Arithmetic liftings and $2$d TQFT
for dormant opers of higher level

Let us consider the linear differential equation $Dy=0$ on a smooth complex algebraic curve $X$ associated to an operator $D$ expressed locally as

$\displaystyle D:=\frac{d^{n}}{dx^{n}}+q_{1}\frac{d^{n-1}}{dx^{n-1}}+\cdots+q_{n-1}\frac{d}{dx}+q_{n}$

(1.1)

($n>1$). Here, $x$ denotes a local coordinate in $X$ and $q_{1},\cdots,q_{n}$ are variable coefficients. To each such differential equation, one can associate a flat connection on a vector bundle whose matrix representation is locally given by

$\displaystyle\nabla=\frac{d}{dx}-\begin{pmatrix}-q_{1}&-q_{2}&-q_{3}&\cdots&-q_{n-1}&-q_{n}\\ 1&0&0&\cdots&0&0\\ 0&1&0&\cdots&0&0\\ 0&0&1&\cdots&0&0\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&0&\cdots&1&0\end{pmatrix}.$

(1.2)

A vector bundle equipped with such a flat connection is known as a $\mathrm{GL}_{n}$-oper. Conversely, any flat bundle (i.e., a vector bundle equipped with a flat connection) defining a $\mathrm{GL}_{n}$-oper becomes a flat connection of that form after possibly carrying out a suitable gauge transformation. See  [BD1] or  [BD2] for the precise definition of a $\mathrm{GL}_{n}$-oper, or more generally a $G$-oper for an algebraic group $G$ of a certain sort.

Note that the assignment $y\mapsto{{}^{t}(}\frac{d^{n-1}y}{dx^{n-1}},\cdots,\frac{dy}{dx},y)$ gives a bijective correspondence between solutions of the equation $Dy=0$ with $D$ as above and horizontal sections of the associated $\mathrm{GL}_{n}$-oper. In particular, such a differential equation has a full set of algebraic solutions (i.e., has $n$ linearly independent algebraic solutions) precisely when the corresponding $\mathrm{GL}_{n}$-oper has finite monodromy.

The problem of classifying and counting linear ODE’s with a full set of algebraic solutions has long been one of the fundamental topics in mathematics. The study of them was tackled and developed from the 19th century onwards by many mathematicians: H. A. Schwarz (for the hypergeometric equations), L. I. Fuchs, P. Gordan, and C. F. Klein (for the second order equations), C. Jordan (for the $n$-th order equations) et al.

Regarding the relationship with the case of characteristic $p$ (where $p$ is a prime number), there is a well-known conjecture, i.e., the so-called Grothendieck-Katz $p$-curvature conjecture; it predicts the existence of a full set of algebraic solutions of a given linear differential equation in terms of its reductions modulo $p$ for various primes $p$. See, e.g.,  [And] or  [Kat4] for detailed accounts of this topic.

We want to focus on the situation where the underlying curve has prime-power characteristic $p^{N}$ ($N\in\mathbb{Z}_{>0}$). One central theme of our study stated in the most primitive form is to answer the following natural question:

How many homogeneous linear differential equations in characteristic $p^{N}$ (associated to differential operators $D$ as in (1.1)) have a full set of solutions?

We can find some previous studies related to this question for $N=1$ under the correspondence with $\mathrm{GL}_{n}$-opers (cf.  [Ihara1],  [JP],  [Jo14],    [LP],  [Mzk2],  [O4],  [Wak],  [Wak3],  [Wak2], and  [Wak8]). Here, note that one may define the notion of a $G$-oper (for an arbitrary $G$) on a curve in positive characteristic because of the algebraic nature of its formulation. For example, $G$-opers in characteristic $p$ have been investigated in the context of $p$-adic Teichmüller theory.

One of the common key ingredients in these developments is the study of $p$-curvature. The $p$-curvature of a flat connection in characteristic $p$ is an invariant that measures the obstruction to the compatibility of $p$-power structures appearing in certain associated spaces of infinitesimal symmetries. A $G$-oper is called dormant if it has vanishing $p$-curvature. It follows from a classical result by Cartier (cf.  [Kal, Theorem (5.1)]) that a $\mathrm{GL}_{n}$-oper, or a $\mathrm{PGL}_{n}$-oper, in characteristic $p$ is dormant if and only if it arises from a differential equation $Dy=0$ having a full set of solutions. Thus, the above question for $N=1$ can be reduced to asking the number of all possible dormant $\mathrm{GL}_{n}$-opers or their projectivizations, i.e., dormant $\mathrm{PGL}_{n}$-opers.

In what follows, we briefly review previous results concerning the counting problem of dormant opers. Let $X$ be a connected proper smooth curve of genus $g>1$ over an algebraically closed field of characteristic $p>2$. In the case of $g=2$, S. Mochizuki (cf.  [Mzk2, Chap. V, Corollary 3.7]), H. Lange-C. Pauly (cf.  [LP, Theorem 2]), and B. Osserman (cf.  [O4, Theorem 1.2]) computed, by applying different methods, the total number of dormant $\mathrm{PGL}_{2}$-opers on a general $X$, as follows:

$\displaystyle\sharp\left\{\begin{matrix}\text{dormant $\mathrm{PGL}_{2}$-opers on $X$}\end{matrix}\right\}=\frac{p^{3}-p}{24}.$

(1.3)

For example, this computation was made by using a correspondence with the base locus of the Verschiebung rational map on the moduli space of rank $2$ semistable bundles (cf. § 9.3).

Moreover, by extending the relevant formulations to the case where $X$ admits marked points or nodal singularities, S. Mochizuki also gave an explicit description of dormant $\mathrm{PGL}_{2}$-opers on each totally degenerate curve in terms of radius (cf.  [Mzk2, Introduction, Theorem 1.3]). The notion of radius is an invariant determining a sort of boundary condition at a marked point to glue together dormant opers in accordance with the attachment of underlying curves at this point. Mochizuki’s description is essentially equivalent to a previous result obtained by Y. Ihara (cf.  [Ihara1, § 1.6]), who investigated the situation when a given Gauss hypergeometric differential equation in characteristic $p$ had a full set of solutions. As a result of this explicit description, we obtained a combinatorial procedure for explicitly computing the number of dormant $\mathrm{PGL}_{2}$-opers.

This description also leads to a work by F. Liu and B. Osserman (cf.  [LO, Theorem 2.1]), who have shown that the value in question may be expressed as a degree $3g-3$ polynomial with respect to “$p$”. They did so by applying Ehrhart’s theory, which concerns computing the cardinality of the set of lattice points inside a polytope.

For general rank cases, K. Joshi and C. Pauly showed that there are only finitely many dormant $\mathrm{PGL}_{n}$-opers on a fixed curve (cf.  [JP, Corollary 6.1.6]), and later Joshi conjectured an explicit description of their total number (cf.  [Jo14, Conjecture 8.1]). As a consequence of developing the moduli theory of dormant $G$-opers on pointed stable curves, we solved affirmatively this conjecture for a general curve (cf.  [Wak8, Theorem H]), which is described as

$\displaystyle\sharp\left\{\begin{matrix}\text{dormant $\mathrm{PGL}_{n}$-opers on}\\ \text{a general stable curve}\\ \text{of genus $g$ in characteristic $p$}\end{matrix}\right\}=\frac{p^{(n-1)(g-1)-1}}{n!}\cdot\sum_{\genfrac{.}{.}{0.0pt}{}{(\zeta_{1},\cdots,\zeta_{n})\in\mathbb{C}^{\times n}}{\zeta_{i}^{p}=1,\ \zeta_{i}\neq\zeta_{j}(i\neq j)}}\frac{(\prod_{i=1}^{n}\zeta_{i})^{(n-1)(g-1)}}{\prod_{i\neq j}(\zeta_{i}-\zeta_{j})^{g-1}}\hskip 8.53581pt$

(1.4)

for $p>n\cdot\mathrm{max}\{g-1,2\}$. Based on the idea of Joshi et al. (cf.  [JP],  [Jo14]) and a work by Holla (cf.  [Hol]), this formula was proved by establishing a relationship with the Gromov-Witten theory of Grassmann varieties and then applying an explicit computation of their Gromov-Witten invariants, called the Vafa-Intriligator formula.

When $n=2$, we can extend this result to pointed stable curves by establishing the relationship with the $\mathfrak{s}\mathfrak{l}_{2}(\mathbb{C})$-WZW (= Wess-Zumino-Witten) conformal field theory (cf.  [Wak8, Theorem 7.41]). As a result, the Verlinde formula for that CFT yields the following formula:

$\displaystyle\sharp\left\{\begin{matrix}\text{dormant $\mathrm{PGL}_{2}$-opers on}\\ \text{a general $r$-pointed stable curve}\\ \text{of genus $g$ in characteristic $p$}\end{matrix}\right\}=\frac{p^{g-1}}{2^{2g-1+r}}\cdot\sum_{j=1}^{p-1}\frac{\left(1-(-1)^{j}\cdot\cos\left(\frac{j\pi}{p}\right)\right)^{r}}{\sin^{2(g-1+r)}\left(\frac{j\pi}{p}\right)},$

(1.5)

which is consistent with (1.4) in the case of $r=0$ and $n=2$.

The purpose of this manuscript is to develop the enumerative geometry of dormant $\mathrm{PGL}_{n}$-opers in prime-power characteristic so that (1.4) and (1.5) are generalized. From now on, let $X$ be a geometrically connected, proper, and smooth curve in characteristic $p^{N}$, where $N$ is a positive integer. Just as in the case of characteristic $p$, an oper, or more generally a flat bundle, on $X$ will be called dormant if it is spanned by its horizontal sections, i.e., isomorphic locally to the trivial flat bundle (cf. Definitions 3.1.4, 5.2.2, and Proposition 3.2.5, (ii)). In particular,

Our central theme posed earlier can be rephrased essentially as the issue of counting dormant opers on a (general) curve in characteristic $p^{N}$.

(Note that we also discuss, at the same time, the case where the underlying curve has nodal singularities or marked points; in such a generalized situation, the dormancy condition has to be formulated in a different and more complicated manner.)

Unfortunately, the lack of a reasonable invariant exactly like the $p$-curvature for $N=1$ makes it difficult to handle with dormant opers in characteristic $p^{N}$. To overcome this difficulty, we relate, partly on the basis of the argument of  [Mzk2, § 2.1, Chap. II], dormant opers on $X$ to certain objects defined on the mod $p$ reduction $X_{0}$ of $X$; we shall refer to the operation resulting from this argument as diagonal reduction/lifting.

One remarkable observation is that each dormant $\mathrm{GL}_{n}$-oper $\mathscr{E}^{\spadesuit}$ on $X$ induces, via diagonal reduction, a dormant $\mathrm{GL}_{n}$-oper ${{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\!\!\mathscr{E}^{\spadesuit}$ on $X_{0}$ equipped with a $\mathcal{D}^{(N-1)}$-module structure extending its flat structure. Here, $\mathcal{D}^{(N-1)}$ denotes the sheaf of differential operators of level $N-1$ introduced by P. Berthelot in  [PBer1] and  [PBer2]. A dormant $\mathrm{GL}_{n}$-oper on $X_{0}$ equipped with such an additional structure is called a dormant $\mathrm{GL}_{n}$-oper of level $N$, or a dormant $\mathrm{GL}_{n}^{(N)}$-oper, for short. Similarly, we obtain the notion of a dormant $\mathrm{PGL}_{n}^{(N)}$-oper such that $\mathrm{PGL}_{n}^{(1)}$-opers are equivalent to $\mathrm{PGL}_{n}$-opers in the classical sense. See Definitions 5.2.1, 5.2.2, 5.3.1, and 5.3.2 for their precise definitions.

We expect that, for a general curve $X$, the assignment $\mathscr{E}^{\spadesuit}\mapsto{{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\!\!\mathscr{E}^{\spadesuit}$ given by taking the diagonal reductions is invertible, i.e., each dormant $\mathrm{PGL}_{n}^{(N)}$-oper $\mathscr{E}_{0}^{\spadesuit}$ on $X_{0}$ may be lifted uniquely to a dormant $\mathrm{PGL}_{n}$-oper ${{}^{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}}}\!\!\mathscr{E}_{0}^{\spadesuit}$ on $X$. One of the consequences in this manuscript shows that this is in fact true for $n=2$. The results of the lifting $\mathscr{E}_{0}^{\spadesuit}\mapsto{{}^{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}}}\!\!\mathscr{E}_{0}^{\spadesuit}$ will be called the canonical diagonal liftings. The bijective correspondence given by $\mathscr{E}^{\spadesuit}\mapsto{{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\!\!\mathscr{E}^{\spadesuit}$ and $\mathscr{E}_{0}^{\spadesuit}\mapsto{{}^{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}}}\!\!\mathscr{E}_{0}^{\spadesuit}$ enables us to obtain a detailed understanding of dormant $\mathrm{PGL}_{2}$-opers in characteristic $p^{N}$ by applying, via diagonal reduction, various methods and perspectives inherent in characteristic-$p$-geometry established in  [Wak8].

In particular, because of a factorization property on the moduli space classifying dormant $\mathrm{PGL}_{2}^{(N)}$-opers in accordance with clutching morphisms, we reduce the counting problem under consideration to the cases of small $g$ and $r$. Some explicit computations based on this argument will be made in, e.g., Corollary 10.4.7 and Example 10.5.3.

In the rest of this Introduction, we shall describe the contents and some (relatively important) results of this manuscript. Note that we substantially apply various results and discussions of  [Wak8], in which the author developed the theory of (dormant) opers on pointed stable curves from the viewpoint of logarithmic geometry. Our study could be placed in a higher-level generalization of that theory.

The discussions in this manuscript may be divided into three parts. The first part is devoted to a general study of flat bundles formulated in terms of logarithmic geometry. In § 2, we discuss flat bundles on a log curve. To do this, we use the logarithmic generalization of the sheaf “$\mathcal{D}^{(m)}$” ($m\in\mathbb{Z}_{\geq 0}$) introduced by C. Montagnon (cf. [Mon]). This formulation is essential in investigating how the related moduli spaces behave when the underlying curve degenerates. Also, the Cartier operator associated to a $p^{m+1}$-flat $\mathcal{D}^{(m)}$-module (i.e., a $\mathcal{D}^{(m)}$-module with vanishing $p^{m+1}$-curvature) is defined by composing the usual Cartier operators of the flat bundles constituting that $\mathcal{D}^{(m)}$-module at the respective levels (cf. Definition 2.6.5).

In § 3, we generalize the classical notion of a dormant flat bundle (i.e., a flat bundle with vanishing $p$-curvature) to characteristic $p^{m+1}$ (cf. Definition 3.1.4). At the same time, we define the diagonal reduction of a dormant flat bundle, as well as a diagonal lifting of a $p^{m+1}$-flat $\mathcal{D}^{(m)}$-module. Roughly speaking, the diagonal reduction ${{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\!\!\mathscr{F}$ of a given dormant flat bundle $\mathscr{F}$ is obtained in such a way that, for each $\ell\leq m$, the reduction of $\mathscr{F}$ modulo $p^{\ell+1}$ corresponds, via the Cartier operator, to the reduction of the level of ${{}^{\rotatebox[origin={c}]{45.0}{$\Leftarrow$}}}\!\!\mathscr{F}$ to $\ell$. If the log structure of the underlying curve is trivial, then the dormancy condition can be characterized by the vanishing of the $p$-curvature of the flat bundles appearing in the inductive procedure for constructing the diagonal reduction (cf. Proposition 3.2.5, (i)). In this sense, a dormant flat bundle in characteristic $p^{m+1}$ may be seen as something like a successive $p$-flat deformation of a $p^{m+1}$-flat $\mathcal{D}^{(m)}$-module.

In §4, we discuss the local description of a dormant flat bundle around a marked/nodal point of the underlying curve. For each scheme $S$ flat over $\mathbb{Z}/p^{m+1}\mathbb{Z}$, we shall set

$\displaystyle U_{\oslash}:=\mathcal{S}pec(\mathcal{O}_{S}[\![t]\!])$

(1.6)

(cf. (4.1)), where $t$ denotes a formal parameter. By equipping $U_{\oslash}$ with the log structure determined by the divisor $t=0$, we obtain a log scheme $U_{\oslash}^{\mathrm{log}}$, regarded as a formal neighborhood of each marked point in a pointed curve. One of the results in § 4 generalizes  [Kin1, Proposition 1.1.12],  [O2, Corollary 2.10], and it tells us by what data a dormant flat bundle can be exactly characterized. The statement is described as follows:

We remark that another main result in that section gives an explicit description of a $p^{m+1}$-flat $\mathcal{D}^{(m)}$-module around a nodal/marked point (cf. Proposition-Definition 4.4.1); in a certain sense, that result commutes with Theorem A via diagonal reduction.

In §§ 5-6, we discuss dormant opers of finite level on pointed curves and their moduli space. For a pair of nonnegative integers $(g,r)$ with $2g-2+r>0$, we denote by $\overline{\mathcal{M}}_{g,r}$ the moduli stack classifying $r$-pointed stable curves of genus $g$ over $\mathbb{F}_{p}$. Also, let us fix integers $n$, $N$ with $1<n<p$ and $N>0$.

An important achievement of our discussion is to construct a compactification of the moduli space by allowing nodal singularities on the underlying curves. This compactification enables us to investigate how the moduli space deforms when the underlying curve degenerates. Additionally, when it has actually occurred, elements of a certain finite set

$\displaystyle\Xi_{n,N}$

(1.8)

(cf. (6.34)) provide a boundary condition (i.e., the coincidence of radii) for gluing dormant $\mathrm{PGL}_{n}^{(N)}$-opers in accordance with an attachment of two curves along respective marked points (cf. Definition 6.3.2).

With that in mind, the moduli space we have to deal with is defined as the category classifying pairs $(\mathscr{X},\mathscr{E}^{\spadesuit})$ consisting of an $r$-pointed stable curve $\mathscr{X}$ of genus $g$ in characteristic $p$ (i.e., an object classified by $\overline{\mathcal{M}}_{g,r}$) and a dormant $\mathrm{PGL}_{n}^{(N)}$-oper $\mathscr{E}^{\spadesuit}$ on $\mathscr{X}$ of fixed radii $\rho\in\Xi_{n,N}^{\times r}\left(=\Xi_{n,N}\times\cdots\times\Xi_{n,N}\right)$; this category will be denoted by

$\displaystyle\mathcal{O}p_{n,N,\rho,g,r,\mathbb{F}_{p}}^{{}^{\mathrm{Zzz...}}},\text{or simply}\ \mathcal{O}p_{\rho,g,r}^{{}^{\mathrm{Zzz...}}}$

(1.9)

(cf. (6.41)). Forgetting the data of dormant $\mathrm{PGL}_{n}^{(N)}$-opers yields the projection

$\displaystyle\Pi_{n,N,\rho,g,r,\mathbb{F}_{p}}\ (\text{or simply}\ \Pi_{\rho,g,r}):\mathcal{O}p_{\rho,g,r}^{{}^{\mathrm{Zzz...}}}\rightarrow\overline{\mathcal{M}}_{g,r}$

(1.10)

(cf. (6.42)). Our results concerning the geometric structures of $\mathcal{O}p_{\rho,g,r}^{{}^{\mathrm{Zzz...}}}$ and $\Pi_{\rho,g,r}$ are summarized as follows.

In §§ 7-8, we focus on the case of $n=2$ and develop the deformation theory of dormant $\mathrm{PGL}_{2}^{(N)}$-opers. The deformation space of a fixed dormant $\mathrm{PGL}_{2}^{(N)}$-oper is described by using the hypercohomology group of a certain complex associated to that oper (cf. Theorem 8.3.5). By this description together with the local study of finite-level $\mathcal{D}$-modules discussed in the first part, we can show more than the facts stated in Theorem B, as described below (cf. [Mzk2, Chap. II, Theorem 2.8] for the case of $N=1$).

The most important of our results would be the generic étaleness of $\Pi_{\rho,g,r}$, since it deduces decompositions of its degree $\mathrm{deg}(\Pi_{\rho,g,r})$ with respect to various clutching morphisms $\mathrm{Clut}_{\mathbb{G},\rho_{\mathbb{G}}}$. According to the discussion in § 8, these decompositions can be collectively explained by the notion of a $2$d TQFT, which is by definition a symmetric monoidal functor from the category of $2$-dimensional cobordisms $2\text{-}\mathcal{C}ob$ to the category of $K$-vector spaces $\mathcal{V}ect_{K}$ for a field $K$, say, $\mathbb{Q}$ or $\mathbb{C}$ (cf. Definition 7.4.1). Applying a well-known generalities on $2$d TQFTs, we obtain an approach to compute $\mathrm{deg}(\Pi_{\rho,g,r})$’s by means of the ring-theoretic structure of the corresponding Frobenius algebra, or equivalently, the fusion ring of the associated fusion rule.

In § 9, the canonical diagonal lifting of a dormant $\mathrm{PGL}_{2}^{(N)}$-oper is constructed by applying again the generic étaleness of $\Pi_{\rho,g,r}$’s. As discussed in Proposition 3.4.4, there exists a direct linkage between raising the level of a dormant $\mathrm{PGL}_{2}$-oper in characteristic $p$ to $N$ and lifting that oper to characteristic $p^{N}$. To be more precise, we show that the space of geometric deformations (which mean deformations classified by the moduli space $\mathcal{O}p^{{}^{\mathrm{Zzz...}}}_{\rho,g,r}$) of a given dormant $\mathrm{PGL}_{2}^{(N)}$-oper is, in a certain sense, dual to the space of its arithmetic deformations (which mean liftings to characteristic $p^{N}$ forgetting the data of higher-level structures). See Corollary 8.3.6, (ii), Propositions 8.4.3, 8.5.5, and 9.1.1, and the picture displayed below.

In particular, when a given smooth curve is $N$-ordinary in the sense of Definition 8.7.4, any dormant $\mathrm{PGL}_{2}^{(N)}$-oper on it admits a canonical lifting. This construction of liftings reverses the operation of taking the diagonal reductions, and it is available to general curves because the $N$-ordinary locus in $\overline{\mathcal{M}}_{g,r}$ is dense.

Now, let $S$ be a flat scheme over $\mathbb{Z}/p^{N}\mathbb{Z}$ (with $p>2$) and $X$ a geometrically connected, proper, and smooth curve over $S$ of genus $g>1$. Denote by $S_{0}$ and $X_{0}$ the mod $p$ reductions of $S$ and $X$, respectively. Also, denote by

$\displaystyle\mathrm{Op}_{1,X}^{{}^{\mathrm{Zzz...}}}\left(\text{resp.,}\ \mathrm{Op}_{N,X_{0}}^{{}^{\mathrm{Zzz...}}}\right)$

(1.20)

(cf. (9.13)) the set of isomorphism classes of dormant $\mathrm{PGL}_{2}$-opers on $X/S$ (resp., dormant $\mathrm{PGL}_{2}^{(N)}$-opers on $X_{0}/S_{0}$). Taking the diagonal reductions induces a map of sets

$\displaystyle\rotatebox[origin={c}]{45.0}{$\Leftarrow$}_{\!\!\spadesuit}:\mathrm{Op}^{{}^{\mathrm{Zzz...}}}_{1,X}\rightarrow\mathrm{Op}^{{}^{\mathrm{Zzz...}}}_{N,X_{0}}.$

(1.21)

(cf. (9.15)). Then, we obtain the following assertion.

For each $\mathscr{E}_{0}^{\spadesuit}$ as in the above theorem, we shall refer to the resulting dormant $\mathrm{PGL}_{2}$-oper ${{}^{\rotatebox[origin={c}]{45.0}{$\Rightarrow$}}}\!\!\mathscr{E}^{\spadesuit}_{0}$ as the canonical diagonal lifting of $\mathscr{E}_{0}^{\spadesuit}$. This fact will also be formulated as the existence of canonical liftings of Frobenius-projective structures (cf. Theorem-Definition 9.3.5), which are characteristic-$p$ analogues of complex projective structures^***A (complex) projective structure is defined as an additional structure on a Riemann surface consisting of local coordinate charts defining its complex structure such that on any two overlapping patches, the change of coordinates may be described as a Möbius transformation. Projective structures are in bijection with $\mathrm{PGL}_{2}$-opers (in our sense) via the Riemann-Hilbert correspondence and the algebraization of the underlying Riemann surface. (cf.  [Hos2, Definition 2.1], or Definition 9.3.2).

We expect that canonical liftings exist even for a general rank $n$ (sufficiently small relative to $p$). To prove it in accordance with the arguments of this manuscript, we will have to prove the generic étaleness of $\Pi_{n,N,\rho,g,r}$’s. (Other constructions of liftings of $\mathrm{PGL}_{2}$-opers in characteristic $p$ can be found in  [LSYZ],  [Mzk1], and  [Mzk2]; but they differ from ours, as mentioned in Remark 9.2.4).

In § 10, we translate the $2$d TQFT $\mathcal{Z}_{2,N}$ into combinatorial objects in order to solve our counting problem in a practical manner. This is possible because the factorization property of $\mathrm{deg}(\Pi_{\rho,g,r})$’s reduces the problem to the case where the underlying curve is totally degenerate (cf. Definition 6.5.1). In this case, the normalizations of its components are $3$-pointed projective lines, so dormant $\mathrm{PGL}_{2}^{(N)}$-opers are determined by purely combinatorial patterns of their radii (cf.  [LO] in the case of $N=1$).

To make the discussion clearer, we set ${{}^{\dagger}}C_{N}$ (cf. (10.34)) to be the collection of triples of nonnegative integers $(s_{1},s_{2},s_{3})$ satisfying the following conditions:

• •

  $\sum_{i=1}^{3}s_{i}\leq p^{N}-2$ and $|s_{2}-s_{3}|\leq s_{1}\leq s_{2}+s_{3}$;

• •

  For every positive integer $N^{\prime}<N$, we can choose a triple of integers $(s^{\prime}_{1},s^{\prime}_{2},s^{\prime}_{3})$ with $s^{\prime}_{i}\in\{[s_{1}]_{N^{\prime}},p^{N^{\prime}}-1-[s_{i}]_{N^{\prime}}\}$ ($i=1,2,3$) that satisfies $\sum_{i=1}^{3}s^{\prime}_{i}\leq p^{N^{\prime}}-2$ and $|s^{\prime}_{2}-s^{\prime}_{3}|\leq s^{\prime}_{1}\leq s^{\prime}_{2}+s^{\prime}_{3}$.

Here, for each nonnegative integer $a$, we set $[a]_{N^{\prime}}$ to be the remainder obtained by dividing $a$ by $p^{N^{\prime}}$. Also, fix trivalent clutching data $\mathbb{G}$ of type $(g,0)$ (cf. Definition 6.2.3, (iii)), and denote by $X_{\mathbb{G}}$ the totally degenerate curve corresponding to $\mathbb{G}$.

A balanced $(p,N)$-edge numbering on $\mathbb{G}$ (cf. Definition 10.5.1, (i)) is a collection of nonnegative integers $(a_{e})_{e\in E}$ indexed by the set $E$ of edges of $\mathbb{G}$ such that, for each triple of edges $(e_{1},e_{2},e_{3})$ (with multiplicity) incident to a common vertex, the integers $(a_{e_{1}},a_{e_{2}},a_{e_{3}})$ belongs to ${{}^{\dagger}}C_{N}$. The set of balanced $(p,N)$-edge numberings on $\mathbb{G}$ is denoted by

$\displaystyle\mathrm{Ed}_{p,N,\mathbb{G}}$

(1.24)

(cf. (10.62)). This set is finite, and it is possible to explicitly find out which combinations of nonnegative integers belong to it.

The study of Gauss hypergeometric differential operators with a full set of solutions shows that dormant $\mathrm{PGL}_{n}^{(N)}$-opers on a $3$-pointed projective line are verified to be parametrized by ${{}^{\dagger}}C_{N}$ (cf. Propositions 10.1.4, 10.3.3). Thus, dormant $\mathrm{PGL}_{n}^{(N)}$-opers on $X_{\mathbb{G}}$ correspond bijectively to elements of $\mathrm{Ed}_{p,N,\mathbb{G}}$. Moreover, such numberings on $\mathbb{G}$ may be identified with lattice points inside a certain subset of a Euclidean space of dimension $(3g-3)\cdot N$ (cf. Proposition 10.6.4). This subset is constructed as the union of a finite number of rational convex polytopes without boundary. It follows that we can apply Ehrhart’s theory for lattice-point counting of polytopes, and obtain a quasi-polynomial realizing its lattice-point function. The consequence of this argument is as follows.

Finally, as a consequence of this fact, a partial answer to the question displayed at the beginning of § 1.2 can be given as follows.

Throughout this manuscript, all schemes are assumed to be locally noetherian. We fix a prime $p$, and write $\mathbb{F}_{p}:=\mathbb{Z}/p\mathbb{Z}$.

Let $S$ be a scheme. Given a sheaf $\mathcal{V}$ on $S$, we use the notation “ $v\in\mathcal{V}$ ” for a local section $v$ of $\mathcal{V}$. If $\mathcal{V}$ is an $\mathcal{O}_{S}$-module, then we denote by $\mathcal{V}^{\vee}$ the dual of $\mathcal{V}$, i.e., $\mathcal{V}^{\vee}:=\mathcal{H}om_{\mathcal{O}_{S}}(\mathcal{V},\mathcal{O}_{S})$. By a vector bundle on $S$, we mean a locally free $\mathcal{O}_{S}$-module of finite rank. If $X$ is a scheme over $S$, then we shall write $\Omega_{X/S}$ for the sheaf of $1$-forms on $X$ over $S$, and $\mathcal{T}_{X/S}$ for its dual.

For the basic properties on log schemes, we refer the reader to  [KaKa],  [ILL], and  [KaFu]. Given a log scheme (or more generally, a log stack) indicated, say, by $S^{\mathrm{log}}$, we shall write $S$ for the underlying scheme (stack) of $S^{\mathrm{log}}$, and $\alpha_{S^{\mathrm{log}}}:\mathcal{M}_{S^{\mathrm{log}}}\rightarrow\mathcal{O}_{S}$ for the morphism of sheaves of monoids defining the log structure of $S^{\mathrm{log}}$. For any morphism of log schemes $f^{\mathrm{log}}:X^{\mathrm{log}}\rightarrow S^{\mathrm{log}}$, we write $\overline{\mathcal{M}}_{X^{\mathrm{log}}/S^{\mathrm{log}}}:=\mathcal{M}_{X^{\mathrm{log}}}/\mathrm{Im}(f^{*}(\mathcal{M}_{S^{\mathrm{log}}})\rightarrow\mathcal{M}_{X^{\mathrm{log}}})$ (cf.  [KaFu, Introduction]), and call it the relative characteristic of $f^{\mathrm{log}}$ (or, of $X^{\mathrm{log}}/S^{\mathrm{log}}$).

Let $\nabla:\mathcal{K}^{0}\rightarrow\mathcal{K}^{1}$ be a morphism of sheaves of abelian groups on a scheme $S$. It may be regarded as a complex concentrated at degrees $0$ and $1$; we denote this complex by $\mathcal{K}^{\bullet}[\nabla]$. Next, let $f:X\rightarrow S$ be a morphism of schemes and $i$ an integer $\geq 0$. Then, one may define the sheaf $\mathbb{R}^{i}f_{*}(\mathcal{K}^{\bullet}[\nabla])$ on $S$ obtained from $\mathcal{K}^{\bullet}[\nabla]$ by applying the $i$-th hyper-derived functor $\mathbb{R}^{i}f_{*}(-)$ of $f_{*}(-)$ (cf.  [Kal], (2.0)). In particular, $\mathbb{R}^{0}f_{*}(\mathcal{K}^{\bullet}[\nabla])=f_{*}(\mathrm{Ker}(\nabla))$. If $S$ is affine, then $H^{0}(S,\mathbb{R}^{i}f_{*}(\mathcal{K}^{\bullet}[\nabla]))$ may be identified with the $i$-th hypercohomology group $\mathbb{H}^{i}(X,\mathcal{K}^{\bullet}[\nabla])$. Given an integer $n$ and a sheaf $\mathcal{F}$ on $S$, we define the complex $\mathcal{F}[n]$ to be $\mathcal{F}$ (considered as a complex concentrated at degree $0$) shifted down by $n$, so that $\mathcal{F}[n]^{-n}=\mathcal{F}$ and $\mathcal{F}[n]^{i}=0$ ($i\neq-n$).

Denote by $\mathbb{G}_{m}$ the multiplicative group. Also, for a positive integer $n$, we shall write $\mathrm{GL}_{n}$ (resp., $\mathrm{PGL}_{n}$) for the general (resp., projective) linear group of rank $n$.

First, we briefly recall sheaves of logarithmic differential operators of finite level discussed in  [PBer1],  [PBer2], and  [Mon].

Let $m$ be a nonnegative integer. Also, let $S^{\mathrm{log}}$ be an fs log scheme over $\mathbb{Z}_{(p)}$ and $X^{\mathrm{log}}$ an fs log scheme equipped with a morphism of log schemes $f^{\mathrm{log}}:X^{\mathrm{log}}\rightarrow S^{\mathrm{log}}$ which is log smooth (i.e., “smooth” in the sense of  [KaKa], (3.3)). Denote by $\Omega_{X^{\mathrm{log}}/S^{\mathrm{log}}}$ the sheaf of logarithmic $1$-forms on $X^{\mathrm{log}}/S^{\mathrm{log}}$ (cf.  [KaKa], (1.7)) and by $\mathcal{T}_{X^{\mathrm{log}}/S^{\mathrm{log}}}$ the sheaf of logarithmic vector fields on $X^{\mathrm{log}}/S^{\mathrm{log}}$, i.e., the dual of $\Omega_{X^{\mathrm{log}}/S^{\mathrm{log}}}$. For simplicity, we occasionally write $\Omega$ and $\mathcal{T}$ instead of $\Omega_{X^{\mathrm{log}}/S^{\mathrm{log}}}$ and $\mathcal{T}_{X^{\mathrm{log}}/S^{\mathrm{log}}}$, respectively. Since $f^{\mathrm{log}}$ is log smooth, both $\Omega$ and $\mathcal{T}$ are vector bundles (cf  [KaKa], Proposition (3.10)).

Suppose that $p$ is locally nilpotent on $S$ and $S$ is equipped with an $m$-PD structure that extends to $X$ via $f$. Denote by $P_{(m)}^{\mathrm{log}}$ the log $m$-PD envelope of the diagonal embedding $X^{\mathrm{log}}\rightarrow X^{\mathrm{log}}\times_{S^{\mathrm{log}}}X^{\mathrm{log}}$ (cf.  [Mon, Proposition 2.1.1]) and by $\mathcal{P}_{(m)}$ the structure sheaf of $P_{(m)}^{\mathrm{log}}$. The defining ideal $\overline{\mathcal{I}}$ of the strict closed immersion $X^{\mathrm{log}}\hookrightarrow P_{(m)}^{\mathrm{log}}$ admits the $m$-PD-adic filtration $\{\overline{\mathcal{I}}^{\{\ell\}}\}_{\ell\in\mathbb{Z}_{\geq 0}}$ constructed in the manner of  [PBer2, Definition A. 3]. For each $\ell\in\mathbb{Z}_{\geq 0}$, the quotient sheaf $\mathcal{P}_{(m)}^{\ell}$ of $\mathcal{P}_{(m)}$ by $\overline{\mathcal{I}}^{\{\ell+1\}}$ determines a strict closed subscheme $P^{\ell,\mathrm{log}}_{(m)}$ of $P_{(m)}^{\mathrm{log}}$. We shall write $\mathrm{pr}_{1}^{\ell}$ and $\mathrm{pr}_{2}^{\ell}$ for the morphisms $P_{(m)}^{\ell,\mathrm{log}}\rightarrow X^{\mathrm{log}}$ induced by the first and second projections $X^{\mathrm{log}}\times_{S^{\mathrm{log}}}X^{\mathrm{log}}\rightarrow X^{\mathrm{log}}$, respectively. Note that $\mathcal{P}^{\ell}_{(m)}$ may be regarded as a sheaf on $X$. Moreover, it has an $\mathcal{O}_{X}$-module structure via $\mathrm{pr}_{1}^{\ell}$ (resp., $\mathrm{pr}_{2}^{\ell}$); it will be applied whenever we are considering an action of $\mathcal{O}_{X}$ by left (resp., right) multiplication. When there is a fear of confusion, we use the notation $\mathrm{pr}_{1*}^{\ell}(\mathcal{P}_{(m)}^{\ell})$ (resp., $\mathrm{pr}_{2*}^{\ell}(\mathcal{P}_{(m)}^{\ell})$) for writing the sheaf $\mathcal{P}_{(m)}^{\ell}$ equipped with this $\mathcal{O}_{X}$-module structure.

Given an integer $m^{\prime}\geq m$, we obtain a natural morphism

$\displaystyle\varsigma^{\ell}_{m^{\prime},m}:\mathcal{P}_{(m^{\prime})}^{\ell}\rightarrow\mathcal{P}_{(m)}^{\ell}$

(2.1)

preserving both the left and right $\mathcal{O}_{X}$-module structures. As mentioned in  [Mon, § 2.3], there exists a canonical $\mathcal{O}_{X}$-algebra morphism

$\displaystyle\delta_{m}^{\ell,\ell^{\prime}}:\mathcal{P}_{(m)}^{\ell+\ell^{\prime}}\rightarrow\mathcal{P}^{\ell}_{(m)}\otimes_{\mathcal{O}_{X}}\mathcal{P}^{\ell^{\prime}}_{(m)}$

(2.2)

for each pair of nonnegative integers $(\ell,\ell^{\prime})$ such that if $m^{\prime}$ is an integer $\geq m$, then the following diagram is commutative:

(2.7)

The morphism $\delta_{m}^{\ell,\ell^{\prime}}$ determines a morphism of log schemes

$\displaystyle\delta_{m}^{\ell,\ell^{\prime}\sharp}:P_{(m)}^{\ell,\mathrm{log}}\times_{X^{\mathrm{log}}}P_{(m)}^{\ell^{\prime},\mathrm{log}}\rightarrow P_{(m)}^{\ell+\ell^{\prime},\mathrm{log}}.$

(2.8)

over $X^{\mathrm{log}}$.

For each $\ell\in\mathbb{Z}_{\geq 0}$, we shall set

$\displaystyle\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}},\leq\ell}^{(m)}:=\mathcal{H}om_{\mathcal{O}_{X}}(\mathrm{pr}_{1*}^{\ell}(\mathcal{P}^{\ell}_{(m)}),\mathcal{O}_{X})$

(2.9)

(cf.  [Mon, Definition 2.3.1]). In particular, we have natural identifications $\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}},\leq 0}^{(m)}=\mathcal{O}_{X}$, $\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}},\leq\ell}^{(m)}/\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}},\leq\ell-1}^{(m)}=\mathcal{T}^{\otimes\ell}$ ($l\geq 1$). The sheaf of logarithmic differential operators of level $m$ is defined by

$\displaystyle\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}}}^{(m)}:=\bigcup_{\ell\in\mathbb{Z}_{\geq 0}}\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}},\leq\ell}^{(m)}.$

(2.10)

For simplicity, we occasionally write $\mathcal{D}_{\leq\ell}^{(m)}$ and $\mathcal{D}^{(m)}$ instead of $\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}},\leq\ell}^{(m)}$ and $\mathcal{D}_{X^{\mathrm{log}}/S^{\mathrm{log}}}^{(m)}$, respectively. The morphisms $\delta_{m}^{\ell,\ell^{\prime}}$ (for $\ell,\ell^{\prime}\in\mathbb{Z}_{\geq 0}$) determine a structure of (possibly noncommutative) $f^{-1}(\mathcal{O}_{S})$-algebra $\mathcal{D}^{(m)}\otimes_{\mathcal{O}_{X}}\mathcal{D}^{(m)}\rightarrow\mathcal{D}^{(m)}$ on $\mathcal{D}^{(m)}$. The collection of morphisms $\varsigma_{m,m^{\prime}}^{\ell}$ (with $m^{\prime}\leq m$) induces an inductive system of sheaves $\{\mathcal{D}^{(m)}_{\leq\ell}\}_{m\in\mathbb{Z}_{\geq 0}}$. We shall write ${{}^{L}\mathcal{D}}^{(m)}_{\leq\ell}$ (resp., ${{}^{R}\mathcal{D}}^{(m)}_{\leq\ell}$) for the sheaf $\mathcal{D}^{(m)}_{\leq\ell}$ endowed with a structure of $\mathcal{O}_{X}$-module arising from left (resp., right) multiplication by sections of $\mathcal{D}_{\leq 0}^{(m)}\left(=\mathcal{O}_{X}\right)$.