{\alpha}_{t_{i+1}}}\mathcal{G}_{\boldsymbol{\Theta}}(\tilde{\boldsymbol{x}}^{(t_{i})}_{0}),(1-\bar{\alpha}_{t_{i+1}})\mathbf{I}).$
(3)

As a result of using Eq. (3), we can achieve the desired outcome by performing
only a few sampling timesteps, also known as Neural Function Evaluations
(NFEs) [19]. This is because the denoising objective depends on the DIP
solver, which relies solely on the sampling sequence. Therefore, our method is
only affected by the number of DIP steps. Fig. 1 summarizes the proposed
scheme of diffusion steps. Notably, some steps are skipped because the DIP
solver regularizes the unconditional denoised image
$\tilde{\boldsymbol{x}}^{(t_{i})}_{0}$ using $\boldsymbol{y}$, thus improving
the diffusion process to a $k$ timesteps.

Figure 1: Illustration with the conditional diffusion sampling. It is
noteworthy how, between $t_{2}$ and $t_{3}$, several steps of the pre-trained
DDPM are skipped by using the DIP solver as a data consistency regularizer.

##  III Simulations and Experiments

###  III-A Diffusion training

We trained the diffusion model with a cosine noise schedule given by
$1-\alpha$ from $10^{-4}$ to $0.02$ and $T=1000$ diffusion steps. We used
$14398$ images with size $128\times 128$ during the training, which provides
diverse seismic features, considering different geological scenarios and
subsurface complexities from 6 databases, including synthetic and field
surveys, as shown in Table I.

TABLE I: Training dataset seismic images from (F) field and (S) synthetic surveys | Dataset | #Training | Size (Mb)
---|---|---|---
S | TGS salt Identification challenge [20] | 4150 | 49.9
SEAM Challenge Phase 1 [21] | 4072 | 26.2
1994 BP [22] | 803 | 4.15
AGL Elastic Marmousi [23] | 152 | 0.8
F | F3 Netherlands [24] | 2000 | 15.9
Kerry 3D [25] | 3221 | 24.7
| Total | 14398 | 121.65

Implementation details: We split randomly the dataset into 90% for training
and 10% for testing (Experiments I and II). The diffusion model
$\boldsymbol{\epsilon}_{\theta}$ was trained for 5000 epochs, with a
computation time of 6 days. For the DIP solver, an attention U-NET was used
[26]. $\mathbf{M}$ simulates irregularly random missing traces for all
experiments. All the experiments were done with an NVIDIA RTX 4090 24 GB GPU.
Further implementation details are provided in the project repository.

###  III-B Experiment I

In this experiment, we analyze the impact of the number of timesteps $k$ and
DIP steps on the quality (PSNR and SSIM) of reconstructing 50% of the traces
in a seismic image. The reconstruction results were compared to state-of-the-
art methods that use conditional constraint diffusion models, namely DiffDPIR
[12] and CCSeis-DDPM [15, 16]; for both methods, the diffusion model was
trained with the same dataset reported in Table I. Table II shows that the
lowest performance is obtained with $10$ DIP steps. Additionally, regardless
of the timestep value, increasing the number of DIP solver steps also improves
the quality of the reconstructions. On the other hand, for $k=25$ and $k=50$
timesteps, increasing the number of DIP solver steps positively impacts the
PSNR, which remains approximately around 37.19 dB on average. The visual
results from Fig. 2 indicate that for smaller DIP steps, the solution is
smoother, and as the steps increase, the solution adjusts more closely to the
noise and curved seismic events ($>25$ DIP steps).

It is essential to note that our method performs similarly to the CCSeis-DDPM
and DiffDPIR methods. However, our method handles internal measurement noise
better. For example, Fig. 2 shows that reducing the number of DIP solver steps
results in smoother reconstructions, whereas the other methods consistently
produce noisy results.

TABLE II: Quantitative reconstruction evaluation for different diffusion sampling timesteps $k$ and DIP solver steps |  | PSNR(dB)$\uparrow$ | SSIM$\uparrow$
---|---|---|---
Timesteps $k$ | 10 | 25 | 50 | 10 | 25 | 50
DIP steps | 15 | 21.624 | 33.194 | 33.804 | 0.497 | 0.861 | 0.874
25 | 29.835 | 34.231 | 36.967 | 0.767 | 0.890 | 0.929
50 | 34.024 | 37.190 | 37.058 | 0.892 | 0.930 | 0.937
100 | 35.650 | 36.830 | 36.253 | 0.919 | 0.923 | 0.919
DiffPIR | 31.065 | 35.315 | 36.112 | 0.819 | 0.860 | 0.870
CCSeis-DDPM | 31.942 | 34.982 | 36.157 | 0.832 | 0.852 | 0.874
Figure 2: Visual results of a seismic reconstruction experiment using a mask
with 50% randomly missing traces for different DIP steps and sampling
timesteps $k$ with a linear/uniform schedule.

We found the best performance when using 50 DIP solver steps and $k=25$, with
an average computation time of 38 seconds. The comparison methods took 36
seconds on average. This provides a suitable balance between computation time
and metrics evaluation. We remark that in other applications with less complex
images, $k$ could decrease.

###  III-C Experiment II

In this experiment, we evaluate the reconstruction capability of our method
over different seismic images containing linear and curved seismic structures
and geological faults. Furthermore, the proposed approach enables the
evaluation of reconstruction uncertainty by estimating diverse realizations.
This is because diffusion sampling starts with random noise, which allows for
the visualization of, for example, the spatial distribution of uncertainties
using variance. This experiment uses $k=25$ and 50 DIP solver steps.

Figure 3: Visual reconstruction results of the average and variance of five
estimations using the proposed method with varying geological scenarios with
50% of missing traces. The four structural complexities increase from top to
bottom rows.

In Fig. 3, our method accurately reconstructs lower structural complexity
levels corresponding to subhorizontal and dipping beds, achieving PSNR values
of $41.469$ dB and $35.536$ dB and SSIM values of $0.981$ and $0.967$,
respectively. However, the method exhibits reduced performance for curved and
highly noisy layers with a PSNR of $27.438$ dB and SSIM of $0.741$. This
discrepancy arises because the employed DIP solver tends to generate smooth
and noise-free images. Nevertheless, visually, our method captures the
pertinent structural features of the seismic image. Our method maintains the
original discontinuities and inclinations for seismic images depicting
geological faults, obtaining a PSNR of $30.251$ dB and SSIM of $0.849$. Across
all scenarios, variance values are high only for the image’s left side, where
$10$ consecutive traces were removed.

###  III-D Experiment III

This experiment aimed to assess the effectiveness of our method on data
outside the training domain. In deep learning approaches, datasets are
typically divided into training and testing subsets within the same domain.
However, we sought to determine the efficacy of our approach when applied to
datasets entirely distinct from the training domain. The seismic datasets
include Stratton, Penobscot, Black Ridge, and Alaska, each with unique
complexities in image structures.

Figure 4: Visual results depict the reconstruction of 60% missing traces using
the proposed method, DiffPIR [12], and CCSeis-DDPM [15] with 25, 100, and 250
timesteps, respectively. For each method, the respective normalized error is
presented below each reconstruction to highlight areas with the greatest
discrepancies relative to the ground truth. It is noteworthy that the mask
pattern remains consistent across all datasets.

This experiment uses $k=25$ and $50$ DIP solver steps for the proposed method.
According to [12, 15, 16], we set $k$ and U to $100$ and $1$ for DiffPIR, and
250 and 10 for the CCSeis-DDPM. U is the number of inner iterations within a
timestep. Table III summarizes the metrics for each method across different
datasets. Notably, our method reduces the number of diffusion timesteps up to
four times, allowing for investment in DIP solver steps. This increases
generalization capability and eliminates the need for retraining, even when
the application includes test data outside the training domain. Fig. 4
illustrates that, for the case of horizontal layers in the Alaska dataset, all
methods achieved high performance exceeding PSNR $30$ dB. However, our method
performed better in reconstructing areas where consecutive traces were
removed. While DiffPIR and CCSeis-DDPM generally demonstrated acceptable
performance in the Black Ridge dataset, our method outperformed them by up to
$2.2$ dB in PSNR and $0.23$ in SSIM, highlighting the effectiveness of the
proposed method in complex scenarios.

TABLE III: Average quantitative results for different seismic surveys. Dataset | Method | PSNR(dB)$\uparrow$ | SSIM$\uparrow$
---|---|---|---
Stratton | Proposed | 28.664 | 0.797
DiffDPIR | 25.697 | 0.689
CCSeis-DDPM | 25.653 | 0.691
Penobscot | Proposed | 32.635 | 0.896
DiffDPIR | 31.291 | 0.855
CCSeis-DDPM | 31.392 | 0.861
Black Ridge | Proposed | 26.692 | 0.785
DiffDPIR | 24.449 | 0.558
CCSeis-DDPM | 24.363 | 0.545
Alaska | Proposed | 33.374 | 0.886
DiffDPIR | 31.728 | 0.849
CCSeis-DDPM | 31.987 | 0.849

##  IV Conclusions

This paper introduces a conditional diffusion model for seismic data
reconstruction, leveraging DIP advantages to implement consistency in image
synthesis from partial measurements during reverse diffusion. Our diffusion
model was trained on field and synthetic datasets to learn the distribution of
post-stack seismic images. Experiments demonstrated that our proposed method
achieves outstanding results compared to state-of-the-art approaches in
similar computation times. This holds for test data within the training
domain, field data outside the training domain, and various complexities of
seismic image structures related to different geological scenarios.

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$\displaystyle\mathscr{P}^{\heartsuit}_{\mathcal{L}}:=(\mathcal{P}_{\mathcal{L}},\nabla_{\mathcal{P}_{\mathcal{L}}},\\{\mathcal{P}^{j}_{\mathcal{L}}\\}_{0\leq
j\leq p^{N}})$

forms a dormant $\mathrm{GL}_{p^{N}}$-oper of level $N$ on $\mathscr{X}$.
Indeed, as discussed in § 2.2, the restriction of this data to
$D_{x}\left(=\mathrm{Spec}(\mathcal{O}_{X,x})\right)$ for each $k$-rational
point $x$ of $X\setminus\bigcup_{i=1}^{r}\\{\sigma_{i}\\}$ (resp.,
$\bigcup_{i=1}^{r}\\{\sigma_{i}\\}$) defines a dormant pinned
$\mathcal{D}_{\mathcal{O}_{X,x}}^{(N-1)}$-module (resp., a dormant pinned
$\breve{\mathcal{D}}_{\mathcal{O}_{X,x}}^{(N-1)}$-module). In particular, we
obtain a dormant $\mathrm{PGL}_{p^{N}}$-oper
$\mathscr{P}^{\heartsuit\Rightarrow\spadesuit}_{\mathcal{L}}$ of level $N$ on
$\mathscr{X}$.

Next, let $(\mathcal{N},\nabla_{\mathcal{N}})$ be a dormant
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-module such that $\mathcal{N}$ is a
line bundle. Since the tensor product $\nabla_{\mathcal{N}}\otimes\nabla$ has
vanishing $p^{N}$-curvature, the composite

(132)
$\displaystyle\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}\otimes(\mathcal{N}\otimes\mathcal{L})\left(=\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}\otimes(\mathcal{N}\otimes\mathcal{P}_{\mathcal{L}}^{p^{N}-1})\right)$
$\displaystyle\xrightarrow{\mathrm{inclusion}}\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}\otimes(\mathcal{N}\otimes\mathcal{P}_{\mathcal{L}})$
$\displaystyle\xrightarrow{\nabla_{\mathcal{N}}\otimes\nabla}\mathcal{N}\otimes\mathcal{P}_{\mathcal{L}}$

factors through the quotient
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}\otimes(\mathcal{N}\otimes\mathcal{L})\twoheadrightarrow\mathcal{P}_{\mathcal{N}\otimes\mathcal{L}}$.
By considering the local description, we can see that the resulting morphism
$\mathcal{P}_{\mathcal{N}\otimes\mathcal{L}}\rightarrow\mathcal{N}\otimes\mathcal{P}_{\mathcal{L}}$
defines an isomorphism

(133)
$\displaystyle\mathscr{P}^{\heartsuit}_{\mathcal{N}\otimes\mathcal{L}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathscr{P}^{\heartsuit}_{\mathcal{L}})_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})}$

of dormant $\mathrm{GL}_{n}$-opers of level $N$.

###  5.2. Duality for dormant $\mathrm{GL}_{n}$ opers of level $N$

Next, let
$\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{j})$ be
a dormant $\mathrm{GL}_{n}$-oper of level $N$ on $\mathscr{X}$ with
$\mathcal{F}^{n-1}=\mathcal{L}$. The inclusion
$\mathcal{L}\left(=\mathcal{D}_{X^{\mathrm{log}},<1}^{(N-1)}\otimes\mathcal{L}\right)\hookrightarrow\mathcal{F}$
extends uniquely to a $\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-linear morphism
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}\otimes\mathcal{L}\rightarrow\mathcal{F}$.
Since $(\mathcal{F},\nabla)$ has vanishing $p^{N}$-curvature, this morphism
factors through the quotient
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}\otimes\mathcal{L}\twoheadrightarrow\mathcal{P}_{\mathcal{L}}$.
Thus, we obtain a morphism of $\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-modules

(134)
$\displaystyle\nu_{\mathscr{F}^{\heartsuit}}:(\mathcal{P}_{\mathcal{L}},\nabla_{\mathcal{P}_{\mathcal{L}}})\rightarrow(\mathcal{F},\nabla).$

By Proposition 4.2.3, (i), the restriction of this morphism to $D_{x}$ for
each point $x$ of $X\setminus\bigcup_{i=1}^{r}\\{\sigma_{i}\\}$ (resp.,
$\bigcup_{i=1}^{r}\\{\sigma_{i}\\}$) may be regarded as a morphism of pinned
$\mathcal{D}_{\mathcal{O}_{X,x}}^{(N-1)}$-modules (resp., pinned
$\breve{\mathcal{D}}_{\mathcal{O}_{X,x}}^{(N-1)}$-modules). Hence, it follows
from Proposition 1.5.4, (i), that $\nu_{\mathscr{F}^{\heartsuit}}$ is verified
to be surjective.

######  Example 5.2.1.

Let us consider the dual
$(\mathcal{P}_{\mathcal{L}}^{\vee},\nabla_{\mathcal{P}_{\mathcal{L}}}^{\vee})$
of $(\mathcal{P}_{\mathcal{L}},\nabla_{\mathcal{P}_{\mathcal{L}}})$. For each
$j=0,\cdots,p^{N}$, we shall write $\mathcal{P}_{\mathcal{L}}^{\vee j}$ for
the $\mathcal{O}_{X}$-submodule of $\mathcal{P}_{\mathcal{L}}^{\vee}$ defined
to be the image of the natural injection
$(\mathcal{P}_{\mathcal{L}}/\mathcal{P}_{\mathcal{L}}^{p^{N}-j})^{\vee}\hookrightarrow\mathcal{P}^{\vee}_{\mathcal{L}}$.
Thus, we obtain a collection of data

(135)
$\displaystyle\mathscr{P}^{\heartsuit\vee}_{\mathcal{L}}:=(\mathcal{P}_{\mathcal{L}}^{\vee},\nabla_{\mathcal{P}_{\mathcal{L}}}^{\vee},\\{\mathcal{P}_{\mathcal{L}}^{\vee
j}\\}_{0\leq j\leq p^{N}}).$

For each $k$-rational point $x$ of $X$, the restriction of
$(\mathcal{P}_{\mathcal{L}}^{\vee},\nabla_{\mathcal{P}_{\mathcal{L}}}^{\vee})$
to $D_{x}$ together with a generator of $\mathcal{P}_{\mathcal{L}}^{\vee
p^{N}-1}|_{D_{x}}$ may be regarded as the data (65). It follows that
$\mathscr{P}^{\heartsuit\vee}_{\mathcal{L}}$ forms a dormant
$\mathrm{GL}_{p^{N}}$-oper of level $N$. Also, the morphism

(136)
$\displaystyle\nu_{\mathscr{P}_{\mathcal{L}}^{\heartsuit\vee}}:(\mathcal{P}_{\mathcal{L}},\nabla_{\mathcal{P}_{\mathcal{L}}})\rightarrow(\mathcal{P}_{\mathcal{L}}^{\vee},\nabla_{\mathcal{P}_{\mathcal{L}}}^{\vee})$

is compatible, via restriction to $D_{x}$, with the morphism (66). It follows
that $\nu_{\mathscr{F}^{\heartsuit\vee}}$ defines an isomorphism
$\mathscr{P}^{\heartsuit}_{\mathcal{L}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{P}^{\heartsuit\vee}_{\mathcal{L}}$
of $\mathrm{GL}_{p^{N}}$-opers of level $N$.

Let us write
$\mathcal{F}^{\blacktriangledown}:=\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}})^{\vee}$.
Since $\nu_{\mathscr{F}^{\heartsuit}}$ preserves the
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-action,
$\nabla_{\mathcal{P}_{\mathcal{L}}}$ is restricted to a
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-module structure
$\nabla_{\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}})}$ on the kernel
$\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}})$; it induces a
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-module structure on
$\mathcal{F}^{\blacktriangledown}$, which we denote by
$\nabla^{\blacktriangledown}$. For each $j=0,\cdots,p^{N}-n$, let us define
$\gamma_{j}$ to be the composite

(137)
$\displaystyle\gamma_{j}:\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}})\xrightarrow{\mathrm{inclusion}}\mathcal{P}_{\mathcal{L}}\twoheadrightarrow\mathcal{P}_{\mathcal{L}}/\mathcal{P}^{p^{N}-n-j}_{\mathcal{L}}.$

By letting $\mathcal{F}^{\blacktriangledown
j}:=\mathrm{Im}(\gamma^{\vee}_{j})\left(\subseteq\mathcal{F}^{\blacktriangledown}\right)$,
we obtain a collection of data

(138)
$\displaystyle\mathscr{F}^{\heartsuit\blacktriangledown}:=(\mathcal{F}^{\blacktriangledown},\nabla^{\blacktriangledown},\\{\mathcal{F}^{\blacktriangledown
j}\\}_{0\leq j\leq p^{N}-n}).$

######  Proposition 5.2.2.

Let us keep the above notation.

  * (i)

$\mathscr{F}^{\heartsuit\blacktriangledown}$ forms a dormant
$\mathrm{GL}_{p^{N}-n}$-oper of level $N$ on $\mathscr{X}$. Moreover, there
exists a canonical isomorphism
$\mathscr{F}^{\heartsuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{F}^{\heartsuit\blacktriangledown\blacktriangledown}$
of $\mathrm{GL}_{n}$-opers of level $N$.

  * (ii)

Let $(\mathcal{N},\nabla_{\mathcal{N}})$ be a dormant
$\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-module such that $\mathcal{N}$ is a
line bundle. Then, there exists a canonical isomorphism

(139)
$\displaystyle(\mathscr{F}^{\heartsuit\blacktriangledown})_{\otimes(\mathcal{N}^{\vee},\nabla_{\mathcal{N}}^{\vee})}\cong(\mathscr{F}^{\heartsuit}_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})})^{\blacktriangledown}$

of $\mathrm{GL}_{p^{N}-n}$-opers of level $N$.

###### Proof.

First, let us consider assertion (i). The formation of
$\mathscr{F}^{\heartsuit\blacktriangledown}$ is compatible with that of
$(E^{\blacktriangledown},\nabla^{\blacktriangledown},v^{\blacktriangledown})$
(cf. (72)) via restriction to $D_{x}$ for every point $x$ of $X$. This implies
that $\mathscr{F}^{\heartsuit\blacktriangledown}$ forms a dormant
$\mathrm{GL}_{p^{N}-n}$-oper of level $N$ (cf. Proposition 4.2.3). Also, let
us consider the following diagram:

(144)

The right-hand square is commutative because its restriction to $D_{x}$ for
every point $x$ of $X$ is commutative, as observed in the proof of Theorem
2.3.2, (i). Hence, it induces a morphism
$\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit\blacktriangledown}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{F}^{\vee}$.
By considering the local description again, one verifies that the dual of this
isomorphism specifies an isomorphism
$\mathscr{F}^{\heartsuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{F}^{\heartsuit\blacktriangledown\blacktriangledown}$
of $\mathrm{GL}_{n}$-opers of level $N$. This completes the proof of assertion
(i).

Next, we shall prove assertion (ii). Consider the following diagram:

(149)

where the middle vertical arrow denotes the dual of (133) and the left-hand
vertical arrow denotes the canonical isomorphism. Since the left-hand square
diagram is commutative, this diagram induces an isomorphism
$\mathcal{N}^{\vee}\otimes\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}})^{\vee}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})}})^{\vee}$.
This isomorphism specifies an isomorphism
$(\mathscr{F}^{\heartsuit\blacktriangledown})_{\otimes(\mathcal{N}^{\vee},\nabla_{\mathcal{N}}^{\vee})}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathscr{F}^{\heartsuit}_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})})^{\blacktriangledown}$
of $\mathrm{GL}_{p^{N}-n}$-opers of level $N$. This completes the proof of
assertion (ii). ∎

######  Remark 5.2.3.

In this remark, we shall examine the determinant of a $\mathrm{GL}_{n}$-oper.
Let $\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{j})$
be as above. Since
$\mathcal{D}_{X^{\mathrm{log}},<j+1}^{(N-1)}/\mathcal{D}_{X^{\mathrm{log}},<j}^{(N-1)}\cong\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes
j}$ ($j=0,1,2,\cdots$), we obtain the composite of canonical isomorphisms

(150) $\displaystyle\mathrm{det}(\mathcal{P}_{\mathcal{L}})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigotimes_{j=0}^{p^{N}-1}\mathcal{P}_{\mathcal{L}}^{j}/\mathcal{P}_{\mathcal{L}}^{j+1}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\bigotimes_{j=0}^{p^{N}-1}\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes(p^{N}-j-1)}\otimes\mathcal{L}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes
p^{N}(p^{N}-1)/2}\otimes\mathcal{L}^{\otimes p^{N}}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{(N)*}_{X/k}(\mathcal{N}_{\mathcal{L}}),$

where $\mathcal{N}_{\mathcal{L}}$ denotes the line bundle on $X^{(N)}$
corresponding to
$\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes(p^{N}-1)/2}\otimes\mathcal{L}$ via
base-change $X^{(N)}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X$ by
$F_{k}^{N}$. Similarly, there exists an isomorphism

(151)
$\displaystyle\mathrm{det}(\mathcal{F})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes
n(n-1)/2}\otimes\mathcal{L}^{\otimes n}.$

The $\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-module structure on the
determinant bundle $\mathrm{det}(\mathcal{P}_{\mathcal{L}})$ induced by
$\nabla_{\mathcal{P}_{\mathcal{L}}}$ corresponds to
$\nabla_{\mathcal{N}_{\mathcal{L}}}^{\mathrm{can}}$ via (150). Hence, the
determinant of $\nabla^{\blacktriangledown}$ corresponds to
$(\nabla^{\mathrm{can}}_{\mathcal{N}_{\mathcal{L}}})^{\vee}\otimes\mathrm{det}(\nabla)$
via the following composite of natural isomorphisms:

(152) $\displaystyle\mathrm{det}(\mathcal{F}^{\blacktriangledown})$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{det}(\mathrm{Ker}(\nu_{\mathscr{F}^{\heartsuit}}))^{\vee}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{det}(\mathcal{P}_{\mathcal{L}})^{\vee}\otimes\mathrm{det}(\mathcal{F})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{(N)*}_{X/k}(\mathcal{N}_{\mathcal{L}})^{\vee}\otimes\mathrm{det}(\mathcal{F}).$

###  5.3. Duality for dormant $\mathrm{PGL}_{n}$ opers of level $N$

By applying Proposition 5.2.2, (i), we obtain a bijective correspondence

(153) $\displaystyle\begin{pmatrix}\text{the set of isomorphism classes of}\\\
\text{dormant $\mathrm{GL}_{n}$-opers of level $N$ on
$\mathscr{X}$}\end{pmatrix}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\begin{pmatrix}\text{the
set of isomorphism classes of}\\\ \text{dormant $\mathrm{GL}_{p^{N}-n}$-opers
of level $N$ on $\mathscr{X}$}\\\ \end{pmatrix}.$

Moreover, this correspondence and Proposition 5.2.2, (ii), together imply the
following assertion, which is a part of Theorem B.

######  Theorem 5.3.1.

  * (i)

The assignment
$\mathscr{F}^{\heartsuit\Rightarrow\spadesuit}\mapsto\mathscr{F}^{\heartsuit\blacktriangledown\Rightarrow\spadesuit}$
defines a well-defined bijection of sets

(154)
$\displaystyle\rotatebox[origin={c}]{180.0}{$D$}_{n,\mathscr{X}}:{{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{n,\mathscr{X}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{{}^{N}}\mathcal{O}p_{p^{N}-n,\mathscr{X}}^{{}^{\mathrm{Zzz...}}}$

satisfying
$\rotatebox[origin={c}]{180.0}{$D$}_{p^{N}-n,\mathscr{X}}\circ\rotatebox[origin={c}]{180.0}{$D$}_{n,\mathscr{X}}=\mathrm{id}$.

  * (ii)

Suppose further that $r>0$. Let $\vec{\rho}:=(\rho_{i})_{i=1}^{r}$ be an
element of $(\mathfrak{S}_{n}\backslash(\mathbb{Z}/p^{N}\mathbb{Z})^{\times
n}/\Delta)^{\times r}$. Then,
$\rotatebox[origin={c}]{180.0}{$D$}_{n,\mathscr{X}}$ is restricted to a
bijection

(155)
$\displaystyle\rotatebox[origin={c}]{180.0}{$D$}_{n,\mathscr{X},\vec{\rho}}:{{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{n,\mathscr{X},\vec{\rho}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{{}^{N}}\mathcal{O}p_{p^{m}-n,\mathscr{X},\vec{\rho}^{\,\blacktriangledown}}^{{}^{\mathrm{Zzz...}}},$

where
$\vec{\rho}^{\,\blacktriangledown}:=(\rho_{i}^{\blacktriangledown})_{i=1}^{r}$
(cf. (111)).

###### Proof.

Assertions follow from Propositions 3.4.3 and 5.2.2. ∎

Moreover, the duality theorem established above implies the following
assertion, which is the remaining portion of Theorem B; note that this
assertion generalizes results proved by Y. Hoshi (cf. [Hos1, Theorem A]) and
the author (cf. [Wak1, § 4, Corollary 4.3.3]).

######  Corollary 5.3.2.

  * (i)

Let $\mathcal{L}$ be a line bundle on $X$ and $\nabla_{\circledcirc}$ a
dormant $\mathcal{D}_{X^{\mathrm{log}}}^{(N-1)}$-module structure on
$\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes
n(n-1)/2}\otimes\mathcal{L}^{\otimes n}$. Then, there exists exactly one
isomorphism class of dormant $\mathrm{GL}_{p^{N}-1}$-oper
$\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\\{\mathcal{F}^{j}\\}_{j})$ of
level $N$ on $\mathscr{X}$ such that $\mathcal{F}^{n-1}=\mathcal{L}$ and
$\mathrm{det}(\nabla)=\nabla_{\circledcirc}$ under the identification
$\mathrm{det}(\mathcal{F})=\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes
n(n-1)/2}\otimes\mathcal{L}^{\otimes n}$ given by (151).

  * (ii)

There exists exactly one isomorphism class of dormant
$\mathrm{PGL}_{p^{N}-1}$-oper of level $N$, i.e., the following equality
holds:

(156)
$\displaystyle\sharp(\mathcal{O}p_{p^{N}-1,\mathscr{X}}^{{}^{\mathrm{Zzz...}}})=1.$

Moreover, if $r>0$, then the radius of the unique dormant
$\mathrm{PGL}_{p^{N}-1}$-oper of level $N$ at any marked point coincides with
$\pi([1,2,\cdots,p^{N}-1])$ (cf. (127)).

###### Proof.

Assertion (i) follows from the observation that the desired
$\mathrm{GL}_{p^{N}-1}$-oper is the unique one corresponding, via (153), to
the level-$N$ dormant $\mathrm{GL}_{1}$-oper

(157)
$\displaystyle(F^{(N)*}_{X/k}(\mathcal{N}_{\mathcal{L}})^{\vee}\otimes\mathcal{T}_{X^{\mathrm{log}}/k}^{\otimes
n(n-1)/2}\otimes\mathcal{L}^{\otimes
n},(\nabla^{\mathrm{can}}_{\mathcal{N}_{\mathcal{L}}})^{\vee}\otimes\nabla_{\circledcirc})$

(cf. Remarks 4.2.2, (i), and 5.2.3). Assertion (ii) follows from the bijection
$\rotatebox[origin={c}]{180.0}{$D$}_{1,\mathscr{X}}$ asserted in Theorem
5.3.1, (i), and the equalities
$\sharp({{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,\mathscr{X}})=\sharp({{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{1,\mathscr{X},(\pi([0]),\cdots,\pi([0]))})=1$.
∎

##  6\. Tamely ramified coverings and dormant $\mathrm{PGL}_{2}$-opers

Recall (cf. [Moc1], [Oss4], [Oss2]) that certain tamely ramified coverings
with ramification indices $<p$ between two copies of the projective line can
be described in terms of dormant $\mathrm{PGL}_{2}$-opers (of level $1$). That
description is the starting point of the enumerative geometry of dormant opers
because it allows us to translate dormant $\mathrm{PGL}_{2}$-opers on a
$3$-pointed projective line into simple combinatorial data. In this section,
the situation is generalized to the case of higher level in order to deal with
tamely ramified covering having large ramification indices. Theorem C will be
proved at the end of this section.

###  6.1. Dormant $\mathrm{PGL}_{2}$-opers arising from tamely ramified
coverings

Denote by $\mathbb{P}$ the projective line over $k$, i.e.,
$\mathbb{P}:=\mathrm{Proj}(k[x_{1},x_{2}])$. Let
$\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i=1}^{r})$ be as before, and take an
$r$-tuple of integers $\vec{\lambda}:=(\lambda_{1},\cdots,\lambda_{r})$ with
$0<\lambda_{i}<p^{N}$ ($i=1,\cdots,r$). We shall write
$\vec{\rho}:=(\rho_{1},\cdots,\rho_{r})$, where
$\rho_{i}:=\frac{1}{2}\cdot\overline{\lambda}_{i}\in(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm
1\\}$. Under the identification $(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm
1\\}=\mathfrak{S}_{2}\backslash(\mathbb{Z}/p^{N}\mathbb{Z})^{\times 2}/\Delta$
defined in (128)), $\vec{\rho}$ may be regarded as an element of
$(\mathfrak{S}_{2}\backslash(\mathbb{Z}/p^{N}\mathbb{Z})^{\times
2}/\Delta)^{\times r}$. We shall denote by

(158) $\displaystyle\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{X},\vec{\lambda}}$

the set of equivalence classes of finite, separable, and tamely ramified
coverings $\phi:X\rightarrow\mathbb{P}$ that are ramified at $\sigma_{i}$ with
index $\lambda_{i}$ and étale elsewhere. Here, the equivalence relation is
defined in such a way that two coverings
$\phi_{1},\phi_{2}:X\rightarrow\mathbb{P}^{1}$ are equivalent if there exists
an element $h\in\mathrm{PGL}_{2}(k)\left(=\mathrm{Aut}_{k}(\mathbb{P})\right)$
with $\phi_{2}=h\circ\phi_{1}$. For each $\phi$ as above, we shall denote by
$[\phi]$ the element of
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{X},\vec{\lambda}}$ (i.e., the
equivalence class) represented by $\phi$.

In what follows, let us construct a map of sets
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{X},\vec{\lambda}}\rightarrow{{}^{N}}\mathcal{O}p_{2,\mathscr{X},\vec{\rho}}^{{}^{\mathrm{Zzz...}}}$.
Let us take an element $[\phi]$ of
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{X},\vec{\lambda}}$, and choose a
tamely ramified covering $\phi$ representing $[\phi]$. Let
$\sigma^{\prime}_{1},\cdots,\sigma^{\prime}_{r^{\prime}}$ ($0<r^{\prime}\leq
r$) be the mutually distinct points of $\mathbb{P}$ such that
$\bigcup_{i=1}^{r}\\{\phi(\sigma_{i})\\}=\\{\sigma^{\prime}_{j}\\}_{j=1}^{r^{\prime}}$.
In particular,
$\mathscr{P}^{\prime}:=(\mathbb{P},\\{\sigma^{\prime}_{j}\\}_{j=1}^{r^{\prime}})$
defines an $r^{\prime}$-pointed genus-$0$ curve; we denote the induced log
curve by $\mathbb{P}^{\mathrm{log}^{\prime}}$. Since $\phi$ is tamely
ramified, the morphism $\phi$ extends to a log étale morphism
$\phi^{\mathrm{log}}:X^{\mathrm{log}}\rightarrow\mathbb{P}^{\mathrm{log}^{\prime}}$.
Write
$\mathcal{L}:=\mathcal{O}_{\mathbb{P}}(-1)\otimes\mathcal{O}_{\mathbb{P}}(\sum_{j=1}^{r^{\prime}}\sigma_{j})$,
and write $\tau_{0}$ for the $\mathcal{O}_{\mathbb{P}}$-linear injection
$\mathcal{O}_{\mathbb{P}}(-1)\hookrightarrow\mathcal{O}_{\mathbb{P}}^{\oplus
2}$ given by $w\mapsto(wx_{1},wx_{2})$ for each local section
$w\in\mathcal{O}_{\mathbb{P}}(-1)$. Also, let $\mathcal{F}$ be a rank $2$
vector bundle on $\mathbb{P}$ which makes the following square diagram
cocartesian:

(163)

The trivial $\mathcal{D}^{(N-1)}_{\mathbb{P}^{\mathrm{log}^{\prime}}}$-module
structure on $\mathcal{O}_{\mathbb{P}}^{\oplus 2}$ extends uniquely to a
$\mathcal{D}^{(N-1)}_{\mathbb{P}^{\mathrm{log}^{\prime}}}$-module structure
$\nabla_{\mathcal{F}}$ on $\mathcal{F}$. It follows from the various
definitions involved that the composite

(164)
$\displaystyle\mathcal{L}\xrightarrow{\mathrm{inclusion}}\mathcal{F}\xrightarrow{\overline{\nabla}_{\mathcal{F}}}\Omega_{\mathbb{P}^{\mathrm{log}^{\prime}}/k}\otimes\mathcal{F}\twoheadrightarrow\Omega_{\mathbb{P}^{\mathrm{log}^{\prime}}/k}\otimes(\mathcal{F}/\mathcal{L})$

is $\mathcal{O}_{\mathbb{P}}$-linear and injective. Moreover, since
$\mathrm{deg}(\mathcal{L})=\mathrm{det}(\Omega_{\mathbb{P}^{\mathrm{log}^{\prime}}/k}\otimes(\mathcal{F}/\mathcal{L}))\left(=r^{\prime}-1\right)$,
this morphism turns out to be an isomorphism. This means that the triple
$(\mathcal{F},\nabla_{\mathcal{F}},\mathcal{L})$ forms a dormant
$\mathrm{GL}_{2}$-oper of level $N$ on $\mathscr{P}^{\prime}$ (cf. Remark
4.2.2, (ii)). Hence, the pull-back of this data via the log étale morphism
$\phi^{\mathrm{log}}$ defines a dormant $\mathrm{GL}_{2}$-oper

(165)
$\displaystyle\mathscr{F}^{\heartsuit}_{\phi}:=\phi^{\mathrm{log}*}(\mathcal{F},\nabla,\mathcal{L})$

of level $N$ on $\mathscr{X}$.

######  Proposition 6.1.1.

The level-$N$ dormant $\mathrm{PGL}_{2}$-oper
$\mathscr{F}^{\heartsuit\Rightarrow\spadesuit}_{\phi}$ on $\mathscr{X}$
represented by $\mathscr{F}_{\phi}^{\heartsuit}$ is of radii $\vec{\rho}$.

###### Proof.

The problem is the computation of the radii of
$\mathscr{F}^{\spadesuit}_{\phi}$. Let us take $i\in\\{1,\cdots,r\\}$, and
choose $j\in\\{1,\cdots,r^{\prime}\\}$ such that
$\sigma^{\prime}_{j}=\phi(\sigma_{i})$. Also, choose a local function $t$ on
$\mathbb{P}$ defining $\sigma^{\prime}_{j}$. This local function allows us to
identify the formal neighborhood $\widehat{D}_{\sigma^{\prime}_{j}}$ of
$\sigma^{\prime}_{j}$ in $\mathbb{P}$ with $\mathrm{Spec}(k[\\![t]\\!])$.
Since the ramification index of $\phi$ at $\sigma_{i}$ is $\lambda_{i}$, the
formal neighborhood $\widehat{D}_{\sigma_{i}}$ of $\sigma_{i}$ in $X$ may be
identified with $\mathrm{Spec}(k[\\![t^{1/\lambda_{i}}]\\!])$ and the
restriction of $\phi$ to $\widehat{D}_{\sigma_{i}}$ may be identified with the
morphism
$\mathrm{Spec}(k[\\![t^{1/\lambda_{i}}]\\!])\rightarrow\mathrm{Spec}(k[\\![t]\\!])$
induced by the natural inclusion $k[\\![t]\\!]\hookrightarrow
k[\\![t^{1/\lambda_{i}}]\\!]$. The
$\breve{\mathcal{D}}_{k[\\![t]\\!]}^{(N-1)}$-module corresponding to the
restriction of $(\mathcal{F},\nabla)$ to $\widehat{D}_{\sigma^{\prime}_{j}}$
is isomorphic to $(k[\\![t]\\!],\widehat{\nabla}_{0})\oplus(t^{-1}\cdot
k[\\![t]\\!],\widehat{\nabla}_{0})$. It follows that the pull-back of
$(\mathcal{F},\nabla)$ to $\phi^{\mathrm{log}}$ restricted to
$\widehat{D}_{\sigma_{i}}$ is isomorphic to
$(k[\\![t^{1/\lambda_{i}}]\\!],\widehat{\nabla}_{0})\oplus((t^{1/\lambda_{i}})^{-\lambda_{i}}\cdot
k[\\![t^{1/\lambda_{i}}]\\!],\widehat{\nabla}_{0})$ (which is also isomorphic
to
$(k[\\![s]\\!],\widehat{\nabla}_{0})\oplus(k[\\![s]\\!],\widehat{\nabla}_{\overline{\lambda}_{i}})$
by putting $s:=t^{1/\lambda_{i}}$). Hence, the exponent of
$\mathscr{F}_{\phi}^{\heartsuit}$ at $\sigma_{i}$ coincides with
$[0,\overline{\lambda}_{i}]$, which implies
$\rho_{\mathscr{F}_{\phi}^{\heartsuit\Rightarrow\spadesuit},i}=\rho_{i}$. This
completes the proof of this proposition. ∎

Since $\tau_{0}$ is invariant under pull-back by automorphisms of
$\mathbb{P}$, the isomorphism class of $\mathscr{F}^{\heartsuit}_{\phi}$ does
not depend on the choice of the representative $\phi$ of $[\phi]$. Hence, the
above proposition implies that the assignment
$[\phi]\mapsto\mathscr{F}_{\phi}^{\spadesuit}$ gives a well-defined map of
sets

(166)
$\displaystyle\Upsilon_{\mathscr{X},\vec{\lambda}}:\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{X},\vec{\lambda}}\rightarrow{{}^{N}}\mathcal{O}p_{2,\mathscr{X},\vec{\rho}}^{{}^{\mathrm{Zzz...}}}.$

###  6.2. Tamely ramified endomorphisms of a $3$-pointed projective line

Denote by $[0]$, $[1]$, and $[\infty]$ the $k$-rational points of $\mathbb{P}$
determined by the values $0$, $1$, and $\infty$ respectively. After ordering
the three points $[0],[1],[\infty]$, we obtain a unique (up to isomorphism)
$3$-pointed proper smooth curve

(167) $\mathscr{P}:=(\mathbb{P}/k,\\{[0],[1],[\infty]\\})$

of genus $0$ over $k$. In particular, we obtain a log curve
$\mathbb{P}^{\mathrm{log}}$ over $k$.

Next, let us take a triple $(\rho_{0},\rho_{1},\rho_{\infty})$ of elements of
$(\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm 1\\}$. There exists the triple
of integers $(\lambda_{0},\lambda_{1},\lambda_{\infty})$ satisfying the
following conditions:

  * (a)

$2\cdot\rho_{x}=\lambda_{x}$ as elements of
$(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm 1\\}$ and $0<\lambda_{x}<p^{N}$ for every
$x=0,1,\infty$;

  * (b)

The sum $\lambda_{0}+\lambda_{1}+\lambda_{\infty}$ is odd $<2\cdot p^{N}$.

Let us write
$\mathcal{O}_{\mathbb{P}}^{+}:=\mathcal{O}_{\mathbb{P}}(\lambda_{0}\cdot[0]+\lambda_{1}\cdot[1]+\lambda_{\infty}\cdot[\infty])$.
Note that there is a unique
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-module structure
$\nabla^{+}$ on $\mathcal{O}_{\mathbb{P}}^{+}$ whose restriction to
$U:=\mathbb{P}\setminus\\{[0],[1],[\infty]\\}$ coincides with the trivial
$\mathcal{D}_{U}^{(N-1)}$-module structure on
$\mathcal{O}_{\mathbb{P}}^{+}|_{U}=\mathcal{O}_{U}$ (cf. Remark 2.1.1).

The following assertion is a special case of [Oss1, § 3, Theorem 3.3, (ii)];
we shall prove it by a relatively elementary argument.

######  Proposition 6.2.1.

Let $\phi:\mathbb{P}\rightarrow\mathbb{P}$ be a tamely ramified covering
classified by
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$.
Then, the points $\phi([0]),\phi([1]),\phi([\infty])$ are mutually distinct.

###### Proof.

First, we shall suppose that the set $\phi(\\{[0],[1],[\infty]\\})$ consists
of one point. By considering the fiber of $\phi$ over this point, we see that
$\mathrm{deg}(\phi)\geq\lambda_{0}+\lambda_{1}+\lambda_{\infty}$. It follows
from the Riemann-Hurwitz formula that

(168) $\displaystyle-2\left(=2\cdot(\text{genus of
$\mathbb{P}$})-2\right)=-2\cdot\mathrm{deg}(\phi)+\sum_{x=0,1,\infty}(\lambda_{x}-1)\leq-(\lambda_{0}+\lambda_{1}+\lambda_{\infty})-3.$

Thus, we obtain a contradiction. Next, suppose that
$\phi(\\{[0],[1],[\infty]\\})$ consists of two points. After possibly applying
a linear transformation, we may assume that these two elements coincide with
$\\{[0],[\infty]\\}$, and that $\\{[0]\\}\subseteq\phi^{-1}([0])$ and
$\\{[1],[\infty]\\}\subseteq\phi^{-1}([\infty])$. Hence, $\phi$ defines a
tamely ramified covering of
$\mathbb{G}_{m}\left(=\mathbb{P}\setminus\\{[0],[\infty]\\}\right)$. Recall
that the tame fundamental group $\pi^{\mathrm{tame}}_{1}(\mathbb{G}_{m})$ of
$\mathbb{G}_{m}$ is isomorphic to $\widehat{\mathbb{Z}}^{p^{\prime}}$, the
maximal prime-to-$p$ quotient of
$\widehat{\mathbb{Z}}:=\varprojlim_{n\in\mathbb{Z}_{\geq
0}}\mathbb{Z}/n\mathbb{Z}$. A topological generator $\sigma$ of
$\pi^{\mathrm{tame}}_{1}(\mathbb{G}_{m})$ acts on the fiber over the point
near $[0]$ as a cyclic permutation. On the other hand, $\sigma^{-1}$ acts on
the fiber over the point near $[\infty]$ as a product of two disjoint cyclic
permutations. This is a contradiction. Hence, the the image
$\phi(\\{[0],[1],[\infty]\\})$ consists of three points. This completes the
proof of this assertion. ∎

######  Remark 6.2.2.

Because of Proposition 6.2.1, each element of
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$
has a unique representative $\phi:\mathbb{P}\rightarrow\mathbb{P}$ satisfying
$\phi([x])=[x]$ for every $x=0,1,\infty$. Following [ABEGKM] (or [BEK]), we
call such a covering a dynamical Belyi map.

Now, let us take a dynamical Belyi map $\phi:\mathbb{P}\rightarrow\mathbb{P}$
classified by
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$;
this covering corresponds to a representation of the tame fundamental group
$\pi_{1}^{\mathrm{tame}}(\mathbb{P}\setminus\\{[0],[1],[\infty]\\})$ (which is
obtained as a quotient of the profinite completion $\widehat{\Pi}_{0,3}$ of
the group
$\Pi_{0,3}:=\langle\gamma_{0},\gamma_{1},\gamma_{\infty}\,|\,\gamma_{0}\gamma_{1}\gamma_{\infty}=1\rangle$).
Hence, by the above proposition, $\phi$ determines three cyclic permutations
$\sigma_{0}$, $\sigma_{1}$, and $\sigma_{\infty}$ (in the symmetric group
$\mathfrak{S}_{d}$ of $d$ letters for some $d\geq 1$) of orders $\lambda_{0}$,
$\lambda_{1}$, and $\lambda_{\infty}$, respectively, satisfying
$\sigma_{0}\circ\sigma_{1}=\sigma_{\infty}$. A trivial elementary argument
shows that this condition implies the following inequalities:

(169)
$\displaystyle|\lambda_{0}-\lambda_{1}|<\lambda_{\infty}<\lambda_{0}+\lambda_{1}.$

These inequalities also can be obtained by the inequality
$\mathrm{deg}(\phi)\left(=\frac{\lambda_{0}+\lambda_{1}+\lambda_{\infty}-1}{2}\right)\geq\lambda_{0},\lambda_{1},\lambda_{\infty}$.

Conversely, suppose that a subgroup of $\mathfrak{S}_{d}$ generated by three
cyclic permutations $\sigma_{0}$, $\sigma_{1}$, $\sigma_{\infty}$ with
$\sigma_{0}\circ\sigma_{1}=\sigma_{\infty}$ has order prime to $p$. Then, the
assignment $\gamma_{x}\mapsto\sigma_{x}$ ($x=0,1,\infty$) induces a
representation
$\pi_{1}^{\mathrm{tame}}(\mathbb{P}\setminus\\{[0],[1],[\infty]\\})\rightarrow\mathfrak{S}_{d}$
because the surjection
$\widehat{\Pi}_{0,3}\twoheadrightarrow\pi_{1}^{\mathrm{tame}}(\mathbb{P}\setminus\\{[0],[1],[\infty]\\})$
becomes bijective after taking their maximal prime-to-$p$ quotients. In
particular, the corresponding tamely ramified covering is classified by
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$.
See [BEK] for the study concerning the relationship between such cyclic
permutations and dynamical Belyi maps in characteristic $p$.

###  6.3. Dormant $\mathrm{PGL}_{2}$-opers on a $3$-pointed projective line

In what follows, we shall prove that the map
$\Upsilon_{\mathscr{X},\vec{\lambda}}$ defined in (166) becomes a bijection in
the case where $\mathscr{X}=\mathscr{P}$. To do this, we construct its inverse
map. We first prove the following proposition.

######  Proposition 6.3.1.

Let $\mathscr{F}^{\spadesuit}$ be a dormant $\mathrm{PGL}_{2}$-oper of level
$N$ on $\mathscr{P}$. Then, there exists a dormant $\mathrm{GL}_{2}$-oper
$\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\mathcal{L})$ of level $N$ on
$\mathscr{P}$ satisfying
$\mathscr{F}^{\heartsuit\Rightarrow\spadesuit}\cong\mathscr{F}^{\spadesuit}$
and
$\mathrm{det}(\mathcal{F},\nabla)\cong(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})$,
Moreover, such a $\mathrm{GL}_{2}$-oper is uniquely determined up isomorphism.

###### Proof.

First, we prove the existence portion. Let us take a dormant
$\mathrm{GL}_{2}$-oper
$\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\mathcal{L})$ of level $N$ on
$\mathscr{P}$ with
$\mathscr{F}^{\heartsuit\Rightarrow\spadesuit}\cong\mathscr{F}^{\spadesuit}$.
Since
$\mathcal{L}\cong\Omega_{\mathbb{P}^{\mathrm{log}}}\otimes(\mathcal{F}/\mathcal{L})$
(cf. (123)), the following equalities hold:

(170)
$\displaystyle\mathrm{deg}(\mathcal{O}^{+}_{\mathbb{P}}\otimes\mathrm{det}(\mathcal{F})^{\vee})$
$\displaystyle=\lambda_{0}+\lambda_{1}+\lambda_{\infty}-\mathrm{deg}(\mathcal{L})-\mathrm{deg}(\mathcal{F}/\mathcal{L})$
$\displaystyle=\lambda_{0}+\lambda_{1}+\lambda_{\infty}+1-2\cdot\mathrm{deg}(\mathcal{L}).$

In particular, the degree of
$\mathcal{O}^{+}_{\mathbb{P}}\otimes\mathrm{det}(\mathcal{F})^{\vee}$ is even.
Hence, it follows from Lemma 4.1.4 that there exists a dormant
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-bundle
$(\mathcal{N},\nabla_{\mathcal{N}})$ such that $\mathcal{N}$ is a line bundle
with $\mathcal{N}^{\otimes
2}\cong(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})\otimes\mathrm{det}(\mathcal{F},\nabla_{\mathcal{F}})^{\vee}$.
By putting
$(\mathcal{F}^{\prime},\nabla^{\prime},\mathcal{L}^{\prime}):=\mathscr{F}^{\heartsuit}_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})}$,
we have

(171)
$\displaystyle\mathrm{det}(\mathcal{F}^{\prime},\nabla^{\prime})\cong(\mathcal{N},\nabla_{\mathcal{N}})^{\otimes
2}\otimes\mathrm{det}(\mathcal{F},\nabla)\cong(\mathcal{O}^{+}_{\mathbb{P}},\nabla^{+}).$

Thus, $(\mathcal{F}^{\prime},\nabla^{\prime},\mathcal{L}^{\prime})$ specifies
the required $\mathrm{GL}_{2}$-oper of level $N$.

Next, we shall prove the uniqueness portion. Let
$\mathscr{F}^{\heartsuit}_{i}:=(\mathcal{F}_{i},\nabla_{i},\mathcal{L}_{i})$
($i=1,2$) be dormant $\mathrm{GL}_{2}$-opers of level $N$ on $\mathscr{P}$
satisfying the required conditions. Since
$\mathscr{F}_{1}^{\heartsuit\Rightarrow\spadesuit}=\mathscr{F}_{2}^{\heartsuit\Rightarrow\spadesuit}$,
there exists a dormant
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-module
$(\mathcal{N},\nabla_{\mathcal{N}})$ such that $\mathcal{N}$ is a line bundle
and
$\mathscr{F}_{2}^{\heartsuit}\cong(\mathscr{F}_{1}^{\heartsuit})_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})}$.
If $\nabla^{\mathrm{triv}}$ denotes the trivial
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-module structure on
$\mathcal{O}_{\mathbb{P}}$, then we have

(172) $\displaystyle(\mathcal{O}_{\mathbb{P}},\nabla^{\mathrm{triv}})$
$\displaystyle\cong\mathrm{det}(\mathcal{F}_{2},\nabla_{2})\otimes(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})^{\vee}$
$\displaystyle\cong\mathrm{det}((\mathcal{N},\nabla_{\mathcal{N}})\otimes(\mathcal{F}_{1},\nabla_{1}))\otimes(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})^{\vee}$
$\displaystyle\cong(\mathcal{N},\nabla_{\mathcal{N}})^{\otimes
2}\otimes\mathrm{det}(\mathcal{F}_{1},\nabla_{1})\otimes(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})^{\vee}$
$\displaystyle\cong(\mathcal{N},\nabla_{\mathcal{N}})^{\otimes 2}.$

Since
$\mathrm{Pic}(\mathbb{P})\cong[\mathcal{O}_{\mathbb{P}}(1)]\cdot\mathbb{Z}$,
the line bundle $\mathcal{N}$ may be identified with
$\mathcal{O}_{\mathbb{P}}$. Moreover, by the uniqueness portion of Proposition
4.1.4, $\nabla_{\mathcal{N}}$ coincides with $\nabla^{\mathrm{triv}}$ via a
fixed identification $\mathcal{N}=\mathcal{O}_{\mathbb{P}}$. Thus, we have
$(\mathscr{F}_{1}^{\heartsuit})_{\otimes(\mathcal{N},\nabla_{\mathcal{N}})}\cong\mathscr{F}_{1}^{\heartsuit}$,
which implies that $\mathscr{F}^{\heartsuit}_{2}$ is isomorphic to
$\mathscr{F}^{\heartsuit}_{1}$. This completes the proof of the uniqueness
portion. ∎

Now, let us take a dormant $\mathrm{PGL}_{2}$-oper $\mathscr{F}^{\spadesuit}$
on $\mathscr{P}$ of level $N$ and radii $(\rho_{0},\rho_{1},\rho_{\infty})$.
Also, let $\mathscr{F}^{\heartsuit}:=(\mathcal{F},\nabla,\mathcal{L})$ be the
dormant $\mathrm{GL}_{2}$-oper of level $N$ resulting from Proposition 6.3.1
applied to $\mathscr{F}^{\spadesuit}$. In particular, we have
$\mathrm{deg}(\mathcal{L})=\frac{\lambda_{0}+\lambda_{1}+\lambda_{\infty}+1}{2}$
and
$\mathrm{deg}(\mathcal{F}/\mathcal{L})=\frac{\lambda_{0}+\lambda_{1}+\lambda_{\infty}-1}{2}$.
Denote by $\tau$ the $\mathcal{O}_{\mathbb{P}}$-linear morphism
$F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}^{\nabla})\rightarrow\mathcal{F}$
extending the ($\mathcal{O}_{\mathbb{P}^{(N)}}$-linear) inclusion
$\mathcal{F}^{\nabla}\hookrightarrow\mathcal{F}$; the morphism $\tau$ is
compatible with $\nabla^{\mathrm{can}}_{\mathcal{F}^{\nabla}}$ (cf. Remark
4.1.2) and $\nabla$. We shall write
$\mathcal{L}^{\natural}:=\mathcal{L}\cap\mathrm{Im}(\tau)$. Since the
restriction of $\tau$ to $U:=\mathbb{P}\setminus\\{[0],[1],[\infty]\\}$ is an
isomorphism, the quotient sheaf $\mathcal{L}/\mathcal{L}^{\natural}$ is a
torsion sheaf supported on $\\{[0],[1],[\infty]\\}$.

######  Lemma 6.3.2.

The length of $\mathcal{L}/\mathcal{L}^{\natural}$ at the marked point
$x\in\\{[0],[1],[\infty]\\}$ is $\lambda_{x}$. Moreover, the natural inclusion
$\mathcal{L}/\mathcal{L}^{\natural}\hookrightarrow\mathrm{Coker}(\tau)$ is an
isomorphism.

###### Proof.

Let us fix $x\in\\{[0],[1],[\infty]\\}$, and choose a local function $t$
defining $x$. The restriction of $(\mathcal{F},\nabla)$ to the formal
neighborhood $\widehat{D}_{x}$ of $x$ may be expressed as
$(k[\\![t]\\!],\widehat{\nabla}_{\overline{a}})\oplus(k[\\![t]\\!],\widehat{\nabla}_{\overline{b}})$
for some integers $a,b$ with $0\leq b\leq a\leq p^{N}-1$. The radius of
$\nabla$ at $x$ coincides with $\rho_{x}$ by assumption, so the equality
$2\cdot\rho_{x}=\overline{a-b}$ of elements in
$(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm 1\\}$ holds. Since
$\mathrm{det}(\mathcal{F},\nabla)\cong(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})$,
a computation of the lengths at $x$ of $\mathrm{det}(\mathcal{F},\nabla)$ and
$(\mathcal{O}_{\mathbb{P}}^{+},\nabla^{+})$ implies $a+b\equiv\lambda_{x}$
(mod $p^{N}$). Hence, since $a+b\leq 2\cdot p^{N}$, it follows that $a+b$ is
either $\lambda_{x}$ or $\lambda_{x}+p^{N}$. We shall prove the claim that
$a+b=\lambda_{x}$. Suppose, on the contrary, that $a+b=p^{N}+\lambda_{x}$. The
condition that $0\leq a-b\leq p^{N}-1$ and $2\cdot\rho_{x}=\overline{a-b}$ in
$(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm 1\\}$ implies that $a-b$ is either
$\lambda_{x}$ or $p^{N}-\lambda_{x}$. Since $a+b$ and $a-b$ have the same
parity, the equality $a-b=p^{N}-\lambda_{x}$ holds. Hence, we have

(173) $\displaystyle 2\cdot
a=(a+b)+(a-b)=(p^{N}+\lambda_{x})+(p^{N}-\lambda_{x})=2\cdot p^{N}.$

This implies the equality $a=p^{N}$, which is a contradiction. This proves the
claim.

By the claim proved just now, we have $a+b=\lambda_{x}$ and $a-b=\lambda_{x}$.
In particular, $a=\lambda_{x}$ and $b=0$, which means that the restriction of
$(\mathcal{F},\nabla)$ to $\widehat{D}_{x}$ is isomorphic to
$(k[\\![t]\\!],\widehat{\nabla}_{\overline{\lambda}_{x}})\oplus(k[\\![t]\\!],\widehat{\nabla}_{0})$.
It follows that the restriction of $\mathrm{Im}(\tau)$ to $\widehat{D}_{x}$
coincides with $t^{\lambda_{x}}\cdot k[\\![t]\\!]\oplus
k[\\![t]\\!]\left(\subseteq k[\\![t]\\!]^{\oplus 2}\right)$. On the other
hand, according to the proof in Proposition 3.4.1, the inclusion
$\mathcal{L}\hookrightarrow\mathcal{F}$ corresponds, after choosing a suitable
trivialization
$\Gamma(\widehat{D}_{x},\mathcal{L}|_{\widehat{D}_{x}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k[\\![t]\\!]$,
to the $k[\\![t]\\!]$-linear morphism $k[\\![t]\\!]\rightarrow
k[\\![t]\\!]^{\oplus 2}$ given by $1\mapsto(u_{1},u_{2})$ for some
$u_{1},u_{2}\in k[\\![t]\\!]^{\times}$. By taking account of this observation,
we see that the length of $\mathcal{L}/\mathcal{L}^{\natural}$ at $x$
coincides with $\lambda_{x}$. Moreover, since the length of
$\mathrm{Coker}(\tau)$ at $x$ is $a+b=\lambda_{x}$, the inclusion
$\mathcal{L}/\mathcal{L}^{\natural}\hookrightarrow\mathrm{Coker}(\tau)$ turns
out to be an isomorphism. ∎

######  Lemma 6.3.3.

There exists an $\mathcal{O}_{\mathbb{P}^{(N)}}$-linear isomorphism
$\gamma:\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{F}^{\nabla}$.

###### Proof.

Since $\mathcal{F}^{\nabla}$ is a rank $2$ vector bundle on the genus-$0$
curve $\mathbb{P}^{(N)}$, it is isomorphic to the direct sum of two line
bundles. Let us fix an isomorphism
$\gamma:\mathcal{O}_{\mathbb{P}^{(N)}}(a)\oplus\mathcal{O}_{\mathbb{P}^{(N)}}(b)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{F}^{\nabla}$,
where $a$ and $b$ are some integers with $a\geq b$. The pull-back of $\gamma$
via $F^{(N)}_{\mathbb{P}/k}$ defines an isomorphism
$\gamma^{F}:\mathcal{O}_{\mathbb{P}}(p^{N}\cdot
a)\oplus\mathcal{O}_{\mathbb{P}}(p^{N}\cdot
b)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}^{\nabla})$.
It follows from Lemma 6.3.2 that

(174) $\displaystyle\lambda_{0}+\lambda_{1}+\lambda_{\infty}$
$\displaystyle=\mathrm{length}(\mathcal{L}/\mathcal{L}^{\natural})$
$\displaystyle=\mathrm{length}(\mathrm{Coker}(\tau))$
$\displaystyle=\mathrm{deg}(\mathcal{F})-\mathrm{deg}(F^{N*}(\mathcal{F}^{\nabla}))$
$\displaystyle=\lambda_{0}+\lambda_{1}+\lambda_{\infty}+p^{N}(a+b).$

This implies $b=-a$. Next, let us consider the composite

(175) $\displaystyle\mathcal{O}_{\mathbb{P}}(p^{N}\cdot
a)\xrightarrow{v\mapsto(v,0)}\mathcal{O}_{\mathbb{P}}(p^{N}\cdot
a)\oplus\mathcal{O}_{\mathbb{P}}(-p^{N}\cdot
a)\xrightarrow{\gamma^{F}}F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}^{\nabla})\xrightarrow{\tau}\mathcal{F}\twoheadrightarrow\mathcal{F}/\mathcal{L}.$

We shall suppose that $a>0$. The assumption
$\lambda_{0}+\lambda_{1}+\lambda_{\infty}<2\cdot p^{N}$ implies

(176) $\displaystyle\mathrm{deg}(\mathcal{O}_{\mathbb{P}}(p^{N}\cdot a))\geq
p^{N}>\frac{\lambda_{0}+\lambda_{1}+\lambda_{\infty}-1}{2}=\mathrm{deg}(\mathcal{F}/\mathcal{L}).$

Hence, the composite (175) must be the zero map. It follows that the image
$\mathcal{I}$ of the inclusion into the first factor
$\mathcal{O}_{\mathbb{P}}(p^{N}\cdot
a)\hookrightarrow\mathcal{O}_{\mathbb{P}}(p^{N}\cdot
a)\oplus\mathcal{O}_{\mathbb{P}}(-p^{N}\cdot a)$ is contained in
$\mathcal{L}\left(\subseteq\mathcal{F}\right)$ via $\tau\circ\gamma^{F}$. But,
since $\mathcal{I}$ is closed under
$\nabla_{\mathcal{O}_{\mathbb{P}^{(N)}}(a)}^{\mathrm{can}}\oplus\nabla_{\mathcal{O}_{\mathbb{P}^{(N)}}(b)}^{\mathrm{can}}$,
the line subbundle $\mathcal{L}$ must be closed under $\nabla$. This
contradicts the fact that $\mathrm{KS}_{\mathscr{F}^{\heartsuit}}^{1}$ (cf.
(122)) is an isomorphism. Thus, we conclude that $a=0$, i.e., $\gamma$ defines
an isomorphism $\mathcal{O}_{\mathbb{P}}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{F}^{\nabla}$. ∎

Let $\gamma$ be as asserted in the above lemma. For convenience, we
occasionally use the notation $X$ to denote the underlying projective line
$\mathbb{P}$. The pull-back $\gamma^{F}:\mathcal{O}_{X}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{(N)*}_{X/k}(\mathcal{F}^{\nabla})$
of $\gamma$ by $F^{(N)}_{X/k}$ induces a trivialization
$\mathbb{P}(\gamma^{F}):\mathbb{P}\times_{k}X\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{P}(F^{(N)*}_{X/k}(\mathcal{F}^{\nabla}))$
of the $\mathbb{P}$-bundle $\mathbb{P}(F^{(N)*}_{X/k}(\mathcal{F}^{\nabla}))$
associated to $F^{(N)*}_{X/k}(\mathcal{F}^{\nabla})$. The sheaf
$\mathcal{L}^{\natural}$, regarded as a line bundle of
$F^{(N)*}_{X/k}(\mathcal{F}^{\nabla})$ via $\tau$, defines a global section
$\sigma:X\rightarrow\mathbb{P}(F^{(N)*}_{X/k}(\mathcal{F}^{\nabla}))$. Thus,
we obtain the composite

(177)
$\displaystyle\phi_{\mathscr{F}^{\spadesuit}}:X\xrightarrow{\sigma}\mathbb{P}(F^{(N)*}_{X/k}(\mathcal{F}^{\nabla}))\xrightarrow{\mathbb{P}(\gamma^{F})^{-1}}\mathbb{P}\times_{k}X\xrightarrow{\mathrm{pr}_{1}}\mathbb{P}.$

######  Lemma 6.3.4.

The morphism $\phi_{\mathscr{F}^{\spadesuit}}:X\rightarrow\mathbb{P}^{1}$
defines a tamely ramified covering classified by the set
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$.

###### Proof.

Let $x$ be a $k$-rational point of $X$. To complete the proof, we shall
describe the morphism $\phi_{\mathscr{F}^{\spadesuit}}$ restricted to the
formal neighborhood $\widehat{D}_{x}$ of $x$.

First, suppose that $x\in\\{[0],[1],[\infty]\\}$. As observed in the proof of
Lemma 6.3.2, the $\breve{\mathcal{D}}_{k[[t]]}^{(N-1)}$-module corresponding
to the restriction of $(\mathcal{F},\nabla)$ to $\widehat{D}_{x}$ is
isomorphic to $(t^{-\lambda_{x}}\cdot
k[\\![t]\\!],\widehat{\nabla}_{0})\oplus(k[\\![t]\\!],\widehat{\nabla}_{0})\left(\cong(k[\\![t]\\!],\widehat{\nabla}_{\overline{\lambda}_{x}})\oplus(k[\\![t]\\!],\widehat{\nabla}_{0})\right)$.
According to the proof of Proposition 3.4.1, the inclusion
$\mathcal{L}\hookrightarrow\mathcal{F}$ restricted to $\widehat{D}_{x}$ may be
identified, after choosing a suitable trivialization
$\Gamma(\widehat{D}_{x},\mathcal{L}|_{\widehat{D}_{x}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k[\\![t]\\!]$,
with the $k[\\![t]\\!]$-linear morphism $k[\\![t]\\!]\rightarrow
t^{-\lambda_{x}}\cdot k[\\![t]\\!]\oplus k[\\![t]\\!]$ determined by
$1\mapsto(t^{-\lambda_{x}}\cdot 1,u)$ for some $u\in k[\\![t]\\!]^{\times}$.
Under this identification, the inclusion
$\mathcal{L}^{\natural}\hookrightarrow F_{X/k}^{(N)*}(\mathcal{F}^{\nabla})$
restricted to $\widehat{D}_{x}$ corresponds to the inclusion
$t^{\lambda_{x}}\cdot k[\\![t]\\!]\rightarrow k[\\![t]\\!]^{\oplus 2}$
determined by $t^{\lambda_{x}}\cdot 1\mapsto(1,t^{\lambda_{x}}\cdot u)$. This
implies that the restriction of $\phi_{\mathscr{F}^{\spadesuit}}$ to
$\widehat{D}_{x}$ arises from the $k$-algebra endomorphism of $k[\\![t]\\!]$
given by $t\mapsto t^{\lambda_{x}}\cdot u$. Hence, the ramification index of
$\phi_{\mathscr{F}^{\spadesuit}}$ at $x$ coincides with $\lambda_{x}$.

Next, suppose that $x\in X\setminus\\{[0],[1],[\infty]\\}$. The inclusion
$\mathcal{L}\hookrightarrow\mathcal{F}$ restricted to $\widehat{D}_{x}$ may be
described, after choosing suitable trivializations of
$\mathcal{L}|_{\widehat{D}_{x}}$ and $\mathcal{F}|_{\widehat{D}_{x}}$, as the
$k[\\![t]\\!]$-linear morphism $k[\\![t]\\!]\rightarrow k[\\![t]\\!]^{\oplus
2}$ determined by $1\mapsto(1,t\cdot v)$ for some $v\in
k[\\![t]\\!]^{\times}$. Hence, by the same reason as above, the ramification
index of $\phi_{\mathscr{F}^{\spadesuit}}$ at $x$ turns out to be $1$, which
means that $\phi_{\mathscr{F}^{\spadesuit}}$ is étale at $x$. This completes
the proof of this lemma. ∎

Note that the resulting element
$[\phi_{\mathscr{F}^{\spadesuit}}]\in\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$
does not depend on the choice of the trivialization
$\gamma:\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{F}^{\nabla}$ because
$\gamma$ is uniquely determined up to forward composition with an element of
$\mathrm{Aut}_{\mathcal{O}_{\mathbb{P}^{(N)}}}(\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2})\left(=\mathrm{PGL}_{2}(k)\right)$. Thus, the assignment
$\mathscr{F}^{\spadesuit}\mapsto[\phi_{\mathscr{F}^{\spadesuit}}]$ determines
a well-defined map of sets
${{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\rho_{0},\rho_{1},\rho_{\infty})}\rightarrow\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$.
One may verify that this map specifies, by construction, the inverse to the
map $\Upsilon_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$. Thus,
we have obtained the following assertion.

######  Proposition 6.3.5.

The assignments $[\phi]\mapsto\mathscr{F}^{\spadesuit}_{\phi}$ (i.e,
$\Upsilon_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}$) and
$\mathscr{F}^{\spadesuit}\mapsto[\phi_{\mathscr{F}^{\spadesuit}}]$ constructed
above give a bijective correspondence

(178)
$\displaystyle{{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\rho_{0},\rho_{1},\rho_{\infty})}\cong\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},(\lambda_{0},\lambda_{1},\lambda_{\infty})}.$

###  6.4. Correspondence between tamely ramified coverings and dormant
$\mathrm{PGL}_{2}$-opers

The following proposition (together with Proposition 6.3.5) may be regarded as
a variant of the rigidity assertion for dynamical Belyi maps proved in [LiOs2,
§ 2, Lemma 2.1].

######  Proposition 6.4.1.

Let $(\rho_{0},\rho_{1},\rho_{\infty})$ be an element of
$((\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm 1\\})^{\times 3}$. Then, a dormant
$\mathrm{PGL}_{2}$-oper on $\mathscr{P}$ of level $N$ and radii
$(\rho_{0},\rho_{1},\rho_{\infty})$ is, if it exists, uniquely determined.
That is to say, the following inequality holds:

(179)
$\displaystyle\sharp({{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\rho_{0},\rho_{1},\rho_{\infty})})\leq
1.$

###### Proof.

Suppose that
${{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\rho_{0},\rho_{1},\rho_{\infty})}\neq\emptyset$.
By the canonical morphism
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(0)}\rightarrow\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$,
each element of
${{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\rho_{0},\rho_{1},\rho_{\infty})}$
induces a dormant $\mathrm{PGL}_{2}$-oper of level $1$ and radii
$(\overline{\rho}_{0},\overline{\rho}_{1},\overline{\rho}_{\infty})$, where
$\overline{\rho}_{x}$ (for $x=0,1,\infty$) denotes the image of $\rho_{x}$ via
the natural surjection $(\mathbb{Z}/p^{N}\mathbb{Z})/\\{\pm
1\\}\twoheadrightarrow\mathbb{F}_{p}/\\{\pm 1\\}$. Hence,
${{}^{1}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\overline{\rho}_{0},\overline{\rho}_{1},\overline{\rho}_{\infty})}$
must be nonempty. This fact together with a comment in Remark 4.3.3 implies
that
$(\overline{\rho}_{0},\overline{\rho}_{1},\overline{\rho}_{\infty})\in(\mathbb{F}_{p}^{\times}/\\{\pm
1\\})^{\times 3}$, or equivalently,
$(\rho_{0},\rho_{1},\rho_{\infty})\in((\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm
1\\})^{\times 3}$. In particular, there exists a triple of integers
$(\lambda_{0},\lambda_{1},\lambda_{\infty})$ associated to
$(\rho_{0},\rho_{1},\rho_{\infty})$ satisfying the conditions (a), (b)
described in § 6.2.

Now, suppose that there exist two dormant $\mathrm{PGL}_{2}$-opers
$\mathscr{F}_{1}^{\spadesuit}$, $\mathscr{F}^{\spadesuit}_{2}$ on
$\mathscr{P}$ of level $N$ and radii $(\rho_{0},\rho_{1},\rho_{\infty})$. For
each $j=1,2$, denote by
$\mathscr{F}^{\heartsuit}_{j}:=(\mathcal{F}_{j},\nabla_{j},\mathcal{L}_{j})$
the dormant $\mathrm{GL}_{n}$-oper of level $N$ resulting from Proposition
6.3.1 applied to $\mathscr{F}^{\spadesuit}_{j}$. The inclusion
$\mathcal{F}_{j}^{\nabla}\hookrightarrow\mathcal{F}_{j}$ extends to an
$\mathcal{O}_{\mathbb{P}}$-linear morphism
$\tau_{j}:F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}_{j}^{\nabla})\rightarrow\mathcal{F}_{j}$.
Let us write
$\mathcal{L}_{j}^{\natural}:=\mathcal{L}_{j}\cap\mathrm{Im}(\tau_{j})$ and
write $\iota_{j}$ for the natural isomorphism
$\mathcal{L}_{j}/\mathcal{L}_{j}^{\natural}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Coker}(\tau_{j})$
(cf. Lemma 6.3.2). Also, denote by $\overline{\nabla}_{j}$ the
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(0)}$-module structure on
$\mathcal{F}_{j}$ induced from $\nabla_{j}$. The triple
$\overline{\mathscr{F}}_{j}^{\heartsuit}:=(\mathcal{F}_{j},\overline{\nabla}_{j},\mathcal{L}_{j})$
defines a dormant $\mathrm{GL}_{2}$-oper in the classical sense, i.e., of
level $1$. In particular, it induces a dormant $\mathrm{PGL}_{2}$-oper
$\overline{\mathscr{F}}_{i}^{\heartsuit\Rightarrow\spadesuit}$ on
$\mathscr{P}$ of level $1$ and radii
$(\overline{\rho}_{0},\overline{\rho}_{1},\overline{\rho}_{\infty})$. Recall
from [Moc1, Chap. I, § 4, Theorem 4.4] (cf. Remark 4.3.3) that dormant
$\mathrm{PGL}_{2}$-opers of level $1$ on $\mathscr{P}$ are completely
determined by their radii. It follows that
$\overline{\mathscr{F}}_{1}^{\heartsuit\Rightarrow\spadesuit}=\overline{\mathscr{F}}_{2}^{\heartsuit\Rightarrow\spadesuit}$.
By the uniqueness assertion in Proposition 6.3.1, there exists an isomorphism
of $\mathrm{GL}_{2}$-opers
$\alpha:\overline{\mathscr{F}}_{1}^{\heartsuit}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\overline{\mathscr{F}}_{2}^{\heartsuit}$.
This isomorphism is restricted to an isomorphism
$\alpha|_{\mathcal{L}_{1}}:\mathcal{L}_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}_{2}$,
which induces, via taking the respective quotients, an isomorphism
$\alpha|_{\mathcal{L}_{1}/\mathcal{L}^{\natural}_{1}}:\mathcal{L}_{1}/\mathcal{L}_{1}^{\natural}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}_{2}/\mathcal{L}^{\natural}_{2}$.
The composite
$\alpha|_{\mathrm{Coker}(\tau_{1})}:=\iota_{2}\circ\alpha|_{\mathcal{L}_{1}/\mathcal{L}^{\natural}_{1}}\circ\iota_{1}^{-1}$
specifies an isomorphism
$\mathrm{Coker}(\tau_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Coker}(\tau_{2})$.

In what follows, we shall prove the commutativity of the following square
diagram:

(184)

where $\pi_{j}$ ($j=1,2$) denotes the natural projection
$\mathcal{F}_{j}\twoheadrightarrow\mathrm{Coker}(\tau_{j})$. Let us take
$j\in\\{1,2\\}$, $x\in\\{[0],[1],[\infty]\\}$. Also, choose a local function
$t$ on $\mathbb{P}$ defining $x$. Denote by $\widehat{D}_{x}$ the formal
neighborhood of $x$ in $\mathbb{P}$, which may be identified with
$\mathrm{Spec}(k[\\![t]\\!])$. Fix an identification of the restriction of
$(\mathcal{F}_{j},\nabla_{j})$ to $\widehat{D}_{x}$ with
$(k[\\![t]\\!],\widehat{\nabla}_{\overline{\lambda}_{x}})\oplus(k[\\![t]\\!],\widehat{\nabla}_{0})$
(cf. the proof of Lemma 6.3.2). As observed in the proof of Proposition 3.4.1,
the inclusion $\mathcal{L}\hookrightarrow\mathcal{F}$ corresponds, after
choosing a suitable trivialization
$\Gamma(\widehat{D}_{x},\mathcal{L}|_{\widehat{D}_{x}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k[\\![t]\\!]$,
to the $k[\\![t]\\!]$-linear morphism $k[\\![t]\\!]\rightarrow
k[\\![t]\\!]^{\oplus 2}$ given by $1\mapsto(u_{j},1)$ for some $u_{j}\in
k[\\![t]\\!]^{\times}$. The isomorphism $\alpha$ restricted to
$\widehat{D}_{x}$ defines an automorphism of
$(k[\\![t]\\!],\widehat{\nabla}_{\overline{\lambda}_{x}})\oplus(k[\\![t]\\!],\widehat{\nabla}_{0})$.
But, since $\rho_{x}\neq 0$ in $\mathbb{F}_{p}/\\{\pm 1\\}$ (or equivalently,
$\overline{\lambda}_{x}\neq 0$), it follows from Proposition 3.2.1, (ii), that
this automorphism may be expressed as $\mu_{v}\oplus\mu_{w}$ for some $v,w\in
k[\\![t]\\!]^{\times}$, where $\mu_{(-)}$ denotes the endomorphism of
$k[\\![t]\\!]$ given by multiplication by $(-)$. Since
$\alpha|_{\mathcal{L}_{1}}$ is obtained as the restriction of $\alpha$, the
restriction of $\alpha|_{\mathcal{L}_{1}}$ to $\widehat{D}_{x}$ may be
expressed as $\mu_{w}$, and the equality $vu_{1}=wu_{2}$ holds. Hence, for
each $(g,h)\in k[\\![t]\\!]^{\oplus
2}\left(=\mathcal{F}_{1}|_{\widehat{D}_{x}}\right)$, we have

(185) $\displaystyle(\pi_{2}\circ\alpha)((g,h))$
$\displaystyle=\pi_{2}((vg,wh))$ $\displaystyle=\left(\frac{vg}{u_{2}}\cdot
u_{2},\frac{vg}{u_{2}}\cdot 1\right)\text{mod}\ \mathrm{Im}(\tau_{2})$
$\displaystyle=\iota_{2}\left(\frac{vg}{u_{2}}\ \mathrm{mod}\
\mathcal{L}^{\natural}_{2}\right)$
$\displaystyle=(\iota_{2}\circ\alpha|_{\mathcal{L}_{1}/\mathcal{L}^{\natural}_{1}})\left(\frac{vg}{wu_{2}}\
\mathrm{mod}\ \mathcal{L}^{\natural}_{1}\right)$
$\displaystyle=(\alpha|_{\mathrm{Coker}(\tau_{1})}\circ\iota_{1})\left(\frac{g}{u_{1}}\
\mathrm{mod}\ \mathcal{L}^{\natural}_{1}\right)$
$\displaystyle=\alpha|_{\mathrm{Coker}(\tau_{1})}\left(\left(\frac{g}{u_{1}}\cdot
u_{1},\frac{g}{u_{1}}\cdot 1\right)\text{mod}\ \mathrm{Im}(\tau_{1})\right)$
$\displaystyle=(\alpha|_{\mathrm{Coker}(\tau_{1})}\circ\pi_{1})((g,h)).$

This shows the desired commutativity of (184).

Moreover, the commutativity of (184) just proved implies that $\alpha$ is
restricted, vis $\iota_{1}$ and $\iota_{2}$, to an isomorphism
$\alpha^{\prime}:F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}_{1}^{\nabla})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}_{2}^{\nabla})$.
Since
$\mathcal{F}_{1}^{\nabla}\cong\mathcal{F}_{2}^{\nabla}\cong\mathcal{O}_{\mathbb{P}^{(N)}}^{\oplus
2}$ (cf. Lemma 6.3.3) and $k$ is perfect, the morphism

(186)
$\displaystyle\mathrm{Hom}_{\mathcal{O}_{\mathbb{P}^{(N)}}}(\mathcal{F}^{\nabla}_{1},\mathcal{F}_{2}^{\nabla})\rightarrow\mathrm{End}_{\mathcal{O}_{\mathbb{P}}}(F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}_{1}^{\nabla}),F^{(N)*}_{\mathbb{P}/k}(\mathcal{F}_{2}^{\nabla}))$

arising from pull-back by $F^{(N)}_{\mathbb{P}/k}$ is bijective. In
particular, $\alpha^{\prime}$ comes from an isomorphism
$\mathcal{F}_{1}^{\nabla}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{F}_{2}^{\nabla}$,
and hence $\alpha^{\prime}$ is compatible with the respective
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-actions
$\nabla^{\mathrm{can}}_{\mathcal{F}^{\nabla}_{1}}$,
$\nabla^{\mathrm{can}}_{\mathcal{F}^{\nabla}_{2}}$ (cf. Remark 4.1.2). Since
$\nabla_{j}$ is the unique
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-module structure on
$\mathcal{F}_{j}$ extending $\nabla_{\mathcal{F}_{j}^{\nabla}}^{\mathrm{can}}$
via $\tau_{j}$, the isomorphism $\alpha$, being an extension of
$\alpha^{\prime}$, preserves the
$\mathcal{D}_{\mathbb{P}^{\mathrm{log}}}^{(N-1)}$-action. It follows that
$\alpha$ defines an isomorphism of level-$N$ $\mathrm{GL}_{2}$-opers
$\mathscr{F}^{\heartsuit}_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{F}^{\heartsuit}_{2}$,
which induces the equality
$\mathscr{F}_{1}^{\spadesuit}=\mathscr{F}_{2}^{\spadesuit}$. This completes
the proof of this proposition. ∎

######  Remark 6.4.2.

The proof of the above proposition shows that
${{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P},(\rho_{0},\rho_{1},\rho_{\infty})}=\emptyset$
unless
$(\rho_{0},\rho_{1},\rho_{\infty})\in((\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm
1\\})^{\times 3}$. As mentioned in Remark 4.3.3, this fact for $N=1$ was
already verified in [Moc1].

We shall write

(187) $\displaystyle\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},+}\
\left(\text{resp.,}\ \mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P}}\right)$

for the set of equivalence classes of finite, separable, and tamely ramified
coverings $\phi:\mathbb{P}\rightarrow\mathbb{P}$ satisfying the following
conditions:

  * •

The set of ramification points of $\phi$ coincides with
$\\{[0],[1],[\infty]\\}$;

  * •

If $\lambda_{x}$ ($x=0,1,\infty$) denotes the ramification index of $\phi$ at
$[x]$, then $\lambda_{0}$, $\lambda_{1}$, $\lambda_{\infty}$ satisfy the
inequality $\lambda_{0}+\lambda_{1}+\lambda_{\infty}<2\cdot p^{N}$ (resp.,
$\lambda_{0}$, $\lambda_{1}$, $\lambda_{\infty}$ are all odd and satisfy the
inequality $\lambda_{0}+\lambda_{1}+\lambda_{\infty}<2\cdot p^{N}$).

Here, the equivalence relation is defined in such a way that two coverings
$\phi_{1},\phi_{2}:\mathbb{P}\rightarrow\mathbb{P}$ are equivalent if there
exists an element
$h\in\mathrm{PGL}_{2}(k)\left(=\mathrm{Aut}_{k}(\mathbb{P})\right)$ with
$\phi_{2}=h\circ\phi_{1}$. Since the identity morphism
$\mathrm{id}_{\mathbb{P}}$ of $\mathbb{P}$ defines a tamely ramified covering
with ramification indices $(1,1,1)$, the set
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},+}$ (resp.,
$\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P}}$) is nonempty. By applying some of
the results proved so far, we obtain the following assertion.

######  Theorem 6.4.3 (cf. Theorem C).

The assignment $\phi\mapsto\mathscr{F}^{\spadesuit}_{\phi}$ gives a $4$-to-$1$
(resp., a $1$-to-$1$, i.e., bijective) correspondence

(188)
$\displaystyle\Upsilon_{\mathscr{P},+}:\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P},+}\twoheadrightarrow{{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P}}\
\left(\text{resp.,}\
\Upsilon_{\mathscr{P}}:\mathrm{Cov}^{\mathrm{tame}}_{\mathscr{P}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P}}\right).$

In particular, the set
${{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P}}$ is finite and
admits an inclusion into the following set:

(189)
$\displaystyle\mathbb{B}_{N}^{\circledast}:=\left\\{(\lambda_{0},\lambda_{1},\lambda_{\infty})\in\mathbb{B}^{\times
3}\,\big{|}\,\lambda_{0}+\lambda_{1}+\lambda_{\infty}<2\cdot p^{N}\
\text{and}\
|\lambda_{0}-\lambda_{1}|\leq\lambda_{\infty}\leq\lambda_{0}+\lambda_{1}\right\\},$

where $\mathbb{B}$ denotes the set of positive odd integers $a$ with $p\nmid
a$, $a<p^{N}$.

###### Proof.

First, we shall consider the former assertion. Let us take a dormant
$\mathrm{PGL}_{2}$-oper $\mathscr{F}^{\spadesuit}$ of level $N$ classified by
${{}^{N}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P}}$, and denote by
$(\rho_{0},\rho_{1},\rho_{\infty})$ the radii of $\mathscr{F}^{\spadesuit}$.
The result of Proposition 6.4.1 shows that $\mathscr{F}^{\spadesuit}$ is the
unique dormant $\mathrm{PGL}_{2}$-oper on $\mathscr{P}$ of level $N$ and radii
$(\rho_{0},\rho_{1},\rho_{\infty})$. Now, let us choose a triple of integers
$\vec{\lambda}:=(\lambda_{0},\lambda_{1},\lambda_{\infty})$ associated to
$(\rho_{0},\rho_{1},\rho_{\infty})$ as defined in § 6.2. As observed in Remark
6.2.2, this triple satisfies the inequalities in (169). It follows that the
triples

(190) $\displaystyle\vec{\lambda}_{0}$
$\displaystyle:=(\lambda_{0},p^{N}-\lambda_{1},p^{N}-\lambda_{\infty}),$
$\displaystyle\vec{\lambda}_{1}$
$\displaystyle:=(p^{N}-\lambda_{0},\lambda_{1},p^{N}-\lambda_{\infty}),$
$\displaystyle\vec{\lambda}_{\infty}$
$\displaystyle:=(p^{N}-\lambda_{0},p^{N}-\lambda_{1},\lambda_{\infty}),$

respectively, satisfy conditions (a) and (b) in § 6.2, and conversely, each
triple of integers satisfying (a) and (b) is one of the four tripes
$\vec{\lambda}$, $\vec{\lambda}_{0}$, $\vec{\lambda}_{1}$,
$\vec{\lambda}_{\infty}$. Thus, by Proposition 6.3.5, the preimage of the
element defined by $\mathscr{F}^{\spadesuit}$ via $\Upsilon_{\mathscr{P},+}$
coincides with
$\\{[\phi_{\vec{\lambda}}],[\phi_{\vec{\lambda}_{0}}],[\phi_{\vec{\lambda}_{1}}],[\phi_{\vec{\lambda}_{\infty}}]\\}$,
where, for a triple
$\vec{\lambda}^{\prime}:=(\lambda^{\prime}_{0},\lambda^{\prime}_{1},\lambda^{\prime}_{\infty})$,
$[\phi_{\vec{\lambda}^{\prime}}]$ denotes a unique (up to equivalence)
covering classified by $\mathrm{Cov}_{\mathscr{P},+}^{\mathrm{tame}}$ whose
ramification index at $[x]$ ($x=0,1,\infty$) is $\lambda^{\prime}_{x}$. This
proves the non-resp’d assertion. Also, the resp’d assertion follows from the
fact that only one of the four triples
$\vec{\lambda},\vec{\lambda}_{0},\vec{\lambda}_{1},\vec{\lambda}_{\infty}$
satisfies the condition that all factors are odd.

Finally, the latter assertion follows from the resp’d portion of the former
assertion (and its proof). This completes the proof of this theorem. ∎

######  Remark 6.4.4.

In the case of $N=1$, we know that the embedding
${{}^{1}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P}}\hookrightarrow\mathbb{B}_{1}^{\circledast}$
resulting from Theorem 6.4.3 is bijective (cf. [Moc1, Introduction, § 1.2,
Theorem 1.3, (2)]). The resulting correspondence
${{}^{1}}\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{2,\mathscr{P}}\cong\mathbb{B}_{1}^{\circledast}$
allows us to translate dormant $\mathrm{PGL}_{2}$-opers into edge-colorings on
trivalent graphs, as well as lattice points of a rational polytope (cf.
[LiOs1], [Wak2]). However, at the time of writing the present paper, the
author does not know much about the image of this map for a general $N$.

######  Remark 6.4.5.

In [Moc1, Chap. II, § 2, Definition 2.2], S. Mochizuki introduced the notion
of a(n) (dormant) $m$-connection (for each nonnegative integer $m$) on a flat
$\mathbb{P}^{1}$-bundle. Here, we recall its definition briefly. Let
$\mathscr{X}:=(X,\\{\sigma_{i}\\}_{i})$ be as in (115). Also, let
$(\mathcal{P},\nabla)$ be a flat $\mathbb{P}$-bundle on $X^{\mathrm{log}}$,
i.e., a $\mathbb{P}$-bundle $\mathcal{P}$ on $X$ equipped with a logarithmic
connection $\nabla$ (with respect to the log structure of $X^{\mathrm{log}}$).
Denote by $W_{m+1}$ the ring of Witt vectors with coefficients in $k$ of
length $m+1$. Then, a dormant $m$-connection on $(\mathcal{P},\nabla)$ (of
prescribed radii) is defined as a crystal in $\mathbb{P}$-bundles on the log
crystalline site $\mathrm{Crys}(X^{\mathrm{log}}/W_{m+1})$ inducing
$(\mathcal{P},\nabla)$ via reduction module $p$ and satisfying some other
conditions. The condition of being a dormant $m$-connection is described in
terms of $p^{m+1}$-curvature in the sense of [Moc1, Chap. II, the discussion
in § 2.1], which is different from (but closely related to) our definition of
$p^{m+1}$-curvature. According to [Moc1, Chap. IV, § 2, Theorem 2.3], there
exists a bijective correspondence between the elements in
$\mathrm{Cov}_{\mathscr{P}}^{\mathrm{tame}}$ and the set of dormant
$(N-1)$-connections on a torally indigenous bundle (i.e., a
$\mathrm{PGL}_{2}$-oper of level $1$) on $\mathscr{P}$. By combining this fact
with Theorem 6.4.3, we see that each dormant $\mathrm{PGL}_{2}$-oper on
$\mathscr{P}$ of level $N$ and radii
$\vec{\rho}\in((\mathbb{Z}/p^{N}\mathbb{Z})^{\times}/\\{\pm 1\\})^{\times 3}$
may be uniquely extended to a dormant $(N-1)$-connection of the same radii.

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