A combinatorial description of the dormant Miura transformation

TL;DR

This paper provides a combinatorial framework for understanding dormant generic Miura $ ext{sl}_2$-opers on totally degenerate curves, linking algebraic geometry with graph theory to analyze the Miura transformation.

Contribution

It introduces a novel combinatorial description of dormant Miura $ ext{sl}_2$-opers using branch numberings of 3-regular graphs, extending previous work on dormant $ ext{sl}_2$-opers.

Findings

Combinatorial description of dormant Miura $ ext{sl}_2$-opers on degenerate curves.

Identification of these opers on curves of genus greater than zero.

Connection of the Miura transformation to combinatorial objects.

Abstract

A dormant generic Miura $\mathfrak{sl}_2$-oper is a flat $\mathrm{PGL}_2$-bundle over an algebraic curve in positive characteristic equipped with some additional data. In the present paper, we give a combinatorial description of dormant generic Miura $\mathfrak{sl}_2$-opers on a totally degenerate curve. The combinatorial objects that we use are certain branch numberings of $3$-regular graphs. Our description may be thought of as an analogue of the combinatorial description of dormant $\mathfrak{sl}_2$-opers given by S. Mochizuki, F. Liu, and B. Osserman. It allows us to think of the Miura transformation in terms of combinatorics. As an application, we identify the dormant generic Miura $\mathfrak{sl}_2$-opers on totally degenerate curves of genus $>0$.

Equations

$$v_t-6v{v}_x+v_{xxx}=0,$$

$$u_t-6u^2{u}_x+u_{xxx}=0.$$

$$({\partial }_x-u)({\partial }_x+u)={\partial }_x^2-v$$

$$u\mapsto v(=u^2-u_x)$$

$$\nabla ={\partial }_x-\left(\begin{array}{cc}0& v\\ 1& 0\end{array}\right)\left(\text{resp.,}{\nabla }_{\mathrm{Miura}}={\partial }_x-\left(\begin{array}{cc}u& 0\\ 1& -u\end{array}\right)\right).$$

$$\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}(\text{resp.,}\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}})$$

$${\mu }_{\mathfrak{X}}^{{}^{Zzz...}}:\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}\to \overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}},$$

$$p\text{-}\mathfrak{E}{\mathfrak{d}}_{{\Gamma }^{+}}^{\mathrm{bal}}$$

$$p\text{-}\mathfrak{E}{\mathfrak{d}}_{{\Gamma }^{+}}^{\mathrm{bal}}\stackrel{\sim }{\to }\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}$$

$$p\text{-}\mathfrak{B}{\mathfrak{r}}_{{\Gamma }^{+}}^{\mathrm{st}}$$

$${\mu }_{{\Gamma }^{+}}^{\mathrm{comb}}:p\text{-}\mathfrak{B}{\mathfrak{r}}_{{\Gamma }^{+}}^{\mathrm{st}}$$

$$p\text{-}\mathfrak{B}{\mathfrak{r}}_{{\Gamma }^{+}}^{\mathrm{st}}\stackrel{\sim }{\to }\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}$$

$$\begin{array}{ccc}p\text{-}\mathfrak{B}{\mathfrak{r}}_{{\Gamma }^{+}}^{\mathrm{st}}& \underset{\sim }{\overset{(\text{ref}EErh9)}{\to }}& \mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}\\ {\mu }_{{\Gamma }^{+}}^{\mathrm{comb}}\downarrow & & \downarrow {\mu }_{\mathfrak{X}}^{{}^{Zzz...}}\\ p\text{-}\mathfrak{E}{\mathfrak{d}}_{{\Gamma }^{+}}^{\mathrm{bal}}& \underset{(\text{ref}EE016)}{\overset{\sim }{\to }}& \overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}.\end{array}$$

$$\Gamma =(V_{\Gamma },E_{\Gamma },{\zeta }_{\Gamma }),$$

$$b^{\divideontimes }\left(\in e\right)$$

$$B_{\Gamma }:=\coprod _{e\in E_{\Gamma }}e$$

$$B_v:={\zeta }_{\Gamma }^{-1}(\{v\})$$

$$g=1-♯(V_{\Gamma })+♯(E_{\Gamma })-♯(B_{V_{\Gamma }}),r=♯(B_{V_{\Gamma }}).$$

$${\lambda }_{\Gamma }:B_{V_{\Gamma }}\stackrel{\sim }{\to }\{1,\cdots ,r\},$$

$${\Gamma }^{+}:=(V_{\Gamma },E_{\Gamma },{\zeta }_{\Gamma },{\lambda }_{\Gamma }),$$

$${\Gamma }_{\mathfrak{X}}^{+}:=(V_{\mathfrak{X}},E_{\mathfrak{X}},{\zeta }_{\mathfrak{X}},{\lambda }_{\mathfrak{X}})$$

$${\tilde{\mathbb{F}}}_p:=\{0,1,\cdots ,p-1\}\left(\subseteq \mathbb{Z}\right),$$

$$\tau :{\tilde{\mathbb{F}}}_p\hookrightarrow \mathbb{Z}\twoheadrightarrow {\mathbb{F}}_p\left(:=\mathbb{Z}/p\mathbb{Z}\right).$$

$$m^{⊻}:=\{\begin{array}{ll}p-m& \text{if m∈Fp∖{0},}\\ 0& \text{if m=0.}\end{array}$$

$$\mathfrak{X}:=(f:X\to \mathrm{Spec}(k),\{{\sigma }_i:\mathrm{Spec}(k)\to X{\}}_{i=1}^r)$$

$$\mathrm{Spec}(k{)}^{\mathfrak{X}\text{-}\log }\text{and}{X}^{\mathfrak{X}\text{-}\log }$$

$$\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}\left(\text{resp.,}\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}\right)$$

$$\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}\left(\text{resp.,}\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}\right)$$

$${\mu }_{\mathfrak{X}}:\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}\to \overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}},$$

$${\mu }_{\mathfrak{X}}^{{}^{Zzz...}}:\mathfrak{M}\overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}}\to \overline{\mathfrak{O}}{\mathfrak{p}}_{\mathfrak{s}{\mathfrak{l}}_2,\mathfrak{X}}^{{}^{Zzz...}},$$

Full text

††footnotetext: Y. Wakabayashi: Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, JAPAN;††footnotetext: e-mail: wkbysh@math.titech.ac.jp;††footnotetext: 2010 Mathematical Subject Classification: Primary 14H70, Secondary 05C30;††footnotetext: Key words: dormant oper, Miura oper, Miura transformation, $3$-regular graph

1. 1 Dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers
2. 2 Combinatorial description of dormant Miura opers

Introduction

0.1

The purpose of the present paper is to give a combinatorial description of dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers. The combinatorial objects that we use are certain branch numberings of $3$-regular (i.e., trivalent) graphs. Our description will be thought of as an analogue of the combinatorial description of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers given by S. Mochizuki, F. Liu, and B. Osserman (cf.  [11], Introduction, § 1.2, Theorem 1.3;  [9], Theorem 3.9). It allows us to think of the Miura transformation in terms of combinatorics.

The celebrated Miura transformation concerns the Korteweg-de Vries (KdV) and the modified KdV (mKdV) equations. The KdV equation was derived as an equation modeling the behavior of shallow water waves moving in one direction by Korteweg and his student de Vries. The KdV equation reads

[TABLE]

while its modified counterpart, the mKdV equation, equals

[TABLE]

Let us take a solution $u\in\mathbb{C}[[t,x]]$ to the mKdV equation. Then, the function $v$ characterized by the equality of differential operators

[TABLE]

(i.e., $v=u^{2}-u_{x}$) specifies a solution to the KdV equation. The assignment

[TABLE]

is nothing but the Miura transformation.

Recall that the differential operator $\partial_{x}^{2}-v$ (resp., $(\partial_{x}-u)(\partial_{x}+u)$) in the right-hand (resp., the left-hand) side of (3) corresponds, in the usual manner, to the connection on a vector bundle (i.e., the first order matrix differential operator) of the form

[TABLE]

A connection of the form (5) is called an $\mathfrak{s}\mathfrak{l}_{2}$-oper (resp., a generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper). If the underlying space of the vector bundle is a Riemann surface, then such a connection may be identified with a projective connection (resp., an affine connection) on this Riemann surface (cf.  [3]). $\nabla_{\mathrm{Miura}}$ becomes $\nabla$ after gauge transformation by some upper triangular matrix. This implies that the Miura transformation may be thought of as the assignment from generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers $\nabla_{\mathrm{Miura}}$ to $\mathfrak{s}\mathfrak{l}_{2}$-opers $\nabla$ induced by gauge transformations in this way.

0.2

In the present paper, we consider the case of (Miura) opers in positive characteristic. A remarkable point is that unlike the complex case, one may have a generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper on some entire (i.e., proper) smooth curve in positive characteristic of genus $>1$. This fact has already proved in the previous work concerning dormant generic Miura opers and Tango structures (cf.  [15], Theorems A and B). Here, an $\mathfrak{s}\mathfrak{l}_{2}$-oper (resp., a generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper) is called dormant if it has vanishing $p$-curvature. (We refer to  [14] for various discussions and results concerning dormant opers on pointed stable curves.) Each pointed stable curve $\mathfrak{X}$ gives rise to the set

[TABLE]

of isomorphism classes of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers (resp., dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers) on $\mathfrak{X}$. The Miura transformation described in terms of opers (i.e., $\nabla_{\mathrm{Miura}}\mapsto\nabla$ as discussed in the previous subsection) gives a map

[TABLE]

referred to as the dormant Miura transformation. Our main interest of the present paper is to understand, in terms of combinatorics, the behavior of this map for the case where the underlying curve is totally degenerate (cf. the discussion preceding Proposition 2.2.2)

0.3

To this end, we first recall the combinatorial description of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers on such a curve, which was obtained in the work by S. Mochizuki in the context of $p$-adic Teichmüller theory and the work by F. Liu and B. Osserman. Let $(g,r)$ be a pair of nonnegative integers with $2g-2+r>0$ and $\mathfrak{X}$ a totally degenerate curve of type $(g,r)$ over an algebraically closed field $k$ of characteristic $p>2$. $\mathfrak{X}$ associates its dual marked semi-graph $\Gamma^{+}$ (cf. the discussion in Remark 1.1.7), which is $3$-regular. A balanced $p$-edge numbering (cf. Definition 2.1.1, (i) for its precise definition) on $\Gamma^{+}$ is a numbering $(a_{e})_{e\in E_{\Gamma}}$ on the set of edges $E_{\Gamma}$ of $\Gamma^{+}$ satisfying certain triangle inequalities with respect to each triple of numbers incident to one vertex. Denote by

[TABLE]

the set of balanced $p$-branch numberings on $\Gamma^{+}$. Then, there exists a canonical bijection

[TABLE]

(cf. the discussion preceding Proposition 2.2.2 for the precise construction). The inverse of (9) is obtained by taking the radii of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers on $\mathfrak{X}$ restricted to the various rational components of its normalization.

A balanced $p$-edgenumbering (with p\,\ooalign{>$\cr$\ ,>}\hskip 2.84526pt0)

3$$4$$6$$7$$5$$3$$2$$2

9$$10$$1$$2$$9$$2$$3$$8$$7$$4$$5$$6$$3$$8A strict $p$-branchnumbering (with $p=11$)

Next, we introduce the notion of a strict $p$-branch numbering (cf. Definition 2.3.1) on $\Gamma^{+}$, which is defined to be a certain numbering $(\varepsilon_{b})_{b\in B_{\Gamma}}$ on the set of branches $B_{\Gamma}$ of $\Gamma^{+}$ satisfying some conditions, including the condition that the three numbers incident to one vertex amount precisely to $p+1$. Each strict $p$-branch numbering $(\varepsilon_{b})_{b\in B_{\Gamma}}$ on $\Gamma^{+}$ induces a balanced $p$-edge numbering $(\varepsilon^{\mu}_{e})_{e\in E_{\Gamma}}$ on $\Gamma^{+}$ which is well-defined in such a way that for each $e\in E_{\Gamma}$ having a branch $b\in B_{\Gamma}$, $\varepsilon^{\mu}_{e}$ equals $\frac{p-\varepsilon_{e}-1}{2}$ if $\varepsilon_{e}$ is even and $\varepsilon^{\mu}_{e}$ equals $\frac{m-1}{2}$ if $\varepsilon_{e}$ is odd. by

[TABLE]

the set of strict $p$-branch numberings on $\Gamma^{+}$. Then, the assignment $(\varepsilon_{b})_{b\in B_{\Gamma}}\mapsto(\varepsilon^{\mu}_{e})_{e\in E_{\Gamma}}$ defines a map

[TABLE]

The main result of the present paper is as follows (cf. Corollary 2.4.3 for its refinement).

Theorem A**.**

Let $\mathfrak{X}$ and $\Gamma^{+}$ be as above. Then, there exists a canonical bijection

[TABLE]

making the following square diagram commute:

[TABLE]

The above theorem implies that one may consider the morphism $\mu^{\mathrm{comb}}_{\Gamma^{+}}$ as a combinatorial realization of the dormant Miura transformation. By means of this combinatorial description, we prove (cf. Corollary 2.5.2) that there is no dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper on a totally degenerate curve of genus $>1$. This result may be thought of as a combinatorial (and dormant) analogue of the emptiness (proved in  [3], Lemma 1) of the space of complex affine structures on a compact hyperbolic Riemann surface. Also, we identify (in terms of combinatorics) the dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers on totally degenerate curves of genus $1$ (cf. Proposition 2.5.1 (ii) and its proof).

Acknowledgement

   The author cannot express enough his sincere and deep gratitude to all those who give the opportunity or impart the great joy of studying mathematics to him. The author wrote the present paper for a gratitude letter to them. The author was partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385).

Notation and conventions

   Let us introduce some notation and conventions used in the present paper. Throughout the present paper, we fix an algebraically closed field $k$ of characteristic $p>2$ and a pair of nonnegative integers $(g,r)$ with $2g-2+r>0$.

For a log scheme indicated, say, by $Y^{\mathrm{log}}$, we shall write $Y$ for the underlying scheme of $Y^{\mathrm{log}}$. If, moreover, $Z^{\mathrm{log}}$ is a log scheme over $Y^{\mathrm{log}}$, then we shall write $\Omega_{Z^{\mathrm{log}}/Y^{\mathrm{log}}}$ for the sheaf of logarithmic $1$-forms on $Z^{\mathrm{log}}$ over $Y^{\mathrm{log}}$ and write $\mathcal{T}_{Z^{\mathrm{log}}/Y^{\mathrm{log}}}:=\Omega_{Z^{\mathrm{log}}/Y^{\mathrm{log}}}^{\vee}$ for its dual. (Basic references for the notion of a log scheme are  [6],  [4], and  [5].)

Given a set $S$ and a positive integer $r$, we shall denote by $S^{\times r}$ the set of $r$-tuples of elements in $S$.

1. Dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers

1.1. Semi-graphs

First, recall from  [Mo3], Appendix (or  [Mo4], § 1), the definition of a semi-graph, as follows.

Definition 1.1.1**.**

A semi-graph is a triple

[TABLE]

where

• $\bullet$

$V_{\Gamma}$ denotes a set, whose elements are called vertices;

• $\bullet$

$E_{\Gamma}$ denotes a set, whose elements are called edges, consisting of sets with cardinality $2$ satisfying the condition that $e\neq e^{\prime}\in E_{\Gamma}$ implies $e\cap e^{\prime}=\emptyset$;

• $\bullet$

$\zeta_{\Gamma}$ denotes a map $\coprod_{e\in E_{\Gamma}}e\rightarrow V_{\Gamma}\cup\{V_{\Gamma}\}$ (where we note that $V_{\Gamma}\cap\{V_{\Gamma}\}=\emptyset$ since $V_{\Gamma}\notin V_{\Gamma}$), which is called a coincidence map.

Let $e$ be an edge of $\Gamma$ (i.e., $e\in E_{\Gamma}$). Then, we shall refer to any element $b\in e$ as a branch of $e$. If $b$ is a branch of $e$, then we shall denote by

[TABLE]

the branch of $e$ with $\{b,b^{\divideontimes}\}=e$. We shall write

[TABLE]

and, for each $v\in V_{\Gamma}\cup\{V_{\Gamma}\}$, we write

[TABLE]

(hence, $B_{\Gamma}=\coprod_{v\in V_{\Gamma}\cup\{V_{\Gamma}\}}B_{v}$).

Let us fix a semi-graph $\Gamma:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma})$.

Definition 1.1.2**.**

• (i)

We shall say that $\Gamma$ is finite if both $V_{\Gamma}$ and $E_{\Gamma}$ are finite.

• (ii)

Let $m$ be a positive integer. We shall say that $\Gamma$ is $m$-regular if for any vertex $v\in V_{\Gamma}$, the cardinality of $B_{v}$ is precisely $m$.

Definition 1.1.3**.**

• (i)

Let $u$, $v$ be vertices of $\Gamma$. A path from $u$ to $v$ is an ordered collection $(b_{j})_{j=1}^{l}$ (for some positive integer $l$) of branches of $\Gamma$ such that $u=\zeta_{\Gamma}(b_{1})$, $v=\zeta_{\Gamma}(b_{l}^{\divideontimes})$, and $\zeta_{\Gamma}(b_{j}^{\divideontimes})=\zeta_{\Gamma}(b_{j+1})$ for any $j\in\{1,\cdots,l-1\}$.

• (ii)

We shall say that $\Gamma$ is connected if for any two distinct vertices $u$, $v\in V_{\Gamma}$, there exists a path from $u$ to $v$.

Definition 1.1.4**.**

Suppose that $\Gamma$ is finite, connected, and $3$-regular. Then, we shall say that $\Gamma$ is of type $(g,r)$ if the following equalities hold:

[TABLE]

Remark 1.1.5**.**

In a natural way (cf., e.g.,  [Mo4], § 1), the semi-graph $\Gamma$ may be thought of as a topological space. Thus, it makes sense to speak of the Betti number of $\Gamma$, which we shall denote by $\beta(\Gamma)$. If $\Gamma$ is finite, connected, and $3$-regular, then the equality $g=\beta(\Gamma)$ holds.

Definition 1.1.6**.**

• (i)

Suppose that $\Gamma$ is finite. A marking on $\Gamma$ is a bijection of sets

[TABLE]

where $r:=\sharp(B_{V_{\Gamma}})$. In particular, if $\sharp(B_{\Gamma_{V}})=0$, then we consider any semi-graph as being equipped with a unique marking $B_{\Gamma_{V}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\emptyset$.

• (ii)

A marked semi-graph is a quadruple

[TABLE]

where $\Gamma:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma})$ is a finite semi-graph and $\lambda_{\Gamma}$ is a marking on $\Gamma$. We shall refer to $\Gamma$ as the underlying semi-graph of $\Gamma^{+}$.

Remark 1.1.7**.**

Let $\mathfrak{X}:=(X/k,\{\sigma_{i}\}_{i=1}^{r})$ be a pointed stable curve of type $(g,r)$ over $k$. Then, in the usual manner, one can associate to $\mathfrak{X}$ a marked semi-graph

[TABLE]

defined as follows. $V_{\mathfrak{X}}$ is the set of irreducible components of $X$ and $E_{\mathfrak{X}}$ is the disjoint union $N_{\mathfrak{X}}\sqcup\{\sigma_{i}\}_{i=1}^{r}$ of the set of nodal points $N_{\mathfrak{X}}$ ($\subseteq X(k)$) and the set of marked points $\{\sigma_{i}\}_{i=1}^{r}$. Here, note that any node $e\in X(k)$ has two distinct branches $b_{1}$ and $b_{2}$, each of which lies on some well-defined irreducible component of $X$; we shall identify $e$ with $\{b_{1},b_{2}\}$. Moreover, identify each marked point $\sigma_{i}$ with the set $\{\sigma_{i},\{\sigma_{i}\}\}$. In this way, we regard the elements of $E_{\mathfrak{X}}$ as sets with cardinality $2$. Also, we define $\zeta_{\mathfrak{X}}$ to be the map $\coprod_{e\in E_{\mathfrak{X}}}e\rightarrow V_{\mathfrak{X}}\cup\{V_{\mathfrak{X}}\}$ determined as follows:

• $\bullet$

if $b\in e$ (for some $e\in N_{\mathfrak{X}}$) or $b=\sigma_{i}$ (for some $i$), then $\zeta_{\mathfrak{X}}(b)$ is the irreducible component (i.e., an element of $V_{\mathfrak{X}}$) in which $b$ lies;

• $\bullet$

$\zeta_{\mathfrak{X}}(\{\sigma_{i}\})=\{V_{\mathfrak{X}}\}$ for any $i$.

(Hence, $B_{V_{\mathfrak{X}}}=\{\{\sigma_{i}\}\}_{i=1}^{r}$.) Finally, $\lambda_{\mathfrak{X}}$ is given by assigning $\{\sigma_{i}\}\mapsto i$ (for any $i\in\{1,\cdots,r\}$). We shall refer to $\Gamma_{\mathfrak{X}}^{+}$ as the dual marked semi-graph associated with $\mathfrak{X}$.

We shall write

[TABLE]

and write $\tau$ for the natural composite bijection

[TABLE]

Let us define an involution $(-)^{\veebar}$ on $\widetilde{\mathbb{F}}_{p}$ to be the map given as follows:

[TABLE]

In particular, we have $\tau(m^{\veebar})=-\tau(m)$.

Definition 1.1.8**.**

• (i)

A $p$-branch numbering on $\Gamma$ is a collection $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ of elements of $\widetilde{\mathbb{F}}_{p}$ indexed by the set $B_{\Gamma}$ such that for any edge $e:=\{b,b^{\divideontimes}\}$, the equality $m_{b^{\divideontimes}}=m_{b}^{\veebar}$ holds.

• (ii)

Assume further that $\Gamma$ is finite and $B_{V_{\Gamma}}\neq\emptyset$. Also, assume that we are given a marking $\lambda_{\Gamma}:B_{V_{\Gamma}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\{1,\cdots,r\}$ (where $r:=\sharp(B_{V_{\Gamma}})$) on $\Gamma$ and an element $\vec{\varepsilon}:=(\varepsilon_{i})_{i=1}^{r}$ of $\mathbb{F}_{p}^{\times r}$. Then, we shall say that a $p$-branch numbering $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ is of exponent $\vec{\varepsilon}$ if $\tau(m_{b})=\varepsilon_{\lambda_{\Gamma}(b)}$ for any $b\in B_{V_{\Gamma}}$. For convenience, (regardless of whether $B_{V_{\Gamma}}$ is empty or not) we shall refer to any $p$-branch numbering as being of exponent $\emptyset$.

1.2. Opers and Miura opers on a pointed stable curve

Next, we recall the notion of a dormant generic Miura oper defined on a pointed stable curve. We refer to  [15] for various definitions and notation used in this section.

Let

[TABLE]

be a pointed stable curve over $k$ of type $(g,r)$, consisting of a (proper) semi-stable curve $X$ over $k$ of genus $g$ and $r$ marked points $\sigma_{i}$ ($i=1,\cdots,r$) of $X$. Note that there exists natural log structures on $X$ and $\mathrm{Spec}(k)$ (cf.  [15], § 1.1); we denote the resulting log schemes by

[TABLE]

respectively. If there is no fear of causing confusion, we write $\mathrm{Spec}(k)^{\mathrm{log}}$ (or just $k^{\mathrm{log}}$) and $X^{\mathrm{log}}$ instead of $\mathrm{Spec}(k)^{\mathfrak{X}\text{-}\mathrm{log}}$ and $X^{\mathfrak{X}\text{-}\mathrm{log}}$ respectively. The structure morphism $f:X\rightarrow\mathrm{Spec}(k)$ extends to a morphism $f^{\mathrm{log}}:X^{\mathrm{log}}\rightarrow\mathrm{Spec}(k)^{\mathrm{log}}$ of log schemes, by which $X^{\mathrm{log}}$ determines a log-curve over $\mathrm{Spec}(k)^{\mathrm{log}}$ (cf.  [1], Definition 4.5 for the definition of a log-curve).

Let us recall briefly the definitions of an $\mathfrak{s}\mathfrak{l}_{2}$-opers and a Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper. Denote by $\mathrm{PGL}_{2}$ the projective linear group over $\mathbb{F}_{p}$ of rank $2$ (considered as an algebraic group over $k$ via base-change by $\mathbb{F}_{p}\hookrightarrow k$) and by $B$ the Borel subgroup of $\mathrm{PGL}_{2}$ defined to be the image (via the quotient $\mathrm{GL}_{2}\twoheadrightarrow\mathrm{PGL}_{2}$) of upper triangular matrices. Let us identify $\mathfrak{s}\mathfrak{l}_{2}$ with the Lie algebra of $\mathrm{PGL}_{2}$. An $\mathfrak{s}\mathfrak{l}_{2}$-oper on $\mathfrak{X}$ is a pair $\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla_{\mathcal{E}})$ consisting of a (right) $B$-torsor $\mathcal{E}_{B}$ over $X$ and a $k^{\mathrm{log}}$-connection $\nabla_{\mathcal{E}}$ on the $\mathrm{PGL}_{2}$-torsor $\mathcal{E}_{\mathrm{PGL}_{2}}:=\mathcal{E}_{B}\times^{B}\mathrm{PGL}_{2}$ induced by $\mathcal{E}_{B}$ such that $\mathcal{E}_{B}$ is transversal to $\nabla_{\mathcal{E}}$. (Here, we refer to  [15], § 1.3, for the definition of a connection on a torsor in the logarithmic sense, and refer to Definition 3.1.1 (i) in loc. cit. or  [11], Chap. I, § 2, Definition 2.2 for the precise definition of an $\mathfrak{s}\mathfrak{l}_{2}$-oper). Also, a Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper on $\mathfrak{X}$ is defined to be a collection of data $\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E}_{B},\nabla_{\mathcal{E}},\mathcal{E}^{\prime}_{B},\eta_{\mathcal{E}})$, where $(\mathcal{E}_{B},\nabla_{\mathcal{E}})$ is an $\mathfrak{s}\mathfrak{l}_{2}$-oper on $\mathfrak{X}$, $\mathcal{E}^{\prime}_{B}$ is another $B$-torsor $\mathcal{E}^{\prime}_{B}$ over $X$, and $\eta_{\mathcal{E}}$ is an isomorphism $\mathcal{E}^{\prime}_{B}\times^{B}\mathrm{PGL}_{2}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}_{\mathrm{PGL}_{2}}$ of $\mathrm{PGL}_{2}$-torsors via which $\mathcal{E}^{\prime}_{B}$ is preserved by $\nabla_{\mathcal{E}}$ (cf. Definition 3.2.1 in  [15] for the precise definition of a Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper). We shall say that a Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper $\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E}_{B},\nabla_{\mathcal{E}},\mathcal{E}^{\prime}_{B},\eta_{\mathcal{E}})$ is generic (cf. Definition 3.3.1 in loc. cit.) if $\mathcal{E}_{B}$ and $\mathcal{E}^{\prime}_{B}$ are in generic relative position. Moreover, we shall say that an $\mathfrak{s}\mathfrak{l}_{2}$-oper $\mathcal{E}^{\spadesuit}:=(\mathcal{E}_{B},\nabla_{\mathcal{E}})$ (resp., a Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper $\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E}_{B},\nabla_{\mathcal{E}},\mathcal{E}^{\prime}_{B},\eta_{\mathcal{E}})$) is dormant if $\nabla_{\mathcal{E}}$ has vanishing $p$-curvature (cf.  [14], Definition 3.2.1 for the definition of $p$-curvature). Denote by

[TABLE]

the set of isomorphism classes of $\mathfrak{s}\mathfrak{l}_{2}$-opers (resp., the set of isomorphism classes of generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers) on $\mathfrak{X}$. Also, denote by

[TABLE]

the subset of $\overline{\mathfrak{O}}\mathfrak{p}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X}}$ (resp., $\mathfrak{M}\overline{\mathfrak{O}}\mathfrak{p}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X}}$) consisting of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers (resp., dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers). The assignment $(\mathcal{E}_{B},\nabla_{\mathcal{E}},\mathcal{E}^{\prime}_{B},\eta_{\mathcal{E}})\mapsto(\mathcal{E}_{B},\nabla_{\mathcal{E}})$ determines a map of sets

[TABLE]

which is nothing but the Miura transformation discussed in Introduction. This map restricts to a map

[TABLE]

referred to as the dormant Miura transformation for $\mathfrak{X}$.

1.3. Radius and exponent

Next, let $\mathfrak{t}$ be the Lie algebra associated with the maximal torus of $\mathrm{PGL}_{2}$ consisting of the image (via the quotient $\mathrm{GL}_{2}\twoheadrightarrow\mathrm{PGL}_{2}$) of diagonal matrices. Denote by $\mathfrak{c}$ the GIT quotient \mathfrak{s}\mathfrak{l}_{2}\ooalign{/$\cr$,/}\mathrm{PGL}_{2} of $\mathfrak{s}\mathfrak{l}_{2}$ by the adjoint action of $\mathrm{PGL}_{2}$ and by $\chi:\mathfrak{t}\twoheadrightarrow\mathfrak{c}$ the composite quotient $\mathfrak{t}\hookrightarrow\mathfrak{s}\mathfrak{l}_{2}\twoheadrightarrow\mathfrak{c}$. Since $\mathfrak{t}$ and $\mathfrak{c}$ is defined over $\mathbb{F}_{p}$, it makes sense to speak of the sets $\mathfrak{t}(\mathbb{F}_{p})$, $\mathfrak{c}(\mathbb{F}_{p})$ of the $\mathbb{F}_{p}$-rational points of $\mathfrak{t}$, $\mathfrak{c}$ respectively. Let us write

[TABLE]

Then, the assignment $\varepsilon\mapsto\varepsilon\cdot\check{\rho}$ determines a bijection $k\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{t}(k)$.

According to  [14], Definition 2.9.1, the notion of the radii of a given $\mathfrak{s}\mathfrak{l}_{2}$-oper is defined as an element of $\mathfrak{c}(k)^{\times r}$ (if $r>0$). Also, the notion of the exponent (cf.  [15], Definition 3.6.1) of a given generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper is defined as an element of $\mathfrak{t}(k)^{\times r}$ (if $r>0$). For convenience, we shall refer to any $\mathfrak{s}\mathfrak{l}_{2}$-oper (resp., generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper) as being of radii $\emptyset$ (resp., of exponent $\emptyset$). For each $\vec{\rho}\in\mathfrak{c}(k)^{\times r}$ (resp., $\vec{\varepsilon}\in\mathfrak{t}(k)^{\times r}$), where $\vec{\rho}:=\emptyset$ (resp., $\vec{\varepsilon}:=\emptyset$) if $r=0$, we denote by

[TABLE]

the subset of $\overline{\mathfrak{O}}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X}}$ (resp., $\mathfrak{M}\overline{\mathfrak{O}}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X}}$) consisting of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers of radius $\vec{\rho}$ (resp., dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers of exponent $\vec{\varepsilon}$). It follows from  [14], Theorem C, and  [15], Theorem 3.8.3, that these sets are finite. Moreover, the sets $\overline{\mathfrak{O}}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X}}$ and $\mathfrak{M}\overline{\mathfrak{O}}\mathfrak{p}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X}}^{{}^{\mathrm{Zzz...}}}$ decompose into the disjoint unions

[TABLE]

respectively. The map (30) restricts to a map

[TABLE]

where $\chi(\vec{\varepsilon}):=(\chi(\varepsilon_{i}))_{i=1}^{r}$ if $r>0$ (resp., $\chi(\vec{\varepsilon}):=\emptyset$ if $r=0$).

1.4. Pre-Tango structures on a pointed stable curve

In this subsection, we recall the definition of a pre-Tango structure given in  [15], Definition 5.3.1. Let $\mathfrak{X}:=(X,\{\sigma_{i}\}_{i=1}^{r})$ be as above.

Definition 1.4.1**.**

A pre-Tango structure on $\mathfrak{X}$ is defined to be a $k^{\mathrm{log}}$-connection $\nabla_{\Omega}$ on the line bundle $\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}$ (i.e., a $k$-linear morphism $\nabla_{\Omega}:\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}\rightarrow\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}^{\otimes 2}$ satisfying the Leibniz rule: $\nabla_{\Omega}(a\cdot v)=da\otimes v+a\cdot\nabla_{\Omega}(v)$, where $a\in\mathcal{O}_{X}$, $v\in\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}$) satisfying the following two conditions:

• •

It has vanishing $p$-curvature;

• •

If $C_{X^{\mathrm{log}}/k^{\mathrm{log}}}$ denotes the Cartier operator $\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}\rightarrow\Omega_{X^{(1)\mathrm{log}}/k^{\mathrm{log}}}$ induced by (the inverse of) “ $C^{-1}$ ” resulting from  [6], Theorem 4.12 (1), where $X^{(1)}$ is the Frobenius twist of $X$ over $k$, then the inclusion relation $\mathrm{Ker}(\nabla_{\Omega})\subseteq\mathrm{Ker}(C_{X^{\mathrm{log}}/k^{\mathrm{log}}})$ holds.

Note that it makes sense to speak of the monodromy (at each marked point $\sigma_{i}$ of $\mathfrak{X}$) of a pre-Tango structure on $\mathfrak{X}$ (cf.  [14], Definition 1.6.1 for the definition of monodromy). The monodromy of a pre-Tango structure lies in $k$ ($\cong\mathcal{E}nd_{\mathcal{O}_{X}}(\sigma^{*}_{i}(\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}))$). For convenience, we shall refer to any pre-Tango structure as being of monodromy $\emptyset$.

Let $\vec{\varepsilon}:=(\varepsilon_{i})_{i=1}^{r}$ be an element of $\mathbb{F}_{p}^{\times r}$ (where $\vec{\varepsilon}:=\emptyset$ if $r=0$). Denote by

[TABLE]

the set of pre-Tango strcutres on $\mathfrak{X}$ (resp., pre-Tango structures on $\mathfrak{X}$ of monodromy $\vec{\varepsilon}$). The set $\overline{\mathfrak{T}}\mathfrak{a}\mathfrak{n}_{\mathfrak{X}}$ decomposes into the disjoint union

[TABLE]

(cf.  [15], (190)). If $\overline{\mathfrak{C}}\mathfrak{o}_{\mathfrak{X}}$ denotes the set of $k^{\mathrm{log}}$-connections on $\Omega_{X^{\mathrm{log}}/k^{\mathrm{log}}}$, then there exists a natural bijection of sets

[TABLE]

which induces, by restriction, a bijection

[TABLE]

(cf.  [15], Theorems 4.4.1 and 5.4.1). If we write

[TABLE]

(where $\vec{\varepsilon}\cdot\check{\rho}:=\emptyset$ if $r=0$), then it restricts to a bijection

[TABLE]

1.5. Gluing pre-Tango structures

Let us discuss the procedure for gluing pre-Tango structures by means of a clutching data. To begin with, we shall define the notion of a clutching data, as follows.

Definition 1.5.1**.**

A clutching data of type $(g,r)$ is a collection of data:

[TABLE]

where

• $\bullet$

$\Gamma^{+}:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma},\lambda_{\Gamma})$ denotes a marked semi-graph with $r=\sharp(B_{V_{\Gamma}})$ whose underlying semi-graph is (finite and) connected;

• $\bullet$

$(g_{v},r_{v})$ (for each $v\in V_{\Gamma}$) denotes a pair of nonnegative integers with $2g_{v}-2+r_{v}>0$, $r_{v}>0$, and $g=\beta(\Gamma)+\sum_{j=1}^{n}g_{j}$;

• $\bullet$

$\lambda_{v}$ (for each $v\in V_{\Gamma}$) denotes a bijection $\lambda_{v}:B_{v}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\{1,\cdots,r_{v}\}$ of sets.

Let $\mathcal{D}$ be a clutching data of type $(g,r)$ as of (41) and $\{{\mathfrak{X}_{v}}\}_{v\in V_{\Gamma}}$ a collection of pointed stable curves over $k$ indexed by the elements of $V_{\Gamma}$, where each ${\mathfrak{X}_{v}}:=(X_{v},\{\sigma_{v,i}\}_{i=1}^{r_{v}})$ is of type $(g_{v},r_{v})$. Here, we assume that the curves $X_{v}$ are all smooth. The pointed curves ${\mathfrak{X}_{v}}$ may be glued together, by means of $\mathcal{D}$, to a new pointed stable curve $\mathfrak{X}:=(X,\{\sigma_{i}\}_{i=1}^{r})$ of type $(g,r)$ in such a way that

• •

the dual marked semi-graph associated with $\mathfrak{X}$ is given by $\Gamma^{+}$, where each vertex $v\in V_{\Gamma}$ corresponds to the irreducible component $X_{v}$;

• •

if $e=\{b_{1},b_{2}\}$ is an edge of $\Gamma$ with $\zeta_{\Gamma}(b_{1})=v_{1}$, $\zeta_{\Gamma}(b_{2})=v_{2}$ (for some $v_{1}$, $v_{2}\in V_{\Gamma}$), then $e$ corresponds to the node of $X$ obtained by gluing together $X_{v_{1}}$ at the $\lambda_{v_{1}}(b_{1})$-th marked point $\sigma_{v_{1},\lambda_{v_{1}}(b_{1})}$ to $X_{v_{2}}$ at the $\lambda_{v_{2}}(b_{2})$-th marked point $\sigma_{v_{2},\lambda_{v_{2}}(b_{2})}$;

• •

the $i$-th ($i=1,\cdots,r$) marked point of $\mathfrak{X}$ arises from the $\lambda_{\zeta_{\Gamma}(\lambda_{\Gamma}^{-1}(i)^{\divideontimes})}(\lambda_{\Gamma}^{-1}(i)^{\divideontimes})$-th marked point of $\mathfrak{X}_{\zeta_{\Gamma}(\lambda_{\Gamma}^{-1}(i)^{\divideontimes})}$.

One may extend, in the evident way, this construction to the case where $X_{v}$’s are (possibly non-smooth) pointed stable curves.

Denote by $\mathfrak{C}\mathfrak{l}\mathfrak{u}\mathfrak{t}_{v}$ ($v\in V_{\Gamma}$) the resulting morphism $X_{v}\rightarrow X$. We shall write $X_{v}^{\mathfrak{X}\text{-}\mathrm{log}}$ for the log scheme obtained by equipping $X_{v}$ with the log structure pulled-back from the log structure of $X^{\mathrm{log}}$ via $\mathfrak{C}\mathfrak{l}\mathfrak{u}\mathfrak{t}_{v}$. The structure morphism $X_{v}\rightarrow\mathrm{Spec}(k)$ of $X_{v}$ extends to a morphism $X_{v}^{\mathfrak{X}\text{-}\mathrm{log}}\rightarrow\mathrm{Spec}(k)^{\mathfrak{X}\text{-}\mathrm{log}}$ of log schemes. Moreover, the morphism $X_{v}^{\mathfrak{X}\text{-}\mathrm{log}}\rightarrow X_{v}$ extends naturally to a commutative square diagram

[TABLE]

The resulting morphism

[TABLE]

is verified to be log étale. (Note that the underlying morphism of $\mathfrak{e}_{v}$ coincides with the identity morphism of $X_{v}$.) In particular, it induces an isomorphism

[TABLE]

of $\mathcal{O}_{X_{v}}$-modules.

Next, let $\nabla_{\Omega}$ be a $k^{\mathfrak{X}\text{-}\mathrm{log}}$-connection on $\Omega_{X^{\mathfrak{X}\text{-}\mathrm{log}}/k^{\mathfrak{X}\text{-}\mathrm{log}}}$. For each $v\in V_{\Gamma}$, the pull-back $\mathfrak{C}\mathfrak{l}\mathfrak{u}\mathfrak{t}^{*}_{v}(\nabla_{\Omega})$ of $\nabla_{\Omega}$ to $X_{v}$ forms a $k^{\mathfrak{X}\text{-}\mathrm{log}}$-connection on $\Omega_{X_{v}^{\mathfrak{X}\text{-}\mathrm{log}}/k^{\mathfrak{X}\text{-}\mathrm{log}}}$. Moreover, by means of (44), $\mathfrak{C}\mathfrak{l}\mathfrak{u}\mathfrak{t}^{*}_{v}(\nabla_{\Omega})$ may be thought of as a $k^{\mathfrak{X}_{v}\text{-}\mathrm{log}}$-connection

[TABLE]

on $\Omega_{X_{v}^{\mathfrak{X}_{v}\text{-}\mathrm{log}}/k^{\mathfrak{X}_{v}\text{-}\mathrm{log}}}$. We shall refer to $\nabla_{\Omega}^{v}$ as the restriction of $\nabla_{\Omega}$ to $\mathfrak{X}_{v}$.

Proposition 1.5.2**.**

Let us keep the above notation.

• (i)

Suppose that $\nabla_{\Omega}$ specifies a pre-Tango structure on $\mathfrak{X}$. Then, for each $v\in V_{\Gamma}$, the restriction $\nabla_{\Omega}^{v}$ specifies a pre-Tango structure on $\mathfrak{X}_{v}$. If, moreover, $\nabla^{v}_{\Omega}$ ($v\in V_{\Gamma}$) is of monodromy $\vec{\varepsilon}^{\,v}:=(\varepsilon_{i}^{v})_{i=1}^{r_{v}}\in\mathbb{F}_{p}^{\times r_{v}}$, then the collection

[TABLE]

*(where $\varepsilon^{V_{\Gamma}}_{\lambda_{V_{\Gamma}}(b)}:=-\varepsilon^{\zeta_{\Gamma}(b^{\divideontimes})}_{\lambda_{\zeta_{\Gamma}(b^{\divideontimes})}(b^{\divideontimes})}$ for any $b\in B_{V_{\Gamma}}$) forms a $p$-branch numbering on $\Gamma$. *

• (ii)

Conversely, let $\vec{m}_{\mathcal{D}}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ be a $p$-branch numbering on $\Gamma$ of exponent $\vec{\varepsilon}\in\mathbb{F}_{p}^{\times r}$ (where $\vec{\varepsilon}:=\emptyset$ if $r=0$). Suppose that on each ${\mathfrak{X}_{v}}$, we are given a pre-Tango structure $\nabla_{\Omega,v}$ of monodromy $(\tau(m_{\lambda^{-1}_{v}(i)}))_{i=1}^{r_{v}}$. Then, there exists a pre-Tango structure $\nabla_{\Omega}$ on $\mathfrak{X}$ of monodromy $\vec{\varepsilon}$, which is uniquely determined by the condition that for any $v\in V_{\Gamma}$ the restriction of $\nabla_{\Omega}$ to ${\mathfrak{X}_{v}}$ coincides with $\nabla_{\Omega,v}$.

Proof.

Assertions (i) and (ii) follow immediately from  [14], Proposition 7.2.1 (i), (ii) and the fact that for each $v\in V_{\Gamma}$, the restriction of the Cartier operator $C_{X^{\mathfrak{X}\text{-}\mathrm{log}}/k^{\mathfrak{X}\text{-}\mathrm{log}}}$ to $X_{j}$ may be identified, via (44), with $C_{X_{v}^{\mathfrak{X}_{v}\text{-}\mathrm{log}}/k^{\mathfrak{X}_{v}\text{-}\mathrm{log}}}$. ∎

Let $\vec{\varepsilon}$ be an element of $\mathbb{F}_{p}^{\times r}$ (where $\vec{\varepsilon}:=\emptyset$ if $r=0$), $\mathcal{D}$ a clutching data of type $(g,r)$ with underlying marked semi-graph $\Gamma^{+}:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma},\lambda_{\Gamma})$, and $\vec{m}_{\mathcal{D}}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ a $p$-branch numbering on $\Gamma$ of exponent $\vec{\varepsilon}$. If we write $\vec{\varepsilon}_{\mathcal{D}}^{\ v}:=(\tau(m_{\lambda^{-1}_{v}(i)}))_{i=1}^{r_{v}}$ (for each $v\in V_{\Gamma}$), then (by applying Proposition 1.5.2 above) a collection of pre-Tango structures $(\nabla_{\Omega,v})_{v\in V_{\Gamma}}$, where each $\nabla_{\Omega,v}$ is of monodromy $\vec{\varepsilon}_{\mathcal{D}}^{\ v}$, induces a pre-Tango structure $\nabla_{\Omega}$ on $\mathfrak{X}$ of monodromy $\vec{\varepsilon}$. The assignment $(\nabla_{\Omega,v})_{v\in V_{\Gamma}}\mapsto\nabla_{\Omega}$ induces a morphism

[TABLE]

Proposition 1.5.3**.**

*Let $\mathcal{D}$ and $\vec{\varepsilon}$ be as above. Then, the following map is bijective: *

[TABLE]

*where the disjoint union in the left-hand side is taken over the set of $p$-branch numberings on $\Gamma$ of exponent $\vec{\varepsilon}$. *

Proof.

The assertion follows immediately from the decomposition (33), Proposition 1.5.2, and the definition of $\mathfrak{C}\mathfrak{l}\mathfrak{u}\mathfrak{t}_{\mathcal{D},\vec{m}_{\mathcal{D}}}$. ∎

2. Combinatorial description of dormant Miura opers

2.1. Balanced $p$-edge numberings

In this section, we shall study a combinatorial description of dormant Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers (equivalently, pre-Tango structures) on a totally degenerate curve. We first recall the combinatorial description of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers on a totally degenerate curve, which was essentially given in the previous work of $p$-adic Teichmüller theory due to S. Mochizuki (cf.  [11], Chap. V, § 1, (3), p. 232, or  [14], § 7.11). The objects used to describe combinatorially dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers will be referred, in the present paper, to as balanced $p$-edge numberings (cf. Definition 2.1.1 below). Let us fix a marked semi-graph $\Gamma^{+}:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma},\lambda_{\Gamma})$ whose underlying semi-graph is connected, $3$-regular, and of type $(g,r)$.

Definition 2.1.1**.**

• (i)

A balanced $p$-edge numbering on $\Gamma^{+}$ is a collection $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ of elements of $\widetilde{\mathbb{F}}_{p}$ indexed by $B_{\Gamma}$ satisfying the following two conditions:

• $\bullet$

For each edge $e:=\{b,b^{\divideontimes}\}\in E_{\Gamma}$, the equality $m_{b}=m_{b^{\divideontimes}}$ holds.

• $\bullet$

For each vertex $v\in V_{\Gamma}$ (where we write $B_{v}:=\{b_{1},b_{2},b_{3}\}$ and $m_{l}:=m_{b_{l}}$ for each $l=1,2,3$), the inequalities in $\bigstar_{m_{1},m_{2},m_{3}}$ displayed below are satisfied:

• • $\bigstar_{m_{1},m_{2},m_{3}}$ :

$|m_{2}-m_{3}|\leq m_{1}\leq m_{2}+m_{3}$,       $m_{1}+m_{2}+m_{3}\leq p-2$.

(By the first condition, any balanced $p$-edge numbering may be thought of as a numbering described on the set of edges $E_{\Gamma}$, as its name suggests.)

• (ii)

Assume further that $B_{V_{\Gamma}}\neq\emptyset$. Let $\vec{m}:=(m_{b})_{b\in B_{V_{\Gamma}}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ be a balanced $p$-edge numbering on $\Gamma$ and $\vec{\varepsilon}:=(\varepsilon_{i})_{i=1}^{r}$ an element of $\mathbb{F}_{p}^{\times r}$. We shall say that $\vec{m}$ is of radii $\vec{\varepsilon}$ if $\tau(m_{b})=\varepsilon_{\lambda_{\Gamma}(b)}$ for any $b\in B_{V_{\Gamma}}$. For convenience, (regardless of whether $B_{V_{\Gamma}}$ is empty or not) we shall refer to any $p$-edge numbering as being of radii $\emptyset$.

Denote by

[TABLE]

the set of balanced $p$-branch numberings on $\Gamma^{+}$. Also, for each $\vec{\varepsilon}\in\mathbb{F}_{p}^{\times r}$ (where $\vec{\varepsilon}:=\emptyset$ if $r=0$), we shall write

[TABLE]

for the subset of $p\text{-}\mathfrak{E}\mathfrak{d}_{\Gamma^{+}}^{\mathrm{bal}}$ consisting of balanced $p$-branch numberings of radii $\vec{\varepsilon}$. In particular, we have

[TABLE]

2.2. Combinatorial description of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers

Next, let us construct a bijective correspondence between the set of dormant $\mathfrak{s}\mathfrak{l}_{2}$-opers on a totally degenerate curve and the set of balanced $p$-branch numberings on the dual semi-graph of this curve. First, let us consider the case where $(g,r)=(0,3)$. Denote by $[0]$, $[1]$, and $[\infty]$ the $k$-rational points of the projective line $\mathbb{P}^{1}$ over $k$ determined by the values [math], $1$, and $\infty$ respectively. After ordering the points $[0],[1],[\infty]$ suitably (say, $\sigma_{1}:=[0]$, $\sigma_{2}:=[1]$, and $\sigma_{3}:=[\infty]$), we obtain a unique (up to isomorphism) pointed stable curve

[TABLE]

of type $(0,3)$ over $k$. The dual marked semi-graph $\Gamma^{+}_{\mathfrak{P}}:=(V_{\mathfrak{P}},E_{\mathfrak{P}},\zeta_{\mathfrak{P}},\lambda_{\mathfrak{P}})$ associated with $\mathfrak{P}$ is given as follows:

• $\bullet$

$V_{\mathfrak{P}}:=\left\{v\ (:=\mathbb{P}^{1})\right\}$;

• $\bullet$

$E_{\mathfrak{P}}:=\{e_{1},e_{2},e_{3}\}$, where

[TABLE]

• $\bullet$

$\zeta_{\mathfrak{P}}:B_{\Gamma_{\mathfrak{P}}}\rightarrow V_{\mathfrak{P}}\sqcup\{V_{\mathfrak{P}}\}$ is given by

[TABLE]

• $\bullet$

$\lambda_{\mathfrak{P}}:B_{V_{\mathfrak{P}}}\ \left(=\{\{\sigma_{1}\},\{\sigma_{2}\},\{\sigma_{3}\}\}\right)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\{1,2,3\}$ is given by $\{\sigma_{l}\}\mapsto l$ for any $l\in\{1,2,3\}$.

The assignment $(m_{b})_{b\in B_{\Gamma_{\mathfrak{P}}}}\mapsto(m_{\sigma_{1}},m_{\sigma_{2}},m_{\sigma_{3}})$ gives a bijective correspondence between the set of balanced $p$-edge numberings on $\Gamma_{\mathfrak{P}}^{+}$ and the set of triples $(m_{1},m_{2},m_{3})\in\mathbb{Z}^{\times 3}$ of integers satisfying the condition $\bigstar_{m_{1},m_{2},m_{3}}$. The inverse assignment is given by $(m_{1},m_{2},m_{3})\mapsto(m_{b})_{b\in B_{\Gamma}}$, where $m_{\sigma_{l}}:=m_{l}$, $m_{\{\sigma_{l}\}}:=m_{l}$ (for any $l\in\{1,2,3\}$). By passing to this correspondence, we shall identify each balanced $p$-edge numbering on $\Gamma_{\mathfrak{P}}^{+}$ with such a triple $(m_{1},m_{2},m_{3})$.

Now, denote by $\iota$ the bijection defined as

[TABLE]

where $\overline{(-)}$ denotes the image of $(-)$ via the composite $\mathbb{Z}\twoheadrightarrow\mathbb{F}_{p}\hookrightarrow k$. Recall from  [11], Chap. I, § 4.3, p. 117, Theorem 4.4 (or  [14], Theorem A), that for each $(\rho_{1},\rho_{2},\rho_{3})\in\mathfrak{c}(k)^{\times 3}$ there exists a unique $\mathfrak{s}\mathfrak{l}_{2}$-oper $\mathcal{E}^{\spadesuit}_{\rho_{1},\rho_{2},\rho_{3}}$ on $\mathfrak{P}$ of radii $(\rho_{1},\rho_{2},\rho_{3})$. That is to say, we obtain a bijection

[TABLE]

Lemma 2.2.1**.**

*The composite $\mathbb{Z}^{\times 3}\twoheadrightarrow\mathbb{F}_{p}^{\times 3}\hookrightarrow k^{\times 3}\xrightarrow{(\ref{aaa6})}\overline{\mathfrak{O}}\mathfrak{p}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{P}}$ restricts to a bijection *

[TABLE]

Proof.

The assertion follows from  [11], Chap. V, § 1, (3), p. 232 (cf.  [14], the discussion in § 7.11. ∎

Next, we shall extend the above result to the case where the underlying curve is an arbitrary totally degenerate curve. To this end, let us recall the definition of a totally degenerate curve. Let $\mathfrak{X}:=(X,\{\sigma_{i}\}_{i=1}^{r})$ be a pointed stable curve over $k$ of type $(g,r)$. Write $\nu_{\mathfrak{X}}:\coprod_{l=1}^{L_{\mathfrak{X}}}X_{l}\rightarrow X$ for the normalization of $X$, where $L_{\mathfrak{X}}$ denotes some positive integer and each $X_{l}$ ($l=1,\cdots,L_{\mathfrak{X}_{/k}}$) is a proper smooth connected curve over $k$. Then, we shall say that $\mathfrak{X}$ is totally degenerate if, for any $l=1,\cdots,L_{\mathfrak{X}}$, the pointed stable curve

[TABLE]

is isomorphic to $\mathfrak{P}$, where we consider $E_{\mathfrak{X}}:=N_{\mathfrak{X}}\sqcup\{\sigma_{i}\}_{i=1}^{r}$ (cf. (21)) as a subset of $X(k)$.

Now, let $\mathfrak{X}$ be a totally degenerate pointed stable curve over $k$ of type $(g,r)$. Then, $\mathfrak{X}$ may be obtained by gluing together finite copies of $\mathfrak{P}$ by means of some clutching data whose underlying marked semi-graph is $\Gamma_{\mathfrak{X}}^{+}$. For each $v\in V_{\mathfrak{X}}$, we shall denote by $\mathfrak{P}_{v}$ the $3$-pointed projective line corresponding to $v$ (i.e., $\mathfrak{P}_{v}\cong\mathfrak{P}$). Also, for each $v\in V_{\mathfrak{X}}$ and $b\in B_{v}$, we shall denote by $\sigma_{b}$ the marked point of $\mathfrak{P}_{v}$ corresponding to $b$.

Let $(m_{b})_{b\in B_{\Gamma_{\mathfrak{X}}}}$ be a balanced $p$-edge numbering on $\Gamma_{\mathfrak{X}}^{+}$. For each $v\in V_{\mathfrak{X}}$ with $B_{v}:=\{b_{1},b_{2},b_{3}\}$, the triple $(m_{b_{1}},m_{b_{2}},m_{b_{3}})$ specifies a balanced $p$-edge numbering on $\Gamma_{\mathfrak{P}_{v}}^{+}$. This triple corresponds, via (57), to a dormant $\mathfrak{s}\mathfrak{l}_{2}$-oper $\mathcal{E}^{\spadesuit}_{v}$ on $\mathfrak{P}_{v}$. One may assume, without loss of generality, that each $\mathcal{E}_{v}^{\spadesuit}$ is of canonical type (cf.  [14], Definition 2.7.1). If $e:=\{b,b^{\divideontimes}\}\in E_{\Gamma_{\mathfrak{X}}}$ is an edge with $\{V_{\Gamma}\}\notin\zeta_{\Gamma}(e)$, then the radius of $\mathcal{E}^{\spadesuit}_{\zeta_{\Gamma_{\mathfrak{X}}}(b)}$ at $\sigma_{b}$ coincides with the radius of $\mathcal{E}^{\spadesuit}_{\zeta_{\Gamma_{\mathfrak{X}}}(b^{\divideontimes})}$ at $\sigma_{b^{\divideontimes}}$. It follows from  [14], Proposition 7.3.3 (ii) that $\mathcal{E}^{\spadesuit}_{v}$’s may be glued together to a dormant $\mathfrak{s}\mathfrak{l}_{2}$-oper $\mathcal{E}^{\spadesuit}$ on $\mathfrak{X}$ (of canonical type). The bijectivity of (57) implies the following proposition.

Proposition 2.2.2**.**

Let $\mathfrak{X}$ be as above. Then, the assignment $(m_{b})_{b\in B_{\Gamma_{\mathfrak{X}}}}\mapsto\mathcal{E}^{\spadesuit}$ discussed above defines a bijection

[TABLE]

If, moreover, $r>0$, then for each $\vec{\varepsilon}:=(\varepsilon_{i})_{i=1}^{r}\in\mathbb{F}_{p}^{\times r}$, the bijection (59) restricts to a bijection

[TABLE]

where $\iota(\vec{\varepsilon}):=(\iota(\varepsilon_{i}))_{i=1}^{r}$.

Remark 2.2.3**.**

If we use the notation $(-)^{\,\mu}$ defined in (66) below, then (60) may be expressed as

[TABLE]

for each $\vec{\varepsilon}\in\mathbb{F}_{p}^{\times r}$.

2.3. Strict $p$-branch numberings

Next, in order to describe combinatorially dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers (equivalently, pre-Tango structures), we shall introduce the notion of a strict $p$-branch numbering, as follows. Let $\Gamma^{+}:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma},\lambda_{\Gamma})$ be a marked semi-graph whose underlying semi-graph $\Gamma$ is connected, $3$-regular, and of type $(g,r)$.

Definition 2.3.1**.**

A strict $p$-branch numbering on $\Gamma^{+}$ is a $p$-branch numbering $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ on $\Gamma$ with $m_{b}\neq 0$ (for any $b\in B_{\Gamma}$) such that for each vertex $v\in V_{\Gamma}$ (where we shall write $B_{v}:=\{b_{1},b_{2},b_{3}\}$), the equality

[TABLE]

holds.

Denote by

[TABLE]

the set of strict $p$-branch numberings on $\Gamma^{+}$. Also, for each $\vec{\varepsilon}\in\mathbb{F}_{p}^{\times r}$ (where $\vec{\varepsilon}:=\emptyset$ if $r=0$), we shall write

[TABLE]

for the set of strict $p$-branch numberings on $\Gamma^{+}$ of exponent $\vec{\varepsilon}$. The set $p\text{-}\mathfrak{B}\mathfrak{r}_{\Gamma^{+}}^{\mathrm{st}}$ decomposes into the disjoint union

[TABLE]

Now, let us construct an assignment from each strict $p$-branch numbering to a balanced $p$-edge numbering. Given an element $m\in\widetilde{\mathbb{F}}_{p}$, we shall write $m^{\mu}$ for the element of $\widetilde{\mathbb{F}}_{p}$ defined as follows:

[TABLE]

For each strict $p$-branch numbering $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}$ on $\Gamma^{+}$, the collection $\vec{m}^{\mu}:=(m^{\mu}_{b})_{b\in B_{\Gamma}}$ is verify to specify a balanced $p$-edge numbering on $\Gamma^{+}$. Thus, we obtain a map of sets

[TABLE]

which we shall refer to as the combinatorial dormant Miura transformation for $\Gamma$. Given $\vec{\varepsilon}:=(\varepsilon_{i})_{i=1}^{r}\in\mathbb{F}_{p}^{\times r}$ (where $\vec{\varepsilon}:=\emptyset$ if $r=0$), the map $\mu^{\mathrm{comb}}_{\Gamma}$ restricts to a map

[TABLE]

where $\vec{\varepsilon}^{\,\mu}:=(\varepsilon_{i}^{\mu})_{i=1}^{r}$ (and $\vec{\varepsilon}^{\,\mu}:=\emptyset$ if $r=0$).

2.4. Combinatorial description of dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers

We shall describe the relationship between the set of pre-Tango structures and the set of strict $p$-branch numberings. In a fashion analogous to the case of balanced $p$-edge numberings, we shall identify each $p$-branch numbering $(m_{b})_{b\in B_{\Gamma_{\mathfrak{P}}}}$ on $\Gamma_{\mathfrak{P}}$ with a triple $(m_{1},m_{2},m_{3})\in\widetilde{\mathbb{F}}_{p}^{\times 3}$, where $m_{l}:=m_{\sigma_{l}}=m^{\veebar}_{\{\sigma_{l}\}}$ ($l=1,2,3$).

Lemma 2.4.1**.**

Let us consider the set

[TABLE]

Then, an element $(m_{1},m_{2},m_{3})$ of $p\text{-}\mathfrak{B}\mathfrak{r}_{\Gamma^{+}_{\mathfrak{P}}}$ specifies a strict $p$-branch numbering on $\Gamma^{+}_{\mathfrak{P}}$ (in the above sense) if and only if the triple $(m_{1}^{\mu},m^{\mu}_{2},m^{\mu}_{3})$ specifies a balanced $p$-edge numbering on $\Gamma^{+}_{\mathfrak{P}}$ (i.e., satisfies the inequalities in $\bigstar_{m_{1}^{\mu},m^{\mu}_{2},m^{\mu}_{3}}$).

Proof.

Let $(m_{1},m_{2},m_{3})$ be an element of $p\text{-}\mathfrak{B}\mathfrak{r}_{\Gamma^{+}_{\mathfrak{P}}}$ such that $(m_{1}^{\mu},m^{\mu}_{2},m^{\mu}_{3})$ satisfies the inequalities in $\bigstar_{m_{1}^{\mu},m^{\mu}_{2},m^{\mu}_{3}}$. To complete the proof, it suffices to verify that this triple specifies a strict $p$-branch numbering on $\Gamma^{+}_{\mathfrak{P}}$. Define $m$ to be the integer with $\sum_{i=1}^{3}m_{i}=mp+1$. Since $m_{i}\leq p-1$, the inequality $m<3$ holds. Consider the case where $m=0$. Then, after possibly change of ordering, we may assume that $m_{1}=m_{2}=0$ and $m_{3}=1$. Then,

[TABLE]

which is a contradiction. Next, consider the case where $m=2$. As the sum $\sum_{i=1}^{3}m_{i}$ is odd, either one of the following two cases (a), (b) is satisfied: (a) the three integers $m_{1}$, $m_{2}$, $m_{3}$ are all odd; (b) two of the three integers $m_{1}$, $m_{2}$, $m_{3}$ are even and the remaining one is odd. But, in the case (a), we obtain a contradiction since

[TABLE]

On the other hand, we shall consider the case (b). Let us assume, without loss of generality, that $m_{1}$ is odd and both $m_{2}$ and $m_{2}$ are even. But, since

[TABLE]

this is a constradiction. Hence, the equality $m=1$ holds. One verifies that $m_{i}\neq 0$ for any $i$. Indeed, one of them, say $m_{1}$, coincides with [math] and $m_{2}$ is odd (resp., even), then

[TABLE]

This is a contradiction. Consequently, $(m_{1},m_{2},m_{3})$ specifies a strict $p$-branch numbering on $\Gamma^{+}_{\mathfrak{P}}$. This completes the proof of the lemma. ∎

Denote by $\overline{\mathfrak{C}}\mathfrak{o}_{\mathfrak{P}}^{\psi=0}$ the set of (logarithmic) $k$-connections on $\Omega_{\mathbb{P}^{1\mathrm{log}}/k}$ with vanishing $p$-curvature. (Hence, $\overline{\mathfrak{T}}\mathfrak{a}\mathfrak{n}_{\mathfrak{P}}$ specifies a subset of $\overline{\mathfrak{C}}\mathfrak{o}_{\mathfrak{P}}^{\psi=0}$.) Let $(m_{1},m_{2},m_{3})$ be a triple in $p\text{-}\mathfrak{B}\mathfrak{r}_{\Gamma^{+}_{\mathfrak{p}}}$ and let $m$ be the integer with $mp=\sum_{i=1}^{3}m_{i}-1$. Denote by $\mathcal{O}_{\mathbb{P}^{1(1)}}(-m)$ a unique (up to isomorphism) line bundle of degree $-m$ on the Frobenius twist $\mathbb{P}^{1(1)}$ of $\mathbb{P}^{1}$. We obtain the pull-back $F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))$ of $\mathcal{O}_{\mathbb{P}^{1(1)}}(-m)$ via the relative Frobenius morphism $F:\mathbb{P}^{1}\rightarrow\mathbb{P}^{1(1)}$. There exists a $k$-connection $\nabla^{\mathrm{can}}$ on $F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))$ with vanishing $p$-curvature determined uniquely by the condition that the sections of the subsheaf $F^{-1}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))$ ($\subseteq F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))$) are contained in $\mathrm{Ker}(\nabla^{\mathrm{can}})$ (cf.  [15], § 1.7). Also, one may construct uniquely a $k$-connection $\nabla^{\mathrm{can}}_{m_{1},m_{2},m_{3}}$ on $F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))(\sum_{i=1}^{3}m_{i}\sigma_{i})$ whose restriction to $F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))$ coincides with $\nabla^{\mathrm{can}}$. The monodromy of $\nabla^{\mathrm{can}}_{m_{1},m_{2},m_{3}}$ at $\sigma_{i}$ ($i=1,2,3$) is $-\tau(m_{i})$. Since

[TABLE]

we have an isomorphism $F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))(\sum_{i=1}^{3}m_{i}\sigma_{i})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega_{\mathbb{P}^{1(1)}/k}$. $\nabla^{\mathrm{can}}_{m_{1},m_{2},m_{3}}$ corresponds, via this isomorphism, to a $k$-connection $\breve{\nabla}^{\mathrm{can}}_{m_{1},m_{2},m_{3}}$ on $\Omega_{\mathbb{P}^{1(1)}/k}$ of monodromy $(-\tau(m_{1}),-\tau(m_{2}),-\tau(m_{3}))\in k^{\times 3}$ (with vanishing $p$-curvature). Notice that this connection does not depend on the choice of the isomorphism $F^{*}(\mathcal{O}_{\mathbb{P}^{1(1)}}(-m))(\sum_{i=1}^{3}m_{i}\sigma_{i})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega_{\mathbb{P}^{1(1)}/k}$. The resulting assignment $(m_{1},m_{2},m_{3})$ $\mapsto\breve{\nabla}^{\mathrm{can}}_{m_{1},m_{2},m_{3}}$ determines a bijection

[TABLE]

Indeed, its inverse is given by $\nabla\mapsto(\tau^{-1}(-\mu_{1}^{\nabla}),\tau^{-1}(-\mu_{2}^{\nabla}),\tau^{-1}(-\mu_{3}^{\nabla}))$. In particular, a $k$-connection on $\Omega_{\mathbb{P}^{1\mathrm{log}}/k}$ with vanishing $p$-curvature may be uniquely determined by its monodromy.

Lemma 2.4.2**.**

The bijection (75) restricts to a bijection

[TABLE]

which makes the following diagram commute:

[TABLE]

In particular, the set $\mathfrak{M}\overline{\mathfrak{O}}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{P}}$ is in bijection with the set of triples $(m_{1},m_{2},m_{3})$ consisting of positive integers with $\sum_{i=1}^{3}m_{i}=p+1$.

Proof.

It is immediately verified that the following diagram is commutative:

[TABLE]

where

• $\bullet$

the left-hand vertical arrow denotes the map given by $(m_{i})_{i=1}^{3}\mapsto(m_{i}^{\mu})_{i=1}^{3}$;

• $\bullet$

the upper left-hand horizontal arrow is the composite of (75) and the natural injection $\overline{\mathfrak{C}}\mathfrak{o}_{\mathfrak{P}}^{\psi=0}\hookrightarrow\overline{\mathfrak{C}}\mathfrak{o}_{\mathfrak{P}}$;

• $\bullet$

the lower horizontal arrow denotes the composite $\widetilde{\mathbb{F}}_{p}^{\times 3}\xrightarrow{\tau^{\times 3}}\mathbb{F}_{p}^{\times 3}\hookrightarrow k^{\times 3}\xrightarrow{(\ref{aaa6})}\overline{\mathfrak{O}}\mathfrak{p}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{P}}$.

The diagram (81) may be obtained from (86) by restricting $\widetilde{\mathbb{F}}_{p}^{\times 3}$ and $\overline{\mathfrak{O}}\mathfrak{p}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{P}}$ to $p\text{-}\mathfrak{E}\mathfrak{d}_{\Gamma_{\mathfrak{P}}^{+}}^{\mathrm{bal}}$ and $\overline{\mathfrak{O}}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{P}}$ respectively. This completes the proof of the lemma. ∎

By applying Proposition 1.5.3, one may glue together the isomorphisms (76) applied to $\mathfrak{P}_{v}$ for the various $v\in V_{\mathfrak{X}}$ (with the notation following Lemma 2.2.1). The resulting isomorphism, as displayed in (87) below, gives a combinatorial description of pre-Tango structures (as well as dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-opers) on an arbitrary totally degenerate curve. (Theorem A follows from the following assertion together with the decompositions (33), (36), (51), 65.)

Corollary 2.4.3**.**

*Let $\mathfrak{X}$ be a totally degenerate pointed stable curve over $k$ of type $(g,r)$ and $\vec{\varepsilon}$ an element of $\mathbb{F}_{p}^{\times r}$, where $\vec{\varepsilon}:=\emptyset$ if $r=0$. Then, there exists a canonical bijection *

[TABLE]

which makes the following diagram commute:

[TABLE]

In particular, the set $\mathfrak{M}\overline{\mathfrak{O}}\mathfrak{p}^{{}^{\mathrm{Zzz...}}}_{\mathfrak{s}\mathfrak{l}_{2},\mathfrak{X},\vec{\varepsilon}\cdot\check{\rho}}$ is in bijection with the set $p\text{-}\mathfrak{B}\mathfrak{r}_{\Gamma_{\mathfrak{X}}^{+},-\vec{\varepsilon}}^{\mathrm{st}}$.

2.5. Case of $g=1$

Finally, we shall conclude the present paper with proving that there is no strict $p$-branch numbering on $\Gamma^{+}$ unless the underlying semi-graph $\Gamma$ is of type $(g,r)$ with $g\leq 1$ (cf. Proposition 2.5.1 below). Equivalently, there is no dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper on any totally degenerate curve of genus $g>1$ (cf. Corollary 2.5.2).

Proposition 2.5.1**.**

Let $\Gamma^{+}:=(V_{\Gamma},E_{\Gamma},\zeta_{\Gamma},\lambda_{\Gamma})$ be a marked semi-graph whose underlying semi-graph $\Gamma$ is connected, $3$-regular, and of type $(g,r)$.

• (i)

If there exists a strict $p$-branch numbering on $\Gamma^{+}$, then the inequality $g\leq 1$ holds.

• (ii)

If $g=1$, then the inclusion

[TABLE]

is bijective, where $\vec{e}:=(\overline{-1},\overline{-1},\cdots,\overline{-1})\in\mathbb{F}_{p}^{\times r}$ if $r>0$ (resp., $\vec{e}:=\emptyset$ if $r=0$), and the following equalities hold:

[TABLE]

Proof.

First, we shall consider assertion (i). Let $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}\in\widetilde{\mathbb{F}}_{p}^{B_{\Gamma}}$ be a strict $p$-branch numbering on $\Gamma^{+}$. Let us assume that $g>0$. Since $H_{1}(\Gamma,\mathbb{Z})\neq 0$ (where we regard $\Gamma$ as a topological space in the manner mentioned in Remark 1.1.5), one may find a vertex $v_{0}$ of $\Gamma$ and a path $(b_{j})_{j=1}^{l}$ from $v_{0}$ to $v_{0}$ itself such that $b_{j}\neq b_{j-1}^{\divideontimes}$ for any $j\in\{1,\cdots,l\}$, where $b_{0}^{\divideontimes}:=b_{l}^{\divideontimes}$. (We shall refer to such a path as a reduced loop based at $v_{0}$.) For each $j\in\{1,\cdots,l\}$, there exists a unique branch $b^{c}_{j}\in B_{\Gamma}$ with $B_{\zeta_{\Gamma}(b_{j})}:=\{b_{j},b_{j-1}^{\divideontimes},b_{j}^{c}\}$. The assumption that $\vec{m}$ specifies a strict $p$-branch numbering implies the following equalities

[TABLE]

The last equality is equivalent to the equality $l=\sum_{j=1}^{l}m_{b_{j}^{c}}$, which implies the equality $m_{b_{j}^{c}}=1$ for any $j\in\{1,\cdots,l\}$.

Now, suppose further that $g\geq 2$. Then, one verifies from the topological structure of $\Gamma$ that (after replacing $v_{0}$ by another vertex) there exist two reduced loops $(b^{\prime}_{j})_{j=1}^{l^{\prime}}$, $(b^{\prime\prime}_{j})_{j=1}^{l^{\prime\prime}}$ based at $v_{0}$ such that $\{b^{\prime}_{1},{b^{\prime}}_{l}^{\divideontimes}\}=\{b_{l}^{\divideontimes},b_{1}^{c}\}$, $\{b^{\prime\prime}_{1},{b^{\prime\prime}}_{l}^{\divideontimes}\}=\{b_{1}^{c},b_{1}\}$. By applying the above discussion to $(b^{\prime}_{j})_{j=1}^{l^{\prime}}$ and $(b^{\prime\prime}_{j})_{j=1}^{l^{\prime\prime}}$ respectively, we obtain the equalities $(m_{b_{1}^{c}}=)\ m_{b_{1}}=m_{b_{l}^{\divideontimes}}=1$. Hence, $3\ (=m_{b_{1}}+m_{b_{1}^{c}}+m_{b_{l}^{\divideontimes}})=1+p$, and this contradicts the assumption $p>2$. Consequently, the inequality $g\leq 1$ holds.

Next, let us consider assertion (ii). Suppose that $g=1$, and let $v_{0}$ and $(b_{j})_{j=1}^{l}$ be as above. If we are given a strict $p$-branch numbering $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}$ on $\Gamma^{+}$, then since $m_{b_{j}^{c}}=1$ and $m_{b_{j}}+m_{b_{j}^{c}}+m_{b_{j-1}^{\divideontimes}}=1+p$ (for any $j$), there exists a unique $a\in\{1,\cdots,p-1\}$ satisfying the condition $(*)_{a}$ described as follows: $(*)_{a}$ $m_{b_{j}}=a$ and $m_{b_{j}^{\divideontimes}}=p-a$ for any $j$. Conversely, for each $a\in\{1,\cdots,p-1\}$, one may construct a unique strict $p$-branch numbering $\vec{m}:=(m_{b})_{b\in B_{\Gamma}}$ on $\Gamma^{+}$ satisfying the condition $(*)_{a}$ in such a way that for each $v\in B_{\Gamma}\setminus\{b_{j}\}_{j=1}^{l}$, the multiset $[m_{b_{1}},m_{b_{2}},m_{b_{3}}]$ (where $B_{v}=\{b_{1},b_{2},b_{3}\}$) coincides with $[1,1,p-1]$. If $r>0$, then each such strict $p$-branch numbering is verified to be of exponent $\vec{e}\in\mathbb{F}_{p}^{\times r}$. Thus, this completes the proof of assertion (ii). ∎

Corollary 2.5.2**.**

Let $\mathfrak{X}$ be a totally degenerate curve over $k$ of type $(g,r)$. If there exists a dormant generic Miura $\mathfrak{s}\mathfrak{l}_{2}$-oper $\widehat{\mathcal{E}}^{\spadesuit}$ on $\mathfrak{X}$, then the inequality $g\leq 1$ holds. If, moreover, $g=1$ (and $r>0$), then there exist precisely $p-1$ dormant generic Mura $\mathfrak{s}\mathfrak{l}_{2}$-opers on $\mathfrak{X}$, and these are of exponent $\vec{e}$ (cf. Proposition 2.5.1 (ii) for the definition of $\vec{e}$).

Proof.

The assertion follows from Corollary 2.4.3 and Proposition 2.5.1. ∎