# Dormant Miura opers, Tango structures,
and the Bethe ansatz equations modulo $p$

Yasuhiro Wakabayashi

###### Abstract.

In the present paper, we study the Bethe ansatz equations and Miura opers in
characteristic $p>0$. Our study is based on a work by E. Frenkel, in which
solutions to the Bethe ansatz equations are described in terms of Miura opers
on the complex projective line. A result of the present paper provides its
partial analogue in the case of dormant Miura $\mathrm{PGL}_{2}$-opers. It is
known that dormant generic Miura $\mathrm{PGL}_{2}$-opers on an algebraic
curve correspond bijectively to certain line bundles on the curve called Tango
structures, which bring various sorts of pathological phenomena in positive
characteristic (e.g., counter-examples to the Kodaira vanishing theorem). By
applying our result, we can construct new examples of Tango structures by
means of solutions to the Bethe ansatz equations modulo $p$.

††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 14H81;††footnotetext: Key words:
oper, Miura oper, dormant oper, Bethe ansatz, Tango structure, Gaudin model,
pathology, $p$-curvature.

###### Contents

  1. 1 Opers and Miura opers
  2. 2 Dormant Miura opers and the Bethe Ansatz equations
  3. 3 Tango structures

## Introduction

In the present paper, we study the Bethe ansatz equations and Miura opers in
characteristic $p>0$. Recall (cf. [1], § 3.2, (3.5)) that the Bethe ansatz
equations for a simple finite-dimensional Lie algebra $\mathfrak{g}$ over
$\mathbb{C}$ are the equations on the set of distinct complex numbers
$z_{1},\cdots,z_{m}$ ($m\geq 1$) of the form

(1)
$\displaystyle\sum_{i=1}^{r}\frac{\langle\alpha_{j},\check{\lambda}_{i}\rangle}{z_{j}-x_{i}}-\sum_{s\neq
j}\frac{\langle\alpha_{j},\check{\alpha}_{s}\rangle}{z_{j}-z_{s}}=0\hskip
28.45274pt\left(j=1,\cdots,m\right),$

where $\alpha_{1},\cdots,\alpha_{m}$ are simple positive roots,
$\check{\alpha}_{1},\cdots,\check{\alpha}_{m}$ are the corresponding coroots,
$x_{1},\cdots,x_{r}$ ($r\geq 0$) are distinct complex numbers, and
$\check{\lambda}_{1},\cdots,\check{\lambda}_{r}$ are dominant coweights of
$\mathfrak{g}$. It is well-known that each solution to the Bethe ansatz
equations (1) specifies an eigenvector of the Hamiltonian of a certain spin
model called the Gaudin model. It gives an effective way to solve the problem
of simultaneous diagonalization of the Gaudin Hamiltonian. On the other hand,
the Bethe Ansatz equations may be also interpreted as a “no-monodromy”
condition on Miura opers. A Miura oper (cf. § 1.3) is an oper equipped with
additional data, or somewhat more precisely, a torsor over an algebraic curve
equipped with two Borel reductions and a flat connection satisfying some
conditions (including a certain transversality condition). For example,
$\mathrm{PGL}_{2}$-opers and Miura $\mathrm{PGL}_{2}$-opers on an algebraic
curve over $\mathbb{C}$ may be identified with, respectively, projective and
affine connections on the associated Riemann surface. In a work by E. Frenkel
(cf. [1] and [2]), a canonical correspondence between solutions to the Bethe
ansatz equations and certain Miura opers with trivial monodromy on the complex
projective line was constructed. Our study is based on this correspondence.

In the present paper, we consider the Bethe ansatz equations and the
correspondence given by Frenkel in the case of characteristic $p>0$. (For the
previous study of the Bethe ansatz equations, we refer to [12].)
Unfortunately, we are not able to apply the proofs of some results in [1] or
[2] to our discussion because of the positive characteristic nature of the
various objects involved. Indeed, on a flat connection in characteristic $p$,
the property of having trivial monodromy is not sufficient to characterize the
local triviality. Although our situation is rather different from the case
over $\mathbb{C}$ in this sense, we can prove a partial analogue of Frenkel’s
correspondence in positive characteristic (cf. Theorem 2.2.3). Moreover,
solutions to the Bethe ansatz equations for
$\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$ of a certain type turn out to
correspond bijectively to dormant generic Miura $\mathrm{PGL}_{2}$-opers on
the projective line $\mathbb{P}$, i.e., generic Miura $\mathrm{PGL}_{2}$-opers
on $\mathbb{P}$ with vanishing $p$-curvature (cf. Corollary 2.3.2). (For the
previous works concerning dormant Miura opers, we refer to [15] and [16].) On
the other hand, according to a previous work by the author (cf. [15], Theorem
A), dormant generic $\mathrm{PGL}_{2}$-opers are in bijection with (pre-)Tango
structures. A (pre-)Tango structure (cf. Definition 3.1.1 (i) or Definition
3.1.3 for the precise definition) is defined as a certain line bundle on an
algebraic curve (or equivalently, a certain connection on the sheaf of
$1$-forms) and brings various sorts of pathological phenomena in positive
characteristic (e.g., a counter-example to the Kodaira vanishing theorem). By
combining the results mentioned above, we obtain the following assertion,
which is the main result of the present paper.

######  Theorem A.

Let $l$ be a nonnegative integer and denote by $C^{lp+1}(\infty)$ (cf. (20))
the set of ordered collections of $lp+1$ distinct closed points of
$\mathrm{Spec}(k[x])=\mathbb{P}\setminus\\{\infty\\}$. Given each ${\bf
z}:=(z_{1},\cdots,z_{lp+1})\in C^{lp+1}(\infty)$, we write
$\mathrm{Tang}(\mathbb{P};{\bf z};1^{lp+1})$ for the set of pre-Tango
structures on the pointed projective line $(\mathbb{P},{\bf z})$ of monodromy
$1^{lp+1}:=(1,\cdots,1)\in k^{lp+1}$ (cf. (51) and (61)). Also, write
$\mathrm{PGL}_{2}\text{-}\mathrm{BA}_{\infty,0,\alpha^{lp+1}}$ for the subset
of $C^{lp+1}(\infty)$ consisting of elements ${\bf
z}:=(z_{1},\cdots,z_{lp+1})$ satisfying the Bethe ansatz equations of the form
(54), i.e., the equations (1) (divided by $2$) in the case where
$\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$, $r=0$, and
$\langle\alpha_{j},\check{\alpha}_{s}\rangle=2$ for any $j$, $s$. Then, there
exists a canonical bijection

(2) $\displaystyle\coprod_{{\bf z}\in
C^{lp+1}(\infty)}\mathrm{Tang}(\mathbb{P};{\bf
z};1^{lp+1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{PGL}_{2}\text{-}\mathrm{BA}_{\infty,0,\alpha^{lp+1}}.$

In particular, $(\mathbb{P},{\bf z})$ admits a pre-Tango structure of
monodromy $1^{lp+1}$ if and only if the equality $f^{\prime\prime}(x)=0$
holds, where $f(x):=\prod_{i=1}^{lp+1}(x-z_{i})$ (cf. Remark 2.3.3). Moreover,
if $(\mathbb{P},{\bf z})$ admits a pre-Tango structure of monodromy
$1^{lp+1}$, then it is uniquely determined and expressed as
$\partial_{x}+\sum_{i=1}^{lp+1}\frac{1}{x-z_{i}}$ (under the identification
$\Omega_{\mathbb{P}^{D_{{\bf
z}}\text{-}\mathrm{log}}/k}|_{\mathbb{P}\setminus\\{\infty\\}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{\mathbb{P}\setminus\\{\infty\\}}$
given by $dx\mapsto 1$).

As a corollary of the above theorem, we can construct infinitely many examples
of Tango curves, i.e., algebraic curves admitting a Tango structure (cf.
Definition 3.1.1 (ii)). To the author’s knowledge, there are very few examples
of Tango curves described explicitly at the present time. The major ones given
by M. Raynaud (cf. [10], Example, or [6], Example 1.3) may be thought of as
special cases of our construction. In particular, we provide other new Tango
curves. Our result is described as follows.

######  Theorem B (cf. Theorem 3.4.1).

Let $(a,b)$ be a pair of positive integers with $\mathrm{gcd}(a,bp-1)=1$.
Denote by $Y$ the desingularization of the plane curve defined by the equation

(3) $\displaystyle y^{bp-1}=h(x),$

where $h(x)\ \left(\in k[x]\right)$ denotes a monic polynomial of degree $ap$
satisfying that $\mathrm{gcd}(h(x),h^{\prime}(x))=1$ and
$h^{\prime\prime}(x)=0$, and $(x,y)$ are an inhomogenous coordinate of the
projective plane. Then, $Y$ is a Tango curve.

Acknowledgement
The author would like to thank dormant Miura opers and Tango curves for their
helpful advice and heartfelt encouragement. The author was partially supported
by the Grant-in-Aid for Scientific Research (KAKENHI No. 18K13385).

##  1\. Opers and Miura opers

In this section, we recall the definitions of an oper and a (generic) Miura
oper.

###  1.1. Algebraic groups and Lie algebras

Throughout the present paper, let us fix a prime $p$ and an algebraically
closed field $k$ of characteristic $p$. Also, let $G$ be a connected simple
algebraic group over $k$ of adjoint type satisfying the condition $(*)_{G}$
described as follows.

  * $(*)_{G}$ :

$G$ is either equal to $\mathrm{PGL}_{n}$ with $1<n<p$ or satisfies the
inequality $p>2\cdot h$, where $h$ denotes the Coxeter number of $G$.

Next, let us fix a maximal torus $T$ of $G$, a Borel subgroup $B$ of $G$
containing $T$. Write $N=[B,B]$, i.e., the unipotent radical of $B$, and write
$W$ for the Weyl group of $(G,T)$. Denote by $\mathfrak{g}$, $\mathfrak{b}$,
$\mathfrak{n}$, and $\mathfrak{t}$ the Lie algebras of $G$, $B$, $N$, and $T$
respectively (hence
$\mathfrak{t},\mathfrak{n}\subseteq\mathfrak{b}\subseteq\mathfrak{g}$). Denote
by $\Phi^{+}$ the set of positive roots in $B$ with respect to $T$ and by
$\Phi^{-}$ the set of negative roots. Also, denote by $\Gamma$
($\subseteq\Phi^{+}$) the set of simple positive roots. In what follows, given
each character $\varphi:T\rightarrow\mathbb{G}_{m}$ (resp., each cocharacter
$\check{\varphi}:\mathbb{G}_{m}\rightarrow T$), we shall use, by abuse of
notation, the same notation $\varphi$ (resp., $\check{\varphi}$) to denote its
differential $d\varphi\in\mathfrak{t}^{\vee}$ (resp.,
$d\check{\varphi}\in\mathfrak{t}$). For each $\alpha\in\Phi^{+}\cup\Phi^{-}$,
we write

(4) $\mathfrak{g}^{\alpha}:=\big{\\{}x\in\mathfrak{g}\ \big{|}\
\text{$\mathrm{ad}(t)(x)=\alpha(t)\cdot x$ for all $t\in T$}\big{\\}}.$

Then, $\mathfrak{g}$ admits a canonical decomposition

(5)
$\displaystyle\mathfrak{g}=\mathfrak{t}\oplus\left(\bigoplus_{\alpha\in\Phi^{+}}\mathfrak{g}^{\alpha}\right)\oplus\left(\bigoplus_{\beta\in\Phi^{-}}\mathfrak{g}^{\beta}\right)$

(which restricts to a decomposition
$\mathfrak{b}=\mathfrak{t}\oplus(\bigoplus_{\alpha\in\Phi^{+}}\mathfrak{g}^{\alpha})$).
In particular, each $\mathfrak{g}^{-\alpha}$ ($\alpha\in\Gamma$) may be
thought of as a subspace of $\mathfrak{g}/\mathfrak{b}$ closed under the
adjoint $B$-action. We shall write $\check{\rho}$ for the cocharacter
$\mathbb{G}_{m}\rightarrow T$ (as well as its differential) defined as the sum
$\sum_{\alpha\in\Gamma}\check{\omega}_{\alpha}$, where each
$\check{\omega}_{\alpha}$ ($\alpha\in\Gamma$) denotes the fundamental coweight
of $\alpha$. In the following discussion, we shall fix a generator
$f_{\alpha}$ of $\mathfrak{g}^{-\alpha}$ for each $\alpha\in\Gamma$. Hence, we
obtain $p_{-1}:=\sum_{\alpha\in\Gamma}f_{\alpha}$.

Denote by $\mathfrak{t}_{\mathrm{reg}}$ the set of regular elements in
$\mathfrak{t}$. It follows from [5], Chap. VI, Theorem 7.2, that

(6)
$\displaystyle\mathfrak{t}_{\mathrm{reg}}=\left\\{\check{\lambda}\in\mathfrak{t}\
|\ \alpha(\check{\lambda})\neq 0\ \text{for any root
$\alpha\in\Phi^{+}\cup\Phi^{-}$}\right\\}.$

Also, denote by $\mathfrak{t}^{F}_{\mathrm{reg}}$ the set of $F$-invariant
elements in $\mathfrak{t}_{\mathrm{reg}}$, where $F$ denotes the absolute
Frobenius morphism of $\mathfrak{t}_{\mathrm{reg}}$ viewed as a $k$-scheme.

###  1.2. Opers

In this subsection, we recall the definitions of an oper. Let $X$ be a
connected proper smooth curve over $k$. Given a reduced effective divisor $D$
on $X$, one can equip $X$ with a log structure determined by $D$ in the usual
manner; we shall denote the resulting log scheme by
$X^{D\text{-}\mathrm{log}}$. (In particular, if $D=0$, then
$X^{D\text{-}\mathrm{log}}$ coincides with $X$.) Let $G_{0}$ be some algebraic
group $G_{0}$ over $k$, $\mathcal{E}$ a (right) $G_{0}$-torsor over $X$. Given
a $k$-vector space $\mathfrak{h}$ equipped with a (left) $G_{0}$-action, we
shall write $\mathfrak{h}_{\mathcal{E}}$ for the vector bundle on $X$
associated with the relative affine space
$\mathcal{E}\times^{G_{0}}\mathfrak{h}$
($:=(\mathcal{E}\times_{k}\mathfrak{h})/G_{0}$) over $X$. We shall refer to
any logarithmic connection on $\mathcal{E}$ (with respect to the log structure
of $X^{D\text{-}\mathrm{log}}$) as a $D$-log connection on $\mathcal{E}$.

Let $X$, $D$ be as above, and $G$ as in the previous subsection. Also, let
$\mathcal{E}$ be a (right) $G$-torsor over $X$. The adjoint $G$-action on
$\mathfrak{g}$ gives rise to a vector bundle $\mathfrak{g}_{\mathcal{E}}$.
Next, suppose that we are given a $D$-log connection $\nabla$ on $\mathcal{E}$
and a $B$-reduction $\mathcal{E}_{B}$ of $\mathcal{E}$ (i.e., a $B$-torsor
$\mathcal{E}_{B}$ over $X$ together with an isomorphism
$\mathcal{E}_{B}\times^{B}G\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}$
of $G$-torsors). Let us choose, locally on $X$, a $D$-log connection
$\nabla^{\prime}$ on $\mathcal{E}$ preserving $\mathcal{E}_{B}$, and take the
difference $\nabla-\nabla^{\prime}$, which forms a section of
$\mathfrak{g}_{\mathcal{E}_{B}}\otimes\Omega_{X^{D\text{-}\mathrm{log}}/k}\
(\cong\mathfrak{g}_{\mathcal{E}}\otimes\Omega_{X^{D\text{-}\mathrm{log}}/k})$.
The local section of
$(\mathfrak{g}/\mathfrak{b})_{\mathcal{E}_{B}}\otimes\Omega_{X^{D\text{-}\mathrm{log}}/k}$
determined by this section via projection is independent of the choice
$\nabla^{\prime}$. Hence, these sections may be glued together to a global
section of
$(\mathfrak{g}/\mathfrak{b})_{\mathcal{E}_{B}}\otimes\Omega_{X^{D\text{-}\mathrm{log}}/k}$,
denoted by $\nabla/\mathcal{E}_{B}$.

By a $G$-oper on $X^{D\text{-}\mathrm{log}}$, we mean a triple
$\mathcal{E}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B})$, where
$\mathcal{E}$, $\nabla$, and $\mathcal{E}_{B}$ are as above such that the
section $\nabla/\mathcal{E}_{B}$ lies in the submodule
$(\bigoplus_{\alpha\in\Gamma}\mathfrak{g}^{-\alpha})_{\mathcal{E}_{B}}\otimes\Omega_{X^{D\text{-}\mathrm{log}}/k}$
and its image in
$\mathfrak{g}^{-\beta}_{\mathcal{E}_{B}}\otimes\Omega_{X^{D\text{-}\mathrm{log}}/k}$
(for each $\beta\in\Gamma$) via the projection
$\bigoplus_{\alpha\in\Gamma}\mathfrak{g}^{-\alpha}\twoheadrightarrow\mathfrak{g}^{-\beta}$
specifies a nowhere vanishing section. Let
$\mathcal{E}^{\spadesuit}_{1}:=(\mathcal{E}_{1},\nabla_{1},\mathcal{E}_{B,1})$,
$\mathcal{E}^{\spadesuit}_{2}:=(\mathcal{E}_{2},\nabla_{2},\mathcal{E}_{B,2})$
be $G$-opers on $X^{D\text{-}\mathrm{log}}$. Then, we refer to an isomorphism
$\mathcal{E}_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{E}_{2}$
compatible with both the respective connections and $B$-reductions as an
isomorphism of $G$-opers (from $\mathcal{E}_{1}^{\spadesuit}$ to
$\mathcal{E}^{\spadesuit}_{2}$). Denote by

(7) $\displaystyle G\text{-}\mathrm{Op}(X^{D\text{-}\mathrm{log}})$

the set of isomorphism classes of $G$-opers on $X^{D\text{-}\mathrm{log}}$.

Next, let $\widehat{\mathcal{O}}$ be a complete discrete valuation ring over
$k$, and write $\mathbb{D}:=\mathrm{Spec}(\widehat{\mathcal{O}})$ (i.e., the
formal disc). The closed point of $\mathbb{D}$, denoted by $x_{0}$, defines a
reduced effective divisor, and hence, defines a log structure on $\mathbb{D}$;
we denote the resulting log scheme by $\mathbb{D}^{\mathrm{log}}$. As in the
case of the entire $X^{D\text{-}\mathrm{log}}$ discussed above, we have the
definition of a $G$-oper on $\mathbb{D}$ (resp., $\mathbb{D}^{\mathrm{log}}$).
Denote by

(8) $\displaystyle G\text{-}\mathrm{Op}(\mathbb{D})$

the set of isomorphism classes of $G$-opers on $\mathbb{D}$.

Let $\mathcal{E}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B})$ be a
$G$-oper on $\mathbb{D}^{\mathrm{log}}$. After choosing a uniformizer $t$ of
$\widehat{\mathcal{O}}$ (which gives an isomorphism
$k[[t]]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widehat{\mathcal{O}}$) and
applying a suitable gauge transformation, $\nabla$ may be expressed as a log
connection on the trivial $G$-torsor $\mathbb{D}\times G$ of the form

(9) $\displaystyle\partial_{t}+\frac{1}{t}(p_{-1}+u)+{\bf u}(t)$

for some $u\in\mathfrak{b}$, ${\bf u}(t)\in\mathfrak{b}[[t]]\
\left(:=\mathfrak{b}(k[[t]])\right)$, where $\partial_{t}:=\frac{d}{dt}$. In
particular, the monodromy, in the sense of [13], Definition 1.6.1, of $\nabla$
at $x_{0}$ (with respect to this expression) is $p_{-1}+u$. Here, let

(12) $\displaystyle\mathfrak{c}\ (:=\mathfrak{g}\ooalign{$/$\cr$\,/$}G)$

denote the GIT quotient of $\mathfrak{g}$ by the adjoint $G$-action and let

(13) $\displaystyle\chi:\mathfrak{g}\rightarrow\mathfrak{c}$

denote the natural quotient. Then, it is verified that the element
$\rho:=\chi(p_{-1}+u)\in\mathfrak{c}(k)$ is independent of both the choice of
$t$ and the expression (9) of $\nabla$. In this situation, we shall say that
$\mathcal{E}^{\spadesuit}$ is of radius $\rho$. For each
$\rho\in\mathfrak{c}(k)$, we shall write

(14) $\displaystyle G\text{-}\mathrm{Op}(\mathbb{D}^{\mathrm{log}};\rho)$

for the set of isomorphism classes of $G$-opers on $\mathbb{D}^{\mathrm{log}}$
of radius $\rho$. As discussed at the end of [3], § 9.1, there exists a
canonical injection

(15) $\displaystyle G\text{-}\mathrm{Op}(\mathbb{D})\hookrightarrow
G\text{-}\mathrm{Op}(\mathbb{D}^{\mathrm{log}};\chi(-\check{\rho})).$

obtained by applying suitable gauge transformations.

######  Remark 1.2.1.

Since the condition $(*)_{G}$ holds, it follows from a Chevalley’s theorem
(cf. [7], Theorem 1.1.1; [5], Chap. VI, Theorem 8.2) that the composite
$\mathfrak{t}\hookrightarrow\mathfrak{g}\stackrel{{\scriptstyle\chi}}{{\rightarrow}}\mathfrak{c}$
induces an isomorphism
$\mathfrak{t}/W\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{c}$. In
particular, an element $\check{\lambda}\in\mathfrak{t}$ satisfies the equality
$\chi(\check{\lambda})=\chi(-\check{\rho})$ if and only if
$\check{\lambda}=w(-\check{\rho})$ for some $w\in W$.

Also, let $X$ be as before and ${\bf x}:=(x_{i})_{i=1}^{r}$ ($r\geq 1$) be an
ordered collection of distinct closed points of $X$. We shall write $D_{{\bf
x}}:=\sum_{i=1}^{r}[x_{i}]$, where $[x_{i}]$ ($i=1,\cdots,r$) denotes the
reduced effective divisor on $X$ determined by $x_{i}$. For each
$i\in\\{1,\cdots,r\\}$, denote by $\mathbb{D}_{x_{i}}$ the formal disc in $X$
around $x_{i}$. Let
$\boldsymbol{\rho}:=(\rho_{i})_{i=1}^{r}\in\mathfrak{c}(k)^{r}$. We shall say
that a $G$-oper on $X^{D_{{\bf x}}\text{-}\mathrm{log}}$ is of radii
$\boldsymbol{\rho}$ if, for each $i\in\\{1,\cdots,r\\}$, the $G$-oper on
$\mathbb{D}_{x_{i}}$ defined as its restriction is of radius $\rho_{i}$.
Denote by

(16) $\displaystyle G\text{-}\mathrm{Op}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\boldsymbol{\rho})$

the subset of
$G\text{-}\mathrm{Op}(X^{D_{\boldsymbol{x}}\text{-}\mathrm{log}})$ classifying
$G$-opers of radii $\boldsymbol{\rho}$.

###  1.3. Generic Miura opers

Next, let us recall the notion of a (generic) Miura oper on a curve with log
structure (cf. [15], Definition 3.2.1). Let $X$ be a connected proper smooth
curve over $k$ and $D$ a reduced effective divisor on $X$. A Miura $G$-oper on
$X$ (resp., on $X^{D\text{-}\mathrm{log}}$) is a quadruple
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$,
where $(\mathcal{E},\nabla,\mathcal{E}_{B})$ is a $G$-oper on $X$ (resp., on
$X^{D\text{-}\mathrm{log}}$) and $\mathcal{E}^{\prime}_{B}$ is another
$B$-reduction of $\mathcal{E}$ horizontal with respect to $\nabla$. Also, for
two Miura $G$-opers $\widehat{\mathcal{E}}^{\spadesuit}_{1}$,
$\widehat{\mathcal{E}}^{\spadesuit}_{2}$, we shall refer to an isomorphism of
$G$-opers from $\widehat{\mathcal{E}}^{\spadesuit}_{1}$ to
$\widehat{\mathcal{E}}^{\spadesuit}_{2}$ compatible with the respective second
$B$-reductions as an isomorphism of Miura $G$-opers (from
$\widehat{\mathcal{E}}^{\spadesuit}_{1}$ to
$\widehat{\mathcal{E}}^{\spadesuit}_{2}$).

Let
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
be a Miura $G$-oper on either $X$ or $X^{D\text{-}\mathrm{log}}$. By twisting
the flag variety $G/B$ by the $B$-torsor $\mathcal{E}_{B}$, we obtain a proper
scheme $(G/B)_{\mathcal{E}_{B}}:=\mathcal{E}_{B}\times^{B}(G/B)$ over $X$.
Here, denote by $w_{0}$ the longest element of $W$. The Bruhat decomposition
$G=\coprod_{w\in W}BwB$ gives rise to a decomposition

(17) $\displaystyle(G/B)_{\mathcal{E}_{B}}=\coprod_{w\in
W}S_{\mathcal{E}_{B},w}$

of $(G/B)_{\mathcal{E}_{B}}$, where each $S_{\mathcal{E}_{B},w}$ denotes the
$\mathcal{E}_{B}$-twist of $Bw_{0}wB$, i.e.,
$S_{\mathcal{E}_{B},w}:=\mathcal{E}_{B}\times^{B}(Bw_{0}wB)$. Note that the
$B$-reduction $\mathcal{E}^{\prime}_{B}$ determines a section
$\sigma_{\mathcal{E}_{B},\mathcal{E}^{\prime}_{B}}:X\rightarrow(G/B)_{\mathcal{E}_{B}}$
of the natural projection $(G/B)_{\mathcal{E}_{B}}\rightarrow X$. Given an
element $w$ of $W$ and a point $x_{\sharp}$ of $X$, we shall say that
$\mathcal{E}_{B}$ and $\mathcal{E}^{\prime}_{B}$ are in relative position $w$
at $x_{\sharp}$ if
$\sigma_{\mathcal{E}_{B},\mathcal{E}^{\prime}_{B}}(x_{\sharp})$ belongs to
$S_{\mathcal{E}_{B},w}$. In particular, if
$\sigma_{\mathcal{E}_{B},\mathcal{E}^{\prime}_{B}}(x_{\sharp})$ belongs to
$S_{\mathcal{E}_{B},1}$, then we shall say that $\mathcal{E}_{B}$ and
$\mathcal{E}^{\prime}_{B}$ are in generic position at $x$. Moreover, we shall
say that a Miura $G$-oper
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
is generic if $\mathcal{E}_{B}$ and $\mathcal{E}^{\prime}_{B}$ are in generic
position at any point of $X$. We also have the various notions recalled here
in the case where the entire $X$ or $X^{D\text{-}\mathrm{log}}$ is replaced by
either $\mathbb{D}$, $\mathbb{D}^{\times}$, or $\mathbb{D}^{\mathrm{log}}$.

###  1.4. Exponent of Miura opers

Let
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
be a generic Miura $G$-oper on $\mathbb{D}^{\mathrm{log}}$. After choosing a
uniformizer $t$ of $\widehat{\mathcal{O}}$ and applying a gauge
transformation, $\nabla$ may be expressed as a log connection on the trivial
$G$-torsor $\mathbb{D}\times G$ of the form

(18)
$\displaystyle\partial_{t}+\frac{1}{t}\left(p_{-1}+\check{\lambda}\right)+{\bf
u}(t)$

for some $\check{\lambda}\in\mathfrak{t}$, ${\bf u}(t)\in\mathfrak{t}[[t]]$
(cf. [15], Proposition 3.4.3, for the case where the underlying curve is
globally defined and in positive characteristic). It is verified that
$\check{\lambda}$ is independent of both the choice of $t$ and the expression
(18) of $\nabla$. In this situation, we shall say that
$\widehat{\mathcal{E}}^{\spadesuit}$ is of exponent $\check{\lambda}$. If a
generic Miura $G$-oper
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
on $\mathbb{D}^{\mathrm{log}}$ is of exponent $\check{\lambda}\
\left(\in\mathfrak{t}\right)$, then the underlying $G$-oper
$(\mathcal{E},\nabla,\mathcal{E}_{B})$ is of radius
$\chi(p_{-1}+\check{\lambda})\in\mathfrak{c}(k)$.

######  Remark 1.4.1.

Let $\check{\lambda}$ and $w$ be elements of $\mathfrak{t}_{\mathrm{reg}}$ and
$W$ respectively. Then, one may verify immediately that the equality
$\chi(w(\check{\lambda}))=\chi(\check{\lambda})=\chi(p_{-1}+\check{\lambda})$
holds (cf. Remark 1.2.1). In particular, if
$\widehat{\mathcal{E}}^{\spadesuit}$ is a generic Miura $G$-oper on
$\mathbb{D}^{\mathrm{log}}$ of exponent $\check{\lambda}$, then the radius of
its underlying $G$-oper coincides with $\chi(\check{\lambda})$.

######  Remark 1.4.2.

By applying the gauge transformation by $\check{\rho}\circ t\in T(k[[t]])$, we
obtain a bijection from the set of isomorphism classes of Miura $G$-opers on
$\mathbb{D}$ to the set of isomorphism classes of Miura $G$-opers on
$\mathbb{D}^{\mathrm{log}}$ of exponent $-\check{\rho}$. This bijection is
compatible (under the equality $\chi(-\check{\rho})=\chi(p_{-1}-\check{\rho})$
obtained in the above Remark) with the injection (15) via forgetting the data
of second $B$-reductions of Miura $G$-opers.

Next, let ${\bf x}:=(x_{i})_{i=1}^{r}$ ($r\geq 1$) be an ordered collection of
distinct closed points of $X$, and let
$\check{\boldsymbol{\lambda}}:=(\check{\lambda}_{i})_{i=1}^{r}\in(\mathfrak{t}_{\mathrm{reg}})^{r}$.
We shall say that a Miura $G$-oper
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
on $X^{D_{\boldsymbol{x}}\text{-}\mathrm{log}}$ is of exponents
$\check{\boldsymbol{\lambda}}$ if, for each $i\in\\{1,\cdots,r\\}$, the Miura
$G$-oper on $\mathbb{D}_{x_{i}}^{\mathrm{log}}$ defined as its restriction is
generic and of exponent $\check{\lambda}_{i}$. Denote by

(19) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{gen}}$

the set of isomorphism classes of generic Miura $G$-opers on $X^{D_{{\bf
x}}\text{-}\mathrm{log}}$ of exponents $\check{\boldsymbol{\lambda}}$.

For each positive integer $m$, we shall write

(20) $\displaystyle C^{m}({\bf x})$

for the set of ordered collections of $m$ distinct closed points in
$X\setminus\\{x_{1},\cdots,x_{r}\\}$. Given each
$\boldsymbol{w}:=(w_{1},\cdots,w_{m})\in W^{m}$, we shall write

(21) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{gen},+\boldsymbol{w}}$
$\displaystyle:=\coprod_{{\bf z}\in C^{m}({\bf
x})}G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}},$

where
$\boldsymbol{w}(-\check{\rho}):=(w_{j}(-\check{\rho}))_{j=1}^{m}\in\mathfrak{t}^{m}$.

For each ${\bf z}\in C^{m}({\bf x})$, the injection (15) induces an injection

(22) $\displaystyle G\text{-}\mathrm{Op}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\chi(\check{\boldsymbol{\lambda}}))\hookrightarrow
G\text{-}\mathrm{Op}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\chi(\check{\boldsymbol{\lambda}}),\chi(-\check{\rho})^{m})),$

where
$\chi(\check{\boldsymbol{\lambda}}):=(\chi(\check{\lambda}_{1}),\cdots,\chi(\check{\lambda}_{r}))\in\mathfrak{c}(k)^{r}$
and
$\chi(-\check{\rho})^{m}:=(\chi(-\check{\rho}),\cdots,\chi(-\check{\rho}))\in\mathfrak{c}(k)^{m}$.
Given each $\boldsymbol{w}$ as above, we denote by

(23) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+({\bf
z};{\bf w})}$

the inverse image of the image of (22) via the forgetting map

(24) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}}\rightarrow
G\text{-}\mathrm{Op}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\chi(\check{\boldsymbol{\lambda}}),\chi(-\check{\rho})^{m}))$

(cf. Remark 1.4.1). That is to say, the following square diagram is
commutative and cartesian:

(25) $\displaystyle\begin{CD}G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+({\bf
z};{\bf w})}@>{}>{}>G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}}\\\
@V{}V{}V@V{}V{(\ref{W1})}V\\\ G\text{-}\mathrm{Op}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\chi(\check{\boldsymbol{\lambda}}))@>{}>{(\ref{Ee900})}>G\text{-}\mathrm{Op}(X^{D_{({\bf
x},{\bf
z})}\text{-}\mathrm{log}};(\chi(\check{\boldsymbol{\lambda}}),\chi(-\check{\rho})^{m})).\end{CD}$

If we write

(26) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+\boldsymbol{w}}$
$\displaystyle:=\coprod_{{\bf z}\in C^{m}({\bf
x})}G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+({\bf
z};\boldsymbol{w})},$

then the upper horizontal arrow in (25) in the case of each ${\bf z}\in
C^{m}({\bf x})$ gives an inclusion

(27) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+\boldsymbol{w}}\hookrightarrow
G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{gen},+\boldsymbol{w}}.$

We shall consider $G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+\boldsymbol{w}}$
as a subset of $G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{gen},+\boldsymbol{w}}$
via this injection.

##  2\. Dormant Miura opers and the Bethe Ansatz equations

In this section, we prove a partial analogue (cf. Theorem 2.2.3) of the
correspondence given in [1] or [2] between solutions to the Bethe ansatz
equations and certain Miura opers with trivial monodromy on the complex
projective line. Moreover, we show (cf. Corollary 2.3.2) that any solution to
the Bethe ansatz equations for $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$ of
a certain type turns out to come from dormant generic Miura
$\mathrm{PGL}_{2}$-opers, i.e., generic Miura $\mathrm{PGL}_{2}$-opers with
vanishing $p$-curvature.

###  2.1. Dormant Miura opers

We shall show (cf. Proposition 2.1.2) that each dormant generic Miura oper
specifies a Miura oper classified by the set $G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+({\bf
z};\boldsymbol{w})}$ (cf. (23)) defined in the previous section.

First, recall from [15], Definition 3.8.1, that a Miura $G$-oper
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
is dormant if $\nabla$ has vanishing $p$-curvature (cf. e.g., [4], § 5.0, or
[15], § 1.6, for the definition of $p$-curvature). Then, we verifies the
following assertion.

######  Proposition 2.1.1.

Let $\check{\lambda}$ be an element of $\mathfrak{t}$ such that there exists a
dormant generic Miura $G$-oper
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
on $\mathbb{D}^{\mathrm{log}}$ of exponent $\check{\lambda}$. Then,
$\check{\lambda}$ lies in $\mathfrak{t}_{\mathrm{reg}}^{F}$.

###### Proof.

Denote by $\mu\in\mathfrak{g}$ the monodromy (cf. [13], Definition 1.6.1) of
$\nabla$ at the closed point $x_{0}$, and consider a Jordan decomposition
$\mu=\mu_{s}+\mu_{n}$ with $\mu_{s}$ semisimple and $\mu_{n}$ nilpotent.
Denote by $\mathrm{ad}:\mathfrak{g}\rightarrow\mathrm{End}(\mathfrak{g})$ the
adjoint representation of $\mathfrak{g}$, which is injective and compatible
with the respective restricted structures, i.e., $p$-power operations. One may
find an isomorphism
$\alpha:\mathrm{End}(\mathfrak{g})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathfrak{g}\mathfrak{l}_{\mathrm{dim}(\mathfrak{g})}$
of restricted Lie algebras which sends $\alpha(\mathrm{ad}(\mu))\
\left(=\alpha(\mathrm{ad}(\mu_{s}))+\alpha(\mathrm{ad}(\mu_{n}))\right)$ to a
Jordan normal form. Namely, $\alpha(\mathrm{ad}(\mu_{s}))$ is diagonal and
every entry of $\alpha(\mathrm{ad}(\mu_{n}))$ except the superdiagonal is $0$.
Let us observe the following sequence of equalities:

(28) $\displaystyle\alpha(\mathrm{ad}(\mu_{s}))+\alpha(\mathrm{ad}(\mu_{n}))$
$\displaystyle=\alpha(\mathrm{ad}(\mu_{s}+\mu_{n}))=\alpha(\mathrm{ad}((\mu_{s}+\mu_{n})^{[p]}))$
$\displaystyle=\alpha(\mathrm{ad}(\mu_{s}+\mu_{n}))^{p}=(\alpha(\mathrm{ad}(\mu_{s}))+\alpha(\mathrm{ad}(\mu_{n})))^{p},$

where the second equality follows from the assumption that $\nabla$ has
vanishing $p$-curvature and $(-)^{[p]}$ denotes the $p$-power operation on
$\mathfrak{g}$ (cf. [13], § 3.2, Remark 3.2.2). By an explicit computation of
$(\alpha(\mathrm{ad}(\mu_{s}))+\alpha(\mathrm{ad}(\mu_{n})))^{p}$, (28)
implies that $\alpha(\mathrm{ad}(\mu_{n}))=0$ (hence $\mu_{n}=0$), namely,
$\mu$ is conjugate to some $v\in\mathfrak{t}$. On the other hand, $\mu$ is, by
definition, conjugate to $p_{-1}+\check{\lambda}$. It follows that
$p_{-1}+\check{\lambda}$ is conjugate to $v$, and hence,
$\check{\lambda}=w(v)$ for some $w\in W$. Since $p_{-1}+\check{\lambda}$, as
well as $v$, is regular, $\check{\lambda}\ \left(=w(v)\right)$ turns out to be
regular. Moreover, by the equality $\mu_{n}=0$, (28) reads the equality
$\alpha(\mathrm{ad}(\mu))=\alpha(\mathrm{ad}(\mu))^{p}$. This implies the
equality $v^{[p]}=v$, or equivalently, $v\in\mathfrak{t}^{F}_{\mathrm{reg}}$.
This completes the proof of the assertion. ∎

Let $X$ be a connected proper smooth curve over $k$, ${\bf
x}:=(x_{i})_{i=1}^{r}$ ($r\geq 1$) an ordered collection of distinct closed
points of $X$. Also, let
$\check{\boldsymbol{\lambda}}:=(\check{\lambda}_{i})_{i=1}^{r}\in(\mathfrak{t}_{\mathrm{reg}})^{r}$,
$\boldsymbol{w}\in W^{m}$. Given each ${\bf z}\in C^{m}({\bf x})$, we write

(29) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}$

for the subset of $G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}}$
consisting of dormant generic Miura $G$-opers. (According to the above
proposition, this subset is empty unless
$\check{\boldsymbol{\lambda}}\in(\mathfrak{t}_{\mathrm{reg}}^{F})^{r}$.) Also,
we write

(30) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{gen},+\boldsymbol{w}}^{{}^{\mathrm{Zzz...}}}:=\coprod_{{\bf
z}\in C^{m}({\bf x})}G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}.$

Then, the following assertion holds.

######  Proposition 2.1.2.

Let us keep the above notation. Then, the set
$G\text{-}\mathrm{MOp}(X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\boldsymbol{w}(-\check{\rho})))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}$
(for each ${\bf z}\in C^{m}({\bf x})$) is contained in
$G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+({\bf
z};\boldsymbol{w})}$. In particular, we have an inclusion

(31) $\displaystyle G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{gen},+\boldsymbol{w}}^{{}^{\mathrm{Zzz...}}}\hookrightarrow
G\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\check{\boldsymbol{\lambda}})_{\mathrm{triv},+\boldsymbol{w}}.$

###### Proof.

Let $w\in W$ and
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
a dormant generic Miura $G$-oper on $\mathbb{D}^{\mathrm{log}}$ of exponent
$w(-\check{\rho})$. Let us choose a uniformizer $t$ of $\widehat{\mathcal{O}}$
(inducing
$\widehat{\mathcal{O}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}k[[t]]$). To
complete the proof, it suffices to prove that
$(\mathcal{E},\nabla,\mathcal{E}_{B})$ becomes a Miura $G$-oper on
$\mathbb{D}$ after a gauge transformation by some element of $B(k((t)))$.
Here, notice that by the assumption $(*)_{G}$, one may obtain the exponential
map $\mathrm{exp}:\mathfrak{n}\rightarrow N$ given by [13], § 1.4, (55). By
means of this, one may apply an argument similar to the argument in the proof
of [3], Proposition 9.2.1, to our positive characteristic case. Hence, after a
gauge transformation by some element of $B(k((t)))$, $\nabla$ may be expressed
as $\partial_{t}+p_{-1}+{\bf v}(t)+\frac{v}{t}$ for some ${\bf
v}(t)\in\mathfrak{b}[[t]]$ and $v\in\mathfrak{n}$. The mod $t$ reduction of
the $p$-curvature of $\nabla$ is given by $v^{[p]}-v$, which is equal to $0$
because of the dormancy condition on $\widehat{\mathcal{E}}^{\spadesuit}$.
But, since $v\in\mathfrak{n}$, we have $v^{[p]}=0$, which implies that $v=0$.
Therefore, $(\mathcal{E},\nabla,\mathcal{E}_{B})$ forms a $G$-oper on
$\mathbb{D}$, as desired. ∎

###  2.2. The Bethe ansatz equations I

In this section, we consider the case where $X$ is taken to be the projective
line over $k$, denoted by $\mathbb{P}$. Denote by $x$ the natural coordinate
of $\mathbb{P}$, i.e., $\mathbb{P}\setminus\\{\infty\\}=\mathrm{Spec}(k[x])$.
Fix integers $r$, $m$ with $r\geq 0$, $m\geq 1$. Let ${\bf
x}:=(x_{1},\cdots,x_{r},x_{r+1})$ be an ordered collection of distinct closed
points of $\mathbb{P}$ with $x_{r+1}=\infty$ and
$\check{\boldsymbol{\lambda}}:=(\check{\lambda}_{1},\cdots,\check{\lambda}_{r},\check{\lambda}_{r+1})\in(\mathfrak{t}_{\mathrm{reg}})^{r+1}$,
$\check{\boldsymbol{\lambda}}^{\prime}:=(\check{\lambda}^{\prime}_{1},\cdots,\check{\lambda}^{\prime}_{m})\in(\mathfrak{t}_{\mathrm{reg}})^{m}$.

######  Proposition 2.2.1.

Let ${\bf z}\in C^{m}({\bf x})$. Then, the set
$G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))_{\mathrm{gen}}$
is nonempty if and only if the equality

(32)
$\displaystyle\sum_{i=1}^{r+1}(\check{\lambda}_{i}+\check{\rho})+\sum_{j=1}^{m}(\check{\lambda}^{\prime}_{j}+\check{\rho})=2\check{\rho}$

holds. Moreover, if the equality (32) holds, then
$G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))_{\mathrm{gen}}$
consists of a single element.

###### Proof.

Let us write
$\mathcal{E}^{\dagger}_{T,\mathrm{log}}:=\Omega_{\mathbb{P}^{D_{(\boldsymbol{x},\boldsymbol{z})}\text{-}\mathrm{log}}/k}^{\times}\times^{\mathbb{G}_{m},\check{\rho}}T$
and
$\mathcal{E}^{\dagger}_{T}:=\Omega_{\mathbb{P}/k}^{\times}\times^{\mathbb{G}_{m},\check{\rho}}T$,
where for each line bundle $\mathcal{L}$ we denote by $\mathcal{L}^{\times}$
the $\mathbb{G}_{m}$-torsor corresponding to $\mathcal{L}$. Recall from [15],
Proposition 3.7.1, that there exists a canonical bijection

(33) $\displaystyle G\text{-}\mathrm{Conn}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{({\bf
x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))_{\mathrm{gen}},$

where the left-hand side denotes the set of $D_{({\bf x},{\bf z})}$-log
connections on $\mathcal{E}^{\dagger}_{T,\mathrm{log}}$ of monodromy
$(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime})$ (in the
sense of [13], Definition 1.6.1).

First, suppose that there exists a $D_{({\bf x},{\bf z})}$-log connection
$\overline{\nabla}$ classified by $G\text{-}\mathrm{Conn}(\mathbb{P}^{D_{({\bf
x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))$.
One may find a unique connection $\overline{\nabla}^{\prime}$ on
$\mathcal{E}^{\dagger}_{T}$ whose restriction to
$\mathbb{P}^{\circledcirc}:=\mathbb{P}\setminus\\{x_{1},\cdots,x_{r+1},z_{1},\cdots,z_{m}\\}$
is isomorphic to $\overline{\nabla}|_{\mathbb{P}^{\circledcirc}}$.
$\overline{\nabla}^{\prime}$ is verified to have monodromy
$(\check{\lambda}_{1}+\check{\rho},\cdots,\check{\lambda}_{r+1}+\check{\rho},\check{\lambda}^{\prime}_{1}+\check{\rho},\cdots,\check{\lambda}^{\prime}_{m}+\check{\rho})$,
and must be expressed as

(34)
$\displaystyle\partial_{x}+\sum_{i=1}^{r}\frac{\check{\lambda}_{i}+\check{\rho}}{x-x_{i}}+\sum_{j=1}^{m}\frac{\check{\lambda}^{\prime}_{j}+\check{\rho}}{x-z_{j}},$

where $\partial_{x}=\frac{d}{dx}$, via the trivialization of the $T$-torsor
$\mathcal{E}_{T}^{\dagger}|_{\mathbb{P}\setminus\\{\infty\\}}$ given by
$\Omega_{\mathbb{P}/k}|_{\mathbb{P}\setminus\\{\infty\\}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{\mathbb{P}\setminus\\{\infty\\}};dx\mapsto
1$ (cf. [1], § 3.1, (3.1)). In particular,
$G\text{-}\mathrm{Conn}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))$
consists of a single connection. Moreover, according to the discussion in loc.
cit., we have the equality

(35) $\displaystyle
2\check{\rho}-\sum_{i=1}^{r}(\check{\lambda}_{i}+\check{\rho})-\sum_{j=1}^{m}(\check{\lambda}^{\prime}_{j}+\check{\rho})=\check{\lambda}_{r+1}+\check{\rho},$

which is equivalent to the equality (32).

Conversely, if the equality (32) holds, then we obtain a unique connection
$\overline{\nabla}^{\prime}$ on $\mathcal{E}^{\dagger}_{T}$ of the form (34).
Moreover, there exists a unique $D_{({\bf x},{\bf z})}$-log connection
$\overline{\nabla}$ on $\mathcal{E}_{T,\mathrm{log}}^{\dagger}$ whose
restriction to $\mathbb{P}^{\circledcirc}$ is isomorphic to
$\overline{\nabla}^{\prime}|_{\mathbb{P}^{\circledcirc}}$. Since
$\overline{\nabla}$ is verified to belong to
$G\text{-}\mathrm{Conn}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))$,
the bijection (33) implies that $G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{({\bf
x},{\bf
z})}\text{-}\mathrm{log}};(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime}))_{\mathrm{gen}}$
is nonempty. This completes the proof of the assertion. ∎

Let $\boldsymbol{\alpha}:=(\alpha_{1},\cdots,\alpha_{m})\in\Gamma^{m}$. For
each $j=1,\cdots,m$, we shall write $\alpha^{W}_{j}\in W$ for the simple
reflection corresponding to $\alpha_{j}$. The coroot $\check{\alpha}_{j}$
corresponding to $\alpha_{j}$ satisfies the equality
$\check{\alpha}_{j}=\alpha^{W}_{j}(-\check{\rho})+\check{\rho}$. Write
$-\check{\boldsymbol{\lambda}}-\check{\rho}:=(-\check{\lambda}_{1}-\check{\rho},\cdots,-\check{\lambda}_{r+1}-\check{\rho})$
and $\boldsymbol{\alpha}^{W}:=(\alpha^{W}_{1},\cdots,\alpha^{W}_{m})$. Now,
suppose that the equality

(36)
$\displaystyle-\sum_{i=1}^{r+1}\check{\lambda}_{i}+\sum_{j=1}^{m}\check{\alpha}_{j}=2\check{\rho}$

(i.e., the equality (32) in the case where the data
$(\check{\boldsymbol{\lambda}},\check{\boldsymbol{\lambda}}^{\prime})$ is
taken to be
$(-\check{\boldsymbol{\lambda}}-\check{\rho},\boldsymbol{\alpha}^{W}(-\check{\rho}))$)
holds. The assignment from each ${\bf z}\in C^{m}({\bf x})$ to a unique
generic Miura $G$-oper on $\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}}$ of exponents
$(-\check{\boldsymbol{\lambda}}-\check{\rho},\boldsymbol{\alpha}^{W}(-\check{\rho}))$
defines a bijection

(37) $\displaystyle C^{m}({\bf
x})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};-\check{\boldsymbol{\lambda}}-\check{\rho})_{\mathrm{gen},+\boldsymbol{\alpha}^{W}},$

where the right-hand side contains $G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};-\check{\boldsymbol{\lambda}}-\check{\rho})_{\mathrm{triv},+\boldsymbol{\alpha}^{W}}$
(cf. (27)). Now, denote by

(38) $\displaystyle
G\text{-}\mathrm{BA}_{\check{\boldsymbol{\lambda}},\boldsymbol{\alpha}}\
\left(\subseteq C^{m}({\bf x})\right)$

the set of elements ${\bf z}:=(z_{1},\cdots,z_{m})$ of $C^{m}({\bf x})$
satisfying the equations

(39)
$\displaystyle\sum_{i=1}^{r}\frac{\langle\alpha_{j},\check{\lambda}_{i}\rangle}{z_{j}-x_{i}}-\sum_{s\neq
j}\frac{\langle\alpha_{j},\check{\alpha}_{s}\rangle}{z_{j}-z_{s}}=0\hskip
28.45274pt\left(j=1,\cdots,m\right)$

(called the Bethe ansatz equations).

######  Remark 2.2.2.

We shall consider the case where $G=\mathrm{PGL}_{2}$ and $p\geq 3$ (or
equivalently, the condition $(*)_{\mathrm{PGL}_{2}}$ is satisfied). Let us
keep the above notation. We have that
$\check{\rho}=\begin{pmatrix}\frac{1}{2}&0\\\ 0&-\frac{1}{2}\end{pmatrix}$,
$\Gamma=\\{\alpha\\}$, and $\check{\alpha}=2\check{\rho}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$. For each $i\in\\{1,\cdots r\\}$, let $q_{i}$ be the
element of $k$ with $\check{\lambda}_{i}=q_{i}\cdot\check{\rho}$. Then, (39)
read the equations

(40) $\displaystyle\sum_{i=1}^{r}\frac{q_{i}}{z_{j}-x_{i}}-\sum_{s\neq
j}\frac{2}{z_{j}-z_{s}}=0\hskip 28.45274pt\left(j=1,\cdots,m\right)$

of values in $k$.

A stronger assertion of the following theorem in the case where the base field
$k$ is replaced by $\mathbb{C}$ was proved in [1], § 3.2, Theorem 3.2. Also,
Lemma 2.2.4 described later, which will be used in the proof of the following
theorem, corresponds to [1], § 2.6, Lemma 2.10. But, we describe it in the
present paper in order to give its proof without any analytic argument unlike
the proof in loc. cit..

######  Theorem 2.2.3.

(Recall the assumption that equality (36) holds.) The bijection (37) restricts
to an injection

(41) $\displaystyle
G\text{-}\mathrm{BA}_{\check{\boldsymbol{\lambda}},\boldsymbol{\alpha}}\hookrightarrow
G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};-\check{\boldsymbol{\lambda}}-\check{\rho})_{\mathrm{triv},+\boldsymbol{\alpha}^{W}}.$

If, moreover, $G=\mathrm{PGL}_{2}$ (and $p\geq 3$), then this injection
becomes a bijection. In particular, there exists a canonical injection

(42) $\displaystyle\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};-\check{\boldsymbol{\lambda}}-\check{\rho})_{\mathrm{gen},+\boldsymbol{\alpha}^{W}}^{{}^{\mathrm{Zzz...}}}\hookrightarrow\mathrm{PGL}_{2}\text{-}\mathrm{BA}_{\check{\boldsymbol{\lambda}},\boldsymbol{\alpha}}$

(cf. Proposition 2.1.2).

###### Proof.

Let ${\bf z}:=(z_{1},\cdots,z_{m})$ be an element of $C^{m}({\bf x})$ and
denote by
$\widehat{\mathcal{E}}^{\spadesuit}:=(\mathcal{E},\nabla,\mathcal{E}_{B},\mathcal{E}^{\prime}_{B})$
the Miura $G$-oper corresponding to the unique element of
$G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};-\check{\boldsymbol{\lambda}}-\check{\rho},\boldsymbol{\alpha}^{W}(-\check{\rho}))_{\mathrm{gen}}$.
By considering the discussion in the proof of Proposition 2.2.1, we see that
$\nabla|_{\mathbb{P}\setminus\\{\infty\\}}$ may be expressed as

(43)
$\displaystyle\partial_{x}+p_{-1}-\sum_{i=1}^{r}\frac{\check{\lambda}_{i}}{x-x_{i}}+\sum_{j=1}^{m}\frac{\check{\alpha}_{j}}{x-z_{j}}$

under a suitable trivialization of
$\mathcal{E}|_{\mathbb{P}\setminus\\{\infty\\}}$. Let us fix
$j\in\\{1,\cdots,m\\}$. Also, consider the restriction of $\nabla$ to
$\mathbb{D}_{z_{j}}$, which may be expressed, after choosing a uniformizer $t$
at $z_{j}$, as

(44) $\displaystyle\partial_{t}+p_{-1}+\frac{\check{\alpha}_{j}}{t}+{\bf
u}_{j}(t)$

(where $\partial_{t}:=\frac{d}{dt}$) for some ${\bf
u}_{j}(t)\in\mathfrak{t}[[t]]$. One verifies that the equality
$\langle\alpha_{j},{\bf u}_{j}(0)\rangle=0$ holds if and only if ${\bf z}$
satisfies the $j$-th equation in (39). Thus, the assertion follows from the
following lemma. ∎

######  Lemma 2.2.4.

Let $\alpha\in\Gamma$, and denote by $\alpha^{W}$ the simple reflection
corresponding to $\alpha$. Choose a uniformizer $t$ of
$\widehat{\mathcal{O}}$. Also, let us take a $G$-oper
$\mathcal{E}^{\spadesuit}$ on $\mathbb{D}^{\mathrm{log}}$ of the form
$(\mathbb{D}\times G,\nabla,\mathbb{D}\times B)$, where
$\nabla=\partial_{t}+p_{-1}+\frac{\check{\alpha}}{t}+{\bf u}(t)$ (under the
identification
$k[[t]]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widehat{\mathcal{O}}$
determined by $t$) for some ${\bf u}(t)\in\mathfrak{t}[[t]]$. Then,
$\langle\alpha,{\bf u}(0)\rangle=0$ implies that $\mathcal{E}^{\spadesuit}$
belongs to $G\text{-}\mathrm{Op}(\mathbb{D})$ (cf. (15)). Moreover, the
inverse implication holds if $G=\mathrm{PGL}_{2}$.

###### Proof.

Suppose that $\langle\alpha,{\bf u}(0)\rangle=0$. Denote by $e_{\alpha}$ the
unique generator of $\mathfrak{g}^{\alpha}$ such that
$\\{f_{\alpha},2\check{\alpha},e_{\alpha}\\}$ forms an
$\mathfrak{s}\mathfrak{l}_{2}$-triple. Also, denote by
$\mathrm{exp}:\mathfrak{n}\rightarrow N$ the exponential map given by [13], §
1.4, (55). (We can construct this map because of the assumption $(*)_{G}$.) It
follows from [13], § 1.4, Corollary 1.4.2, that after the gauge transformation
by $\mathrm{exp}(\frac{1}{t}\cdot e_{\alpha})^{-1}$, $\nabla$ becomes the
connection $\nabla^{\prime}$ of the following from:

(45)
$\displaystyle\nabla^{\prime}:=\partial_{t}+\frac{d}{dt}\left(\frac{1}{t}\cdot
e_{\alpha}\right)+\sum_{s=0}^{\infty}\frac{1}{s!}\cdot\mathrm{ad}\left(-\frac{1}{t}\cdot
e_{\alpha}\right)^{s}\left(p_{-1}+\frac{\check{\alpha}}{t}+{\bf u}(t)\right).$

Observe that $\frac{d}{dt}(\frac{1}{t}\cdot e_{\alpha})=-\frac{1}{t^{2}}\cdot
e_{\alpha}$ and

(46) $\displaystyle\mathrm{ad}\left(-\frac{1}{t}\cdot
e_{\alpha}\right)\left(p_{-1}+\frac{\check{\alpha}}{t}+{\bf u}(t)\right)$
$\displaystyle=-\frac{1}{t}\cdot\check{\alpha}+\frac{2}{t^{2}}\cdot
e_{\alpha}-\frac{\langle\alpha,{\bf u}(t)\rangle}{t}\cdot e_{\alpha},$
$\displaystyle\mathrm{ad}\left(-\frac{1}{t}\cdot
e_{\alpha}\right)^{2}\left(p_{-1}+\frac{\check{\alpha}}{t}+{\bf u}(t)\right)$
$\displaystyle=-\frac{2}{t^{2}}\cdot e_{\alpha},$
$\displaystyle\mathrm{ad}\left(-\frac{1}{t}\cdot
e_{\alpha}\right)^{l}\left(p_{-1}+\frac{\check{\alpha}}{t}+{\bf u}(t)\right)$
$\displaystyle=0$

($l=3,4,\cdots$). Thus, we have the equality

(47) $\displaystyle\nabla^{\prime}=\partial_{t}+p_{-1}+{\bf
u}(t)-\frac{\langle\alpha,{\bf u}(t)\rangle}{t}\cdot e_{\alpha}.$

Since $\langle\alpha,{\bf u}(0)\rangle=0$, the triple $(\mathbb{D}\times
G,\nabla^{\prime},\mathbb{D}\times B)$, as well as $\mathcal{E}^{\spadesuit}$,
belongs to $G\text{-}\mathrm{Op}(\mathbb{D})$. This completes the proof of the
former assertion.

Next, we shall assume that $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$, where
$\nabla$ can be expressed as

(48) $\displaystyle\nabla=\partial_{t}+\begin{pmatrix}\frac{1}{t}+u(t)&0\\\
1&-\frac{1}{t}-u(t)\end{pmatrix}$

with $u(t):=\frac{1}{2}\cdot\langle\alpha,{\bf u}(t)\rangle\in k[[t]]$.
Suppose further that $\nabla$ belongs to $G\text{-}\mathrm{Op}(\mathbb{D})$.
Then, there exists an element $b\in B(k((t)))$ such that $\nabla$ becomes,
after the gauge transformation by $b$, the connection
$\nabla_{b}=\partial_{t}+\begin{pmatrix}v(t)&w(t)\\\ 1&-v(t)\end{pmatrix}$ for
some $v(t)$, $w(t)\in k[[t]]$. By an explicit computation, $b$ turns out to
belong to $N(k((t)))$, i.e., $b=\mathrm{exp}(\begin{pmatrix}0&-\beta(t)\\\
0&0\end{pmatrix})^{-1}$ for some $\beta(t)\in k((t))$. Hence, we have

(49) $\displaystyle\nabla_{b}:=$ $\displaystyle\
\partial_{t}+\frac{d}{dt}\left(\begin{pmatrix}0&-\beta(t)\\\
0&0\end{pmatrix}\right)+\sum_{s=0}^{\infty}\frac{1}{s!}\cdot\mathrm{ad}\left(\begin{pmatrix}0&\beta(t)\\\
0&0\end{pmatrix}\right)^{s}\left(\begin{pmatrix}\frac{1}{t}+u(t)&0\\\
1&-\frac{1}{t}-u(t)\end{pmatrix}\right)$ $\displaystyle=$ $\displaystyle\
\partial_{t}+\begin{pmatrix}\frac{1}{t}+u(t)+\beta(t)&-\beta(t)^{\prime}-2\cdot\beta(t)\cdot(\frac{1}{t}+u(t))-\beta(t)^{2}\\\
1&-\frac{1}{t}-u(t)-\beta(t)\end{pmatrix}.$

By considering the $(1,1)$-th element of the matrix form of $\nabla_{b}$,
$\beta(t)$ can be expressed as $\beta(t)=-\frac{1}{t}+\beta_{+}(t)$ for some
$\beta_{+}(t)\in k[[t]]$. Hence, the $(1,2)$-th element of $\nabla_{b}$ reads

(50)
$\displaystyle\beta(t)^{\prime}-2\cdot\beta(t)\cdot\left(\frac{1}{t}+u(t)\right)-\beta(t)^{2}=-\beta_{+}(t)^{\prime}-2\cdot\beta_{+}(t)\cdot
u(t)-\beta_{+}^{2}(t)+\frac{2}{t}\cdot u(t).$

Since it must belong to $k[[t]]$, we see that $\frac{2}{t}\cdot u(t)\in
k[[t]]$, or equivalently, $u(0)=0$. This completes the proof of the latter
assertion. ∎

######  Remark 2.2.5.

In [12], A. Varchenko studied the Bethe ansatz equations (for
$\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$) modulo $p$ and showed that the
Bethe vector corresponding to its solution is an eigenvector of the Gaudin
Hamiltonian. By this result together with Thereom 2.2.3, we can construct
eigenvectors by means of dormant generic Miura $\mathrm{PGL}_{2}$-opers. On
the other hand, as explained in § 3.1 later, dormant generic Miura $G$-opers
correspond bijectively to (pre-)Tango structures. Thus, we also can say that a
certain type of (pre-)Tango structures provides eigenvectors of the Gaudin
Hamiltonian.

###  2.3. The Bethe ansatz equations II

We show that the solutions of the Bethe ansatz equations of a certain type
come from dormant Miura $\mathrm{PGL}_{2}$-opers. Let us keep the notation in
the previous subsection. In what follows, for an element $v\in\mathfrak{g}$
and a positive integer $s$, we shall write

(51) $\displaystyle v^{s}:=(v,v,\cdots,v)\in\mathfrak{g}^{s}.$

######  Proposition 2.3.1.

The injection

(52) $\displaystyle G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};(-\check{\rho})^{r+1})^{{}^{\mathrm{Zzz...}}}_{\mathrm{gen},+\boldsymbol{\alpha}^{W}}\hookrightarrow
G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};(-\check{\rho})^{r+1})_{\mathrm{triv},+\boldsymbol{\alpha}^{W}}$

displayed in (42) of the case where the data
$(X,\check{\boldsymbol{\lambda}},\boldsymbol{w})$ is taken to be
$(\mathbb{P},(-\check{\rho})^{r+1},\boldsymbol{\alpha}^{W})$ is an
isomorphism.

###### Proof.

We shall prove that (52) is surjective. Suppose that we are given a Miura
$G$-oper $\widehat{\mathcal{E}}^{\spadesuit}$ classified by
$G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};(-\check{\rho})^{r+1})_{\mathrm{triv},+\boldsymbol{\alpha}^{W}}$,
i.e., by $G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};(-\check{\rho})^{r+1})_{\mathrm{triv},+({\bf
z};\boldsymbol{\alpha}^{W})}$ for some ${\bf z}:=(z_{1},\cdots,z_{m})\in
C^{m}({\bf x})$. According to the comment in Remark 1.4.2, it comes, via gauge
transformation, from a Miura $G$-oper on $\mathbb{P}^{D_{(\infty,{\bf
z})}\text{-}\mathrm{log}}$ of exponent
$(-\check{\rho},\boldsymbol{\alpha}^{W}(-\check{\rho}))$. Moreover, since the
underlying $G$-oper of $\widehat{\mathcal{E}}^{\spadesuit}$ belongs to
$G\text{-}\mathrm{Op}(\mathbb{P})$ $\left(\subseteq
G\text{-}\mathrm{Op}(\mathbb{P}^{D_{\boldsymbol{z}}\text{-}\mathrm{log}};\chi(-\check{\rho})^{m})\right)$,
its restriction to $\mathbb{D}_{z_{j}}$ (for each $j\in\\{1,\cdots,m-1\\}$)
belongs to $G\text{-}\mathrm{Op}(\mathbb{D}_{z_{j}})\ \left(\subseteq
G\text{-}\mathrm{Op}(\mathbb{D}_{z_{j}}^{\mathrm{log}};\chi(-\check{\rho}))\right)$.
Hence, this $G$-oper comes from a $G$-oper $\mathcal{E}^{\spadesuit}_{0}$ on
$\mathbb{P}^{D_{(\infty,z_{m})}}$ of exponent $\chi(-\check{\rho})^{2}$. By
[14], Corollary 2.6.2 and Theorem 3.2.2, (iv), $\mathcal{E}^{\spadesuit}_{0}$
turns out to be dormant. It follows that $\widehat{\mathcal{E}}^{\spadesuit}$
is dormant, and hence, belongs to $G\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};(-\check{\rho})^{r+1})^{{}^{\mathrm{Zzz...}}}_{\mathrm{gen},+\boldsymbol{\alpha}^{W}}$.
This completes the proof of the assertion. ∎

By combining the results of Theorem 2.2.3 and Proposition 2.3.1, we obtain
Corollary 2.3.2 described as follows.

######  Corollary 2.3.2.

Let $l$ be a nonnegative integer. Then, there exists a canonical bijection

(53)
$\displaystyle\mathrm{PGL}_{2}\text{-}\mathrm{BA}_{0^{r+1},\alpha^{lp+1}}$
$\displaystyle\hskip
8.53581pt\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{{\bf
x}}\text{-}\mathrm{log}};(-\check{\rho})^{r+1})^{{}^{\mathrm{Zzz...}}}_{\mathrm{gen},+\alpha^{lp+1}}$
$\displaystyle\left(=\coprod_{{\bf z}\in C^{lp+1}({\bf
x})}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}};(((-\check{\rho})^{r+1},\check{\rho}^{lp+1}))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}\right)$

That is to say, for each ${\bf z}:=(z_{1},\cdots,z_{lp+1})\in C^{lp+1}({\bf
x})$, there exists a dormant Miura $\mathrm{PGL}_{2}$-oper on
$\mathbb{P}^{D_{({\bf x},{\bf z})}\text{-}\mathrm{log}}$ of exponent
$((-\check{\rho})^{r+1},\check{\rho}^{lp+1})$ if and only if ${\bf z}$
satisfies the Bethe ansatz equations of the form

(54) $\displaystyle\sum_{s\neq j}\frac{1}{z_{j}-z_{s}}=0\hskip
14.22636pt\left(j=1,\cdots,lp+1\right).$

######  Remark 2.3.3.

We shall describe, in a somewhat simpler way, the condition $\sum_{s\neq
j}\frac{1}{z_{j}-z_{s}}=0$ appearing in the above corollary. Let us consider

(55) $\displaystyle f(x):=\prod_{j=1}^{lp+1}(x-z_{j})\in k[\tau].$

Since $z_{1},\cdots,z_{lp+1}$ are distinct, we have
$\mathrm{gcd}(f(x),f^{\prime}(x))=1$ (or equivalently, $f^{\prime}(z_{j})\neq
0$ for any $j\in\\{1,\cdots,lp+1\\}$). Also, one verifies immediately that,
for each $j\in\\{1,\cdots,lp+1\\}$,

(56) $\displaystyle\frac{f^{\prime\prime}(z_{j})}{f^{\prime}(z_{j})}$
$\displaystyle=\frac{\sum_{s^{\prime}=1}^{lp+1}\sum_{s^{\prime\prime}\neq
s^{\prime}}\prod_{s^{\prime\prime\prime}\neq
s^{\prime},s^{\prime\prime}}(x-z_{s^{\prime\prime\prime}})|_{x=z_{j}}}{\sum_{s^{\prime}=1}^{lp+1}\prod_{s^{\prime\prime}\neq
s^{\prime}}(x-z_{s^{\prime\prime}})|_{x=z_{j}}}$
$\displaystyle=\frac{2\cdot\sum_{s^{\prime}\neq s}\prod_{s^{\prime\prime}\neq
j,s^{\prime}}(z_{j}-z_{s^{\prime\prime}})}{\prod_{j\neq s}(z_{j}-z_{s})}$
$\displaystyle=2\cdot\sum_{s\neq j}\frac{1}{z_{j}-z_{s}}.$

On the other hand, (since $f(z_{j})=0$ for any $j\in\\{1,\cdots,lp+1\\}$) the
equality $\frac{f^{\prime\prime}(z_{j})}{f^{\prime}(z_{j})}=0$ holds for any
$j\in\\{1,\cdots,lp+1\\}$ if and only if $f^{\prime\prime}(x)=0$. Thus, we
obtain an equivalence of conditions

(57) $\displaystyle\sum_{s\neq j}\frac{1}{z_{j}-z_{s}}=0\hskip
8.53581pt(j\in\\{1,\cdots,lp+1\\})\ \ \Longleftrightarrow\ \
f^{\prime\prime}(x)=0.$

For instance, if $(a,b,c)$ is an element of $k^{3}$ with $ab-c\neq 0$ and
$z_{1},\cdots,z_{p+1}$ are the roots of the polynomial
$f(x):=x^{p+1}+ax^{p}+bx+c$, then the equalities $\sum_{s\neq
j}\frac{1}{z_{j}-z_{s}}$ ($j=1,\cdots,p+1$) are satisfied.

##  3\. Tango structures

In this section, we recall the notion of a Tango structure and consider a
bijective correspondence between Tango structures and solutions to the Bethe
ansatz equations for $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$. Also, by
means of this correspondence, we construct new examples of Tango structures.
In what follows, we suppose that $p\geq 3$.

###  3.1. Tango structures

First, we recall the definition of a (pre-)Tango structure and the bijective
correspondence between dormant Miura $\mathrm{PGL}_{2}$-opers and them.

Let $X$ be a connected proper smooth curve over $k$ of genus $g_{X}\ (\geq
0)$. Denote by $X^{(1)}$ the Frobenius twist of $X$ relative to $k$ and by
$F:X\rightarrow X^{(1)}$ the relative Frobenius morphism of $X$. Also, denote
by $\mathcal{B}_{X/k}\ (\subseteq\Omega_{X/k})$ the sheaf of locally exact
$1$-forms on $X$ (relative to $k$). The direct image $F_{*}(\Omega_{X/k})$
forms a vector bundle on $X^{(1)}$ of rank $p$. $\mathcal{B}_{X/k}$ may be
considered, via the underlying homeomorphism of $F$, as a subbundle of
$F_{*}(\Omega_{X/k})$ of rank $p-1$.

Now, let $\mathcal{L}$ be a line subbundle of $F_{*}(\mathcal{B}_{X/k})$.
Consider the $\mathcal{O}_{X^{(1)}}$-linear composite

(58) $\displaystyle\mathcal{L}\hookrightarrow
F_{*}(\mathcal{B}_{X/k})\hookrightarrow F_{*}(\Omega_{X/k}),$

where the first arrow denotes the natural inclusion and the second arrow
denotes the morphism obtained by applying the functor $F_{*}(-)$ to the
natural inclusion $\mathcal{B}_{X/k}\rightarrow\Omega_{X/k}$. This composite
corresponds to a morphism

(59)
$\displaystyle\xi_{\mathcal{L}}:F^{*}(\mathcal{L})\rightarrow\Omega_{X/k}$

via the adjunction relation “$F^{*}(-)\dashv F_{*}(-)$”.

######  Definition 3.1.1.

  * (i)

We shall say that $\mathcal{L}$ is a Tango structure on $X$ if
$\xi_{\mathcal{L}}$ is an isomorphism.

  * (ii)

A Tango curve is a connected proper smooth curve over $k$ admitting a Tango
structure on it.

######  Remark 3.1.2.

Let $\mathcal{L}$ be a line subbundle of $\mathcal{B}_{X/k}$. Then, one
verifies immediately that $\mathcal{L}$ is a Tango structure on $X$ if and
only if $\mathcal{L}$ has degree $\frac{2g_{X}-2}{p}$. In particular, if
$p\nmid g_{X}-1$, then there is no Tango structure on $X$. At the time of
writing the present paper, the author does not obtain the complete list of
$g$’s such that there is a Tango curve of genus $g$.

Next, we shall recall the definition of a pre-Tango structure in the sense of
[15], Definition 5.3.1. Let $r$ be a nonnegative integer and ${\bf
x}:=(x_{1},\cdots,x_{r})$ an ordered collection of distinct closed points of
$X$. (We take ${\bf x}:=\emptyset$ if $r=0$.) If there is no fear of
confusion, we shall write $X^{\mathrm{log}}$ for $X^{D_{{\bf
x}}\text{-}\mathrm{log}}$. The data ${\bf x}$ induces, via base-change, a
collection of closed points ${\bf x}^{(1)}:=(x_{1}^{(1)},\cdots,x_{r}^{(1)})$
in $X^{(1)}$, which determines a log structure on $X^{(1)}$; denote by
$X^{(1)\mathrm{log}}$ the resulting log scheme. Moreover, let

(60) $\displaystyle
C_{X^{\mathrm{log}}/k}:F_{*}(\Omega_{X^{\mathrm{log}}/k})\rightarrow\Omega_{X^{(1)\mathrm{log}}/k}$

be the Cartier operator of $X^{\mathrm{log}}/k$. That is to say,
$C_{X^{\mathrm{log}}/k}$ is a unique $\mathcal{O}_{X^{(1)}}$-linear morphism
whose composite with the injection
$\Omega_{X^{(1)\mathrm{log}}/k}\rightarrow\Omega_{X^{(1)\mathrm{log}}/k}\otimes
F_{*}(\mathcal{O}_{X})$ induced by the natural injection
$\mathcal{O}_{X^{(1)}}\rightarrow F_{*}(\mathcal{O}_{X})$ coincides with the
Cartier operator associated with $(\mathcal{O}_{X},d)$ in the sense of [8],
Proposition 1.2.4.

######  Definition 3.1.3.

A pre-Tango structure on $(X,{\bf x})$ is a $D_{{\bf x}}$-log connection
$\nabla$ on $\Omega_{X^{\mathrm{log}}/k}$ with vanishing $p$-curvature
satisfying that
$\mathrm{Ker}(\nabla)\subseteq\mathrm{Ker}(C_{X^{\mathrm{log}}/k})$. (Here, we
consider $\mathrm{Ker}(\nabla)$ as an $\mathcal{O}_{X^{(1)}}$-submodule of
$F_{*}(\Omega_{X^{\mathrm{log}}/k})$ via $F$.)

######  Remark 3.1.4.

According to [15], Proposition 5.3.2, the set of pre-Tango structures on
$(X,\emptyset)$ is in bijection with the set of Tango structures on $X$.

Indeed, let $\nabla$ be a pre-Tango structure on $(X,\emptyset)$. The
$\mathcal{O}_{X^{(1)}}$-module $\mathrm{Ker}(\nabla)$ is contained in
$F_{*}(\mathcal{B}_{X/k})\ \left(=F_{*}(\mathrm{Ker}(C_{X/k}))\right)$. Since
$\nabla$ gives an injection from $F_{*}(\Omega_{X/k})/\mathrm{Ker}(\nabla)$ to
$F_{*}(\Omega_{X/k}^{\otimes 2})$ (which implies that
$F_{*}(\Omega_{X/k})/\mathrm{Ker}(\nabla)$ is a vector bundle), the quotient
$F_{*}(\mathcal{B}_{X/k})/\mathrm{Ker}(\nabla)$ $\left(\subseteq
F_{*}(\Omega_{X/k})/\mathrm{Ker}(\nabla)\right)$ is verified to be a vector
bundle. Namely, $\mathrm{Ker}(\nabla)$ specifies a subbundle of
$F_{*}(\mathcal{B}_{X/k})$. Moreover, the condition that $\nabla$ has
vanishing $p$-curvature implies that $\xi_{\mathrm{Ker}(\nabla)}$ is an
isomorphism. Hence, $\mathrm{Ker}(\nabla)$ specifies a Tango structure.

Conversely, let $\mathcal{L}$ be a Tango structure on $X$. The line bundle
$F^{*}(\mathcal{L})$ has uniquely a connection determined by the condition
that the sections in $F^{-1}(\mathcal{L})$ are horizontal. The connection
$\nabla_{\mathcal{L}}$ corresponding, via $\xi_{\mathcal{L}}$, to this
connection specifies a pre-Tango structure on $(X,\emptyset)$ (because of the
equality $F_{*}(\mathcal{B}_{X/k})=F_{*}(\mathrm{Ker}(C_{X/k}))$). One
verifies that the assignments $\nabla\mapsto\mathrm{Ker}(\nabla)$ and
$\mathcal{L}\mapsto\nabla_{\mathcal{L}}$ constructed just above give the
desired bijective correspondence.

Let us fix an ordered collection
$\boldsymbol{l}:=(l_{1},\cdots,l_{r})\in\mathbb{F}_{p}^{r}$. (We take
$\boldsymbol{l}:=\emptyset$ if $r=0$.) Denote by

(61) $\displaystyle\mathrm{Tang}(X;{\bf x};\boldsymbol{l})$

the set of pre-Tango structures on $(X,{\bf x})$ of monodromy
$\boldsymbol{l}$, i.e., whose monodromy at $x_{i}$ (for each
$i\in\\{1,\cdots,r\\}$) is $l_{i}$ .

######  Proposition 3.1.5.

  * (i)

Let $m$ be a positive integer and ${\bf z}:=(z_{1},\cdots,z_{m})\in C^{m}({\bf
x})$. Then, there exists a canonical bijection

(62) $\displaystyle\mathrm{Tang}(X;({\bf x},{\bf
z});(\boldsymbol{l},(-1)^{m}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Tang}(X;{\bf
x};\boldsymbol{l}).$

  * (ii)

There exists a bijective correspondence between the set $\mathrm{Tang}(X;{\bf
x};(-1)^{r})$ and the set of Tango structures on $X$.

###### Proof.

We shall consider assertion (i). Let us take a pre-Tango structure $\nabla$
classified by $\mathrm{Tang}(X;({\bf x},{\bf z});(\boldsymbol{l},(-1)^{m}))$.
Denote by $\nabla_{-{\bf z}}$ the $D_{({\bf x},{\bf z})}$-log connection on
$\Omega_{X^{D_{{\bf x}}\text{-}\mathrm{log}}/k}$ obtained by restricting
$\nabla$ via the inclusion $\Omega_{X^{D_{{\bf
x}}\text{-}\mathrm{log}}/k}\hookrightarrow\Omega_{X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}}/k}$. One verifies that the monodromy of
$\nabla_{-{\bf z}}$ at $z_{j}$ (for each $j\in\\{1,\cdots,m\\}$) coincides
with $0$. Hence, $\nabla_{-{\bf z}}$ may be thought of as a $D_{{\bf x}}$-log
connection. Moreover, since the equality

(63) $\displaystyle\mathrm{Ker}(C_{X^{D_{{\bf
x}}\text{-}\mathrm{log}}/k})=\mathrm{Ker}(C_{X^{D_{({\bf x},{\bf
z})}\text{-}\mathrm{log}}/k})$

holds, $\nabla_{-{\bf z}}$ specifies a pre-Tango structure classified by
$\mathrm{Tang}(X;{\bf x};\boldsymbol{l})$.

Conversely, let us take a pre-Tango structure $\nabla$ classified by
$\mathrm{Tang}(X;{\bf x};\boldsymbol{l})$; it may be thought of as a $D_{({\bf
x},{\bf z})}$-log connection whose monodromy at $z_{j}$ (for each
$j\in\\{1,\cdots,m\\}$) is $0$. Then, there exists uniquely a $D_{({\bf
x},{\bf z})}$-log connection $\nabla_{+{\bf z}}$ on $\Omega_{X^{D_{({\bf
x},{\bf z})}\text{-}\mathrm{log}}/k}$ whose restriction to $\Omega_{X^{D_{{\bf
x}}\text{-}\mathrm{log}}/k}$ coincides with $\nabla$. One verifies (from (63)
again) that $\nabla_{+{\bf z}}$ specifies a pre-Tango structure classified by
$\mathrm{Tang}(X;({\bf x},{\bf z});(\boldsymbol{l},(-1)^{m}))$.

The assignments $\nabla\mapsto\nabla_{-{\bf z}}$ and
$\nabla\mapsto\nabla_{+{\bf z}}$ constructed above give the desired bijective
correspondence. This completes the proof of assertion (i).

Assertion (ii) follows from assertion (i) of the case where $m=0$ together
with the result mentioned in Remark 3.1.4. ∎

###  3.2. Proof of Theorem A

Now, we shall prove Theorem A, as follows. Recall from [15], Theorem A, that
if $\boldsymbol{l}:=(l_{1},\cdots,l_{r})\in(\mathbb{F}_{p}^{\times})^{r}$,
then there exists a canonical bijection

(64) $\displaystyle\mathrm{Tang}(X;{\bf
x};\boldsymbol{l})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(X^{D_{{\bf
x}}\text{-}\mathrm{log}};\boldsymbol{l}\check{\rho})^{{}^{\mathrm{Zzz...}}}_{\mathrm{gen}},$

where $\check{\rho}=\begin{pmatrix}\frac{1}{2}&0\\\
0&-\frac{1}{2}\end{pmatrix}$ and
$\boldsymbol{l}\check{\rho}:=(l_{1}\check{\rho},\cdots,l_{r}\check{\rho})\in(\mathfrak{t}_{\mathrm{reg}})^{r}$.
In particular, by considering the case where $(X,{\bf x},\boldsymbol{l})$ is
taken to be $(\mathbb{P},(\infty,{\bf z}),(-1,1^{lp+1}))$ (for each ${\bf
z}\in C^{lp+1}(\infty)$), we obtain a bijection

(65) $\displaystyle\mathrm{Tang}(\mathbb{P};(\infty,{\bf
z});(-1,1^{lp+1}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{(\infty,{\bf
z})}\text{-}\mathrm{log}};(-\check{\rho},\check{\rho}^{lp+1}))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}.$

Also, it follows from Proposition 3.1.5 (of the case where
$(X,m,\boldsymbol{l})$ is taken to be $(\mathbb{P},1,1^{lp+1})$) that there
exists a canonical bijection

(66) $\displaystyle\mathrm{Tang}(\mathbb{P};(\infty,{\bf
z});(-1,1^{lp+1}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{Tang}(\mathbb{P};{\bf
z};1^{lp+1}).$

The composition of the inverse of (66) and (65) determines a bijection

(67) $\displaystyle\mathrm{Tang}(\mathbb{P};{\bf
z};1^{lp+1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{(\infty,{\bf
z})}\text{-}\mathrm{log}};(-\check{\rho},\check{\rho}^{lp+1}))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}$

(for each ${\bf z}\in C^{lp+1}(\infty)$). Thus, we have

(68) $\displaystyle\coprod_{{\bf z}\in
C^{lp+1}(\infty)}\mathrm{Tang}(\mathbb{P};{\bf z};1^{lp+1})$
$\displaystyle\hskip
8.53581pt\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{\infty}\text{-}\mathrm{log}};-\check{\rho})_{\mathrm{gen},+\alpha^{lp+1}}^{{}^{\mathrm{Zzz...}}}$
$\displaystyle\left(=\coprod_{{\bf z}\in
C^{lp+1}(\infty)}\mathrm{PGL}_{2}\text{-}\mathrm{MOp}(\mathbb{P}^{D_{(\infty,{\bf
z})}\text{-}\mathrm{log}};(-\check{\rho},\check{\rho}^{lp+1}))_{\mathrm{gen}}^{{}^{\mathrm{Zzz...}}}\right).$

The desired bijection (2) may be obtained from this bijection together with
Corollary 2.3.2 of the case where $r=0$. This completes the proof of Theorem
A.

###  3.3. Pull-back via tamely ramified coverings

In what follows, we proceed to our discussion toward Theorem B. First, let us
consider the pull-back of a pre-Tango structure via a tamely ramified
covering. Suppose that we are given a collection of data

(69) $\displaystyle(Y,{\bf y},\pi),$

where

  * $\bullet$

$Y$ denotes another connected proper smooth curve over $k$;

  * $\bullet$

${\bf y}:=(y_{1},\cdots,y_{s})$ ($s>1$) denotes an ordered collection of
distinct closed points of $Y$;

  * $\bullet$

$\pi:Y\rightarrow X$ denotes a tamely ramified covering such that
$\pi^{-1}(\bigcup_{i=1}^{r}\\{x_{i}\\})=\bigcup_{j=1}^{s}\\{y_{j}\\}$ and
$\pi$ is étale away from $\bigcup_{i=1}^{r}\\{x_{i}\\}$.

Denote by $q:\\{1,\cdots,s\\}\rightarrow\\{1,\cdots,r\\}$ the map of sets
determined by $\pi(y_{j})=x_{q(j)}$ ($j=1,\cdots,s$). For each
$j\in\\{1,\cdots,s\\}$, denote by $R_{j}$ the ramification index of $\pi$ at
$y_{j}$ (hence $p\nmid R_{j}$). The morphism $\pi$ extends to a log étale
morphism $\pi^{\mathrm{log}}:Y^{D_{{\bf y}}\text{-}\mathrm{log}}\rightarrow
X^{D_{{\bf x}}\text{-}\mathrm{log}}$, and hence, the natural morphism
$\pi^{*}(\Omega_{X^{D_{{\bf
x}}\text{-}\mathrm{log}}/k})\rightarrow\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}$ is an isomorphism.

Now, let $\nabla$ be an element of $\mathrm{Tang}(X;{\bf x};\boldsymbol{l})$.
The pull-back of $\nabla$ via $\pi$ determines, under the isomorphism
$\pi^{*}(\Omega_{X^{D_{{\bf
x}}\text{-}\mathrm{log}}/k})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}$, a $D_{{\bf y}}$-log connection

(70) $\displaystyle\pi^{*}(\nabla):\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}\rightarrow\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}^{\otimes 2}$

on $\Omega_{Y^{D_{{\bf y}}\text{-}\mathrm{log}}/k}$. Here, observe that both
the relative Frobenius morphisms and the Cartier operators of
$X\setminus\bigcup_{i=1}^{r}\\{x_{i}\\}$ and
$Y\setminus\bigcup_{i=1}^{s}\\{y_{i}\\}$ are compatible (in the evident sense)
via $\pi$. This implies that
$\mathrm{Ker}(\pi^{*}(\nabla))\subseteq\mathrm{Ker}(C_{Y^{D_{\boldsymbol{y}}\text{-}\mathrm{log}}/k})$,
and hence, $\pi^{*}(\nabla)$ specifies a pre-Tango structure on $(Y,{\bf y})$.

######  Proposition 3.3.1.

Write
$\boldsymbol{l}^{\prime}:=(l_{q(1)}R_{1},\cdots,l_{q(s)}R_{s})\in\mathbb{F}_{p}^{s}$.
Then, the assignment $\nabla\mapsto\pi^{*}(\nabla)$ discussed above defines an
injection

(71) $\displaystyle\mathrm{Tang}(X;{\bf
x};\boldsymbol{l})\hookrightarrow\mathrm{Tang}(Y;{\bf
y};\boldsymbol{l}^{\prime}).$

###### Proof.

Since the injectivity may be verified immediately, it suffices to prove that
the monodromy of the pre-Tango structure $\pi^{*}(\nabla)$ at $y_{j}$ (where
$\nabla\in\mathrm{Tang}(X;{\bf x};\boldsymbol{l})$, $j\in\\{1,\cdots,s\\}$) is
$l_{q(j)}R_{j}$. For simplicity, we write $x:=x_{q(j)}$, $y:=y_{j}$,
$R:=R_{j}$, and $l:=l_{q(j)}$. The formally local description of $\pi$ at $y$
may be given by
$\widehat{\mathbb{D}}_{y}:=\mathrm{Spec}(k[[t^{1/R}]])\rightarrow\mathrm{Spec}(k[[t]]):=\widehat{\mathbb{D}}_{x}$
(corresponding to the natural inclusion $k[[t]]\hookrightarrow k[[t^{1/R}]]$),
where $t$ is a formal coordinate in $X$ at $x$. The restriction
$\nabla|_{\widehat{\mathbb{D}}_{x}}$ of $\nabla$ to $\widehat{\mathbb{D}}_{x}$
may be expressed (via a suitable trivialization $\Omega_{X^{D_{{\bf
x}}\text{-}\mathrm{log}}/k}|_{\widehat{\mathbb{D}}_{x}}\cong\mathcal{O}_{\widehat{\mathbb{D}}_{x}}$)
as $\nabla=d+l\cdot\frac{dt}{t}$ (cf. [9], Corollary 2.10). Since

(72)
$\displaystyle\frac{dt}{t}=\frac{d(t^{1/R})^{R}}{(t^{1/R})^{R}}=\frac{R\cdot(t^{1/R})^{R-1}\cdot
dt^{1/R}}{(t^{1/R})^{R}}=R\cdot\frac{dt^{1/R}}{t^{1/R}},$

the restriction $\pi^{*}(\nabla)|_{\widehat{\mathbb{D}}_{y}}$ of
$\pi^{*}(\nabla)$ to $\widehat{\mathbb{D}}_{y}$ may be expressed as
$\pi^{*}(\nabla)|_{\widehat{\mathbb{D}}_{y}}=d+lR\cdot\frac{dt^{1/R}}{t^{1/R}}$.
This implies that the monodromy of $\pi^{*}(\nabla)$ at $y$ is $lR$, and
hence, completes the proof of the assertion. ∎

######  Example 3.3.2.

We shall consider the case where $X=\mathbb{P}$. Let
$\boldsymbol{l}:=(l_{1},\cdots,l_{r+1})$ ($r\geq 0$) be an element of
$\mathbb{F}^{r+1}_{p}$ and ${\bf x}:=(x_{1},\cdots,x_{r+1})$ an ordered
collection of distinct $r+1$ closed points of $\mathbb{P}$ with
$x_{r+1}:=\infty$. Let us consider the desingularization $Y$ of the plane
curve defined by

(73) $\displaystyle y^{n}=a\cdot\prod_{i=1}^{r}(x-x_{i}),$

where $a\in k^{\times}$, $p\nmid n$, and $(x,y)$ are an inhomogeneous
coordinate of the projective plane $\mathbb{P}^{2}$. Denote by
$\pi:Y\rightarrow\mathbb{P}$ the projection given by $(x,y)\mapsto x$. Given
$i\in\\{1,\cdots,r\\}$, we shall write $y_{i}$ for the unique point of $Y$
over $x_{i}$, i.e., $\pi(y_{i})=x_{i}$. (The ramification index of $\pi$ at
$y_{i}$ for each $i\in\\{1,\cdots,r\\}$ is $n$.) Also, let
$y_{r+1},\cdots,y_{r+r^{\prime}}$ ($r^{\prime}\geq 1$) be the set of points of
$Y$ over $x_{r+1}\ (=\infty)$. Write ${\bf
y}:=(y_{1},\cdots,y_{r+r^{\prime}})$. According to [11], § 6.3, Proposition
6.3.1, the ramification index of $\pi$ at $y_{r+i^{\prime}}$ (for each
$i^{\prime}\in\\{1,\cdots,r^{\prime}\\}$) is $\frac{n}{d}$, where
$d:=\mathrm{gcd}(n,r)$. Moreover, the genus of $Y$ is given by
$\frac{(n-1)(r-1)}{2}-\frac{\mathrm{gcd}(n,r)-1}{2}$.

Now, let us fix
$\boldsymbol{l}:=(l_{1},\cdots,l_{r+1})\in\mathbb{F}_{p}^{r+1}$ and write

(74)
$\displaystyle\boldsymbol{l}^{\prime}:=(l_{1}n,l_{2}n,\cdots,l_{r}n,\frac{l_{r+1}n}{d},\cdots,\frac{l_{r+1}n}{d})\in\mathbb{F}_{p}^{r+r^{\prime}}$

(where each of the last $r^{\prime}$ factors is $\frac{l_{r+1}n}{d}$). Then,
it follows from Proposition 3.3.1 that the assignment
$\nabla\mapsto\pi^{*}(\nabla)$ define an injection

(75) $\displaystyle\mathrm{Tang}(\mathbb{P};{\bf
x};\boldsymbol{l})\hookrightarrow\mathrm{Tang}(Y;{\bf
y};\boldsymbol{l}^{\prime}).$

In particular, $\mathrm{Tang}(Y;{\bf y};\boldsymbol{l}^{\prime})$ is nonempty
unless $\mathrm{Tang}(\mathbb{P};{\bf x};\boldsymbol{l})$ is empty.

###  3.4. Examples of Tango structures

In this subsection, we construct examples of Tango structures by applying
Theorem A proved above.

Let $a$, $b$ be positive integers with $\mathrm{gcd}(a,bp-1)=1$ and $h(x)\
\left(\in k[x]\right)$ a monic polynomial of degree $ap$ with
$\mathrm{gcd}(h(x),h^{\prime}(x))=1$. Let

(76) $\displaystyle Y$

be the smooth curve defined by the equation

(77) $\displaystyle y^{bp-1}=h(x).$

(If $a>b$, then the point at infinity is singular, and hence, we need to
replace this plane curve by its desingularization to obtain a smooth curve.)
Since $\mathrm{gcd}(ap,bp-1)=\mathrm{gcd}(a,bp-1)=1$, there is only one point
$\infty_{Y}$ at infinity in $Y$ (cf. [11], § 6.3, Proposition 6.3.1). Here,
let us take an ordered collection ${\bf z}:=(z_{1},\cdots,z_{ap+1})$ of
elements of $k$ (i.e., closed points in
$\mathrm{Spec}(k[x])=\mathbb{P}\setminus\\{\infty\\}$) with $z_{ap+1}=\infty$
satisfying the equality $h(x)=\prod_{i=1}^{ap}(x-z_{i})$. (The assumption
$\mathrm{gcd}(h(x),h^{\prime}(x))=1$ implies that $z_{i}\neq z_{i^{\prime}}$
if $i\neq i^{\prime}$.) Denote by $\pi:Y\rightarrow\mathbb{P}$ the natural
projection $(x,y)\mapsto x$, which is tamely ramified. Also, denote by ${\bf
y}:=(y_{1},\cdots,y_{ap+1})$ the ordered collection of distinct points in $Y$
determined uniquely by $\pi(y_{i})=z_{i}$ for any $i=1,\cdots,ap+1$. (In
particular, $y_{ap+1}=\infty_{Y}$.) Moreover, denote by $\nabla$ the $D_{{\bf
z}}$-log connection on $\Omega_{\mathbb{P}^{D_{{\bf
z}}\text{-}\mathrm{log}}/k}$ defined as

(78) $\displaystyle\nabla=\partial_{x}+\sum_{i=1}^{ap}\frac{1}{x-z_{i}}$

(under the identification $\Omega_{\mathbb{P}^{D_{{\bf
z}}\text{-}\mathrm{log}}/k}|_{\mathbb{P}\setminus\\{\infty\\}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{O}_{\mathbb{P}\setminus\\{\infty\\}}$
given by $dx\mapsto 1$). By passing to the isomorphism
$\pi^{*}(\Omega_{\mathbb{P}^{D_{{\bf
z}}\text{-}\mathrm{log}}/k})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}$ induced by $\pi$, we obtain a $D_{{\bf y}}$-log
connection

(79) $\displaystyle\pi^{*}(\nabla):\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}\rightarrow\Omega_{Y^{D_{{\bf
y}}\text{-}\mathrm{log}}/k}^{\otimes 2}$

on $\Omega_{Y^{D_{{\bf y}}\text{-}\mathrm{log}}/k}$ defined to be the pull-
back of $\nabla$. In particular, we obtain an
$\mathcal{O}_{Y^{(1)}}$-submodule

(80) $\displaystyle\mathrm{Ker}(\pi^{*}(\nabla))$

of $F_{*}(\Omega_{Y^{D_{{\bf y}}\text{-}\mathrm{log}}/k})$, where
$F:Y\rightarrow Y^{(1)}$ denotes the relative Frobenius morphism of $Y$
(relative to $k$). Then, the following assertion, which implies Theorem B,
holds.

######  Theorem 3.4.1 (cf. Theorem B).

Suppose that $h^{\prime\prime}(x)=0$. Then, $\mathrm{Ker}(\pi^{*}(\nabla))$
forms a Tango structure on $Y$. In particular, $Y$ is a Tango curve.

###### Proof.

Let us fix an element $\gamma\in k$ with $h(\gamma)\neq 0$. Consider the
automorphism $\iota$ of $\mathbb{P}$ given by $x\mapsto\frac{1}{x-\gamma}$.
Then, $\iota({\bf z}):=(\iota(z_{1}),\cdots,\iota(z_{ap+1})))$ are a
collection of distinct points in $\mathbb{P}\setminus\\{\infty\\}$. Since
$f(x):=\prod_{i=1}^{ap+1}(x-\iota(z_{i}))$ coincides with $h(\gamma)^{-1}\cdot
h(\frac{1}{x}+\gamma)\cdot x^{ap+1}$, we see that
$\mathrm{gcd}(f(x),f^{\prime}(x))=1$ and $f^{\prime\prime}(x)=0$. It follows
from Theorem A that the $D_{\iota({\bf z})}$-log connection on
$\Omega_{\mathbb{P}^{D_{\iota({\bf z})}\text{-}\mathrm{log}}/k}$ given by
$\partial_{x}+\sum_{i=1}^{ap+1}\frac{1}{x-\iota(z_{i})}$ forms a pre-Tango
structure on $(\mathbb{P},\iota({\bf z}))$ of monodromy $1^{ap+1}$. By
pulling-back via $\iota$, we obtain a pre-Tango structure on $(\mathbb{P},{\bf
z})$ of monodromy $1^{ap+1}$. One verifies immediately that it coincides with
$\nabla$. Hence, (since $bp-1=-1$ mod $p$) the pull-back $\pi^{*}(\nabla)$
forms a pre-Tango structure on $(Z,\boldsymbol{z})$ of monodromy $(-1)^{ap+1}$
(cf. Proposition 3.3.1). It follows from Proposition 3.1.5 (and its proof)
that $\mathrm{Ker}(\pi^{*}(\nabla))$ specifies a Tango structure, as desired.
∎

Since the curve $Y$ constructed above has genus $g=\frac{(ap-1)(bp-2)}{2}$, it
follows from Theorem 3.4.1 and [15], Theorem B, that the following assertion
holds .

######  Corollary 3.4.2.

Let $a$, $b$ be positive integers with $\mathrm{gcd}(a,bp-1)=1$, and write
$g:=\frac{(ap-1)(bp-2)}{2}$. Denote by
$\mathfrak{T}\mathfrak{a}\mathfrak{n}_{g}$ the moduli stack classifying pairs
$(X,\mathcal{L})$ consisting of a connected proper smooth curve $X$ over $k$
of genus $g$ and a Tango structure $\mathcal{L}$ on $X$. Then,
$\mathfrak{T}\mathfrak{a}\mathfrak{n}_{g}$ may be represented by a nonempty
equidimensional smooth Deligne-Mumford stack over $k$ of dimension
$2g-2+\frac{2g-2}{p}\ \left(=abp^{2}+(ab-2a-b)p-2a-b\right)$.

######  Remark 3.4.3.

A well-known example of a Tango structures can be found in some paper (cf.
e.g., [10], Example, or [6], Example 1.3). This is constructed as follows. Let
$l$ be an integer with $lp\geq 4$. Also, let $f(x)$ be a polynomial of degree
$l$ in one variable $x$ and let $Y$ be the plane curve defined by

(81) $\displaystyle y^{lp-1}=f(x^{p})-x.$

One verifies that $Y$ is a smooth curve having only one point $\infty$ at
infinity and $\Omega_{Y/k}=\mathcal{O}_{Y}(lp(lp-3)\cdot\infty)$. Then, the
base-change of the line bundle $\mathcal{O}(l(lp-3)\cdot\infty)$ via the
absolute Frobenius morphism of $\mathrm{Spec}(k)$ turns out to specify a Tango
structure.

Notice that $(f(x^{p})-x)^{\prime}=-1$ (which implies that
$\mathrm{gcd}(f(x^{p})-x,(f(x^{p})-x)^{\prime})=1$) and
$(f(x^{p})-x)^{\prime\prime}=0$. Hence, $Y$ is a special case of Tango curves
constructed preceding Theorem 3.4.1. On the other hand, the curve defined,
e.g., by the equation $y^{2p-1}=x^{2p}+x^{p+1}+ax^{p}+bx+c$ with $c\neq
b(a-b)$ is a Tango curve (by Theorem 3.4.1), which specifies a new example.
Thus, one can obtain infinitely many new (explicit!) examples of Tango curves.

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