Dormant Opers in Positive Characteristic

• Dormant opers are algebraic connections on curves with zero p-curvature in positive characteristic, defined by intricate moduli spaces and explicit degree formulas.
• They bridge deformation theory, duality, and Frobenius descent, thereby linking arithmetic geometry with enumerative and TQFT methods.
• Extensions such as higher-level, Miura, and parabolic dormant opers further reveal connections to classical structures, Gauss maps, and Tango curves.

Dormant opers are opers on algebraic curves in characteristic $p>0$ whose $p$-curvature vanishes identically. They are a specifically positive-characteristic phenomenon and occur for $\mathrm{PGL}_r$, $\mathfrak{sl}_n$, and more general semisimple groups on smooth, pointed, and stable curves. In rank $2$, a dormant $\mathrm{PGL}(2)$-oper is a dormant indigenous bundle in the sense of Mochizuki, and the resulting moduli spaces form finite, geometrically rich loci over the moduli of curves, with explicit degree formulas, deformation-theoretic splittings, dualities, and relations to Quot-schemes, Gromov–Witten theory, Frobenius descent, and $2$d TQFT (Joshi, 2013, Wakabayashi, 2014).

1. Definitions and basic structure

An $(r)$-oper on a smooth projective curve $X$ is a triple $(V, V_\bullet, \nabla)$, where $p$0 is a rank $p$1 vector bundle, $p$2 is a decreasing filtration, and $p$3 is a connection satisfying the usual oper conditions. Its $p$4-curvature $p$5 measures the deviation from being flat in characteristic $p$6. A dormant $p$7-oper is an oper for which $p$8. The scheme or stack of dormant opers is denoted $p$9 in the $\mathrm{PGL}_r$0 setting (Joshi, 2013).

The same pattern extends to $\mathrm{PGL}_r$1-opers for semisimple algebraic groups $\mathrm{PGL}_r$2, including pointed stable curves and logarithmic structures. In the logarithmic formulation, opers on pointed stable curves admit radii at marked or nodal points, and dormant opers are those with vanishing $\mathrm{PGL}_r$3-curvature. This log-geometric formulation is one of the basic structural advances of the subject and supports compactified moduli and clutching constructions (Wakabayashi, 2014).

A recurrent misconception is to identify dormancy with the characteristic-zero notion of trivial monodromy. In the positive-characteristic theory, dormancy is stronger: for Miura opers, vanishing $\mathrm{PGL}_r$4-curvature is described as “a much stronger condition than merely requiring trivial monodromy” (Wakabayashi, 2019). For $\mathrm{PGL}_r$5, the characteristic-$\mathrm{PGL}_r$6 replacement of classical projective structures is expressed in terms of dormant opers and Frobenius-projective structures rather than naive monodromy-free analogues (Wakabayashi, 2014).

The notion also admits higher-level variants. A dormant $\mathrm{PGL}_r$7-oper is defined using an action of the sheaf of differential operators of level $\mathrm{PGL}_r$8 and the vanishing of $\mathrm{PGL}_r$9-curvature. In rank $\mathfrak{sl}_n$0, a dormant $\mathfrak{sl}_n$1-oper is a triple $\mathfrak{sl}_n$2 with vanishing $\mathfrak{sl}_n$3-curvature and a Kodaira–Spencer morphism that is an isomorphism; passage to $\mathfrak{sl}_n$4 is by the usual twisting equivalence (Wakabayashi, 2022, Kondo et al., 4 Sep 2025).

2. Moduli spaces and geometric properties

For pointed stable curves, the moduli functor of opers is representable by a smooth Deligne–Mumford stack, an affine bundle over the moduli of curves. Its dormant locus is proper over the moduli of pointed stable curves, and in the $\mathfrak{sl}_n$5 setting it is finite over the moduli stack of curves. Wakabayashi’s general theory also establishes compactification, comparison with differential operators, and deformation-theoretic control in the logarithmic category (Wakabayashi, 2014).

In enumerative applications, generic étaleness is decisive. Wakabayashi proved that Joshi’s degree formula holds for generic $\mathfrak{sl}_n$6 and $\mathfrak{sl}_n$7 large, and later work extended generic étaleness to additional Lie types. For $\mathfrak{sl}_n$8, if $\mathfrak{sl}_n$9 and $2$0, the moduli stack of dormant $2$1-opers is étale over points classifying totally degenerate curves; consequently, any irreducible component dominating $2$2 contains a dense open substack étale over $2$3 (Joshi, 2013, Wakabayashi, 2024).

Higher-level moduli exhibit analogous features. For $2$4, the stack of dormant $2$5-opers is proper and finite over $2$6, and the higher-level Hitchin–Mochizuki morphism identifies the dormant locus with the zero section of the corresponding Hitchin base. When the dormant stack is nonempty, it is irreducible (Wakabayashi, 4 Sep 2025).

The genus-one case is unusually explicit. On elliptic curves, the moduli stacks of dormant $2$7-opers and dormant generic Miura $2$8-opers are finite, proper Deligne–Mumford stacks over the moduli stack of pointed stable elliptic curves, and they are connected. Over the ordinary locus, the moduli of dormant opers is a finite étale cover described in terms of regular elements in $2$9 modulo Weyl group action (Karube et al., 1 Apr 2025).

3. Degree formulas and enumerative geometry

A central problem is the degree of the dormant operatic locus. For a smooth projective curve $\mathrm{PGL}(2)$0 of genus $\mathrm{PGL}(2)$1 over an algebraically closed field of characteristic $\mathrm{PGL}(2)$2, Joshi proposed a formula for the degree of the scheme of dormant $\mathrm{PGL}(2)$3-opers under the numerical hypothesis

$\mathrm{PGL}(2)$4

With a line bundle $\mathrm{PGL}(2)$5 satisfying

$\mathrm{PGL}(2)$6

the degree is

$\mathrm{PGL}(2)$7

and explicitly

$\mathrm{PGL}(2)$8

where the sum is over $\mathrm{PGL}(2)$9-tuples of distinct $2$0-th roots of unity (Joshi, 2013).

For $2$1 and $2$2, the formula specializes to

$2$3

when $2$4 and $2$5 is ordinary. This agrees with explicit computations of Mochizuki, Lange, and Osserman (Joshi, 2013).

The degree calculation is mediated by Quot-schemes. The scheme of dormant opers is identified with a finite Quot-scheme parametrizing rank $2$6, degree $2$7 subbundles $2$8, where $2$9 is Frobenius pushforward. The resulting degree formula is a special case of the Vafa–Intriligator formula and is related to Holla’s formula for Quot-scheme intersection numbers, thereby importing Gromov–Witten methods into the arithmetic geometry of dormant opers (Joshi, 2013).

The same enumerative structure persists beyond type $(r)$0. For dormant $(r)$1-opers, the generic degree with prescribed radii $(r)$2 satisfies a factorization formula

$(r)$3

where $(r)$4 is the set of complex ring homomorphisms from the associated pseudo-fusion ring and $(r)$5 is the Casimir element. This realizes the degree as a Verlinde-type quantity governed by clutching and fusion (Wakabayashi, 2024).

A further striking relation is the coincidence, up to a factor, with Verlinde numbers: $(r)$6 for generic $(r)$7 and $(r)$8 large in the range established by Wakabayashi. The data explicitly suggest a deep relation between dormant opers and conformal blocks (Joshi, 2013).

4. Deformation theory, ordinariness, and duality

The deformation theory of dormant opers in positive characteristic exhibits a canonical splitting phenomenon. On a general pointed stable curve, the moduli space of $(r)$9-opers and the locus of $X$0-flat connections intersect transversally inside the de Rham moduli space: $X$1 and hence

$X$2

For dormant opers induced from an $X$3-oper, the corresponding de Rham cohomology decomposes into symmetric-product pieces, yielding an Eichler–Shimura-type canonical decomposition on general pointed stable curves (Wakabayashi, 2023).

More concretely, if $X$4 is the rank $X$5 bundle underlying a dormant $X$6-oper and $X$7, then

$X$8

with a parabolic analogue for pointed curves. This provides a positive-characteristic algebraic analogue of the classical Eichler–Shimura decomposition (Wakabayashi, 2023).

Ordinariness supplies a second deformation-theoretic refinement. For a dormant $X$9-oper $(V, V_\bullet, \nabla)$0, ordinariness is defined by the condition that a natural cohomological map

$(V, V_\bullet, \nabla)$1

is an isomorphism. For elliptic curves, dormant-opers-ordinariness is equivalent to classical ordinariness. More generally, if $(V, V_\bullet, \nabla)$2 is a general smooth pointed hyperbolic curve, $(V, V_\bullet, \nabla)$3 is a cyclic étale covering of degree prime to $(V, V_\bullet, \nabla)$4, and $(V, V_\bullet, \nabla)$5 is an ordinary dormant $(V, V_\bullet, \nabla)$6-oper on $(V, V_\bullet, \nabla)$7, then $(V, V_\bullet, \nabla)$8 is ordinary on $(V, V_\bullet, \nabla)$9 (Wakabayashi, 2016).

Duality is another structural feature. For $p$00, there is a canonical involutive isomorphism

$p$01

compatible with radii. As a limiting case, there exists a unique dormant $p$02-oper on a fixed pointed stable curve (Wakabayashi, 2015). More recently, under the numerical condition $p$03, a canonical isomorphism was constructed between the moduli spaces of dormant $p$04-opers and dormant $p$05-opers with prescribed symmetric radii, extending the type-$p$06 duality pattern to types $p$07 and $p$08 (Wakabayashi, 18 May 2026).

5. Miura, parabolic, higher-level, and elliptic variants

Dormant Miura opers refine the oper structure by an additional Borel reduction. For generic Miura $p$09-opers there is a natural correspondence with $p$10-Cartan connections, and the dormant locus is a closed substack finite over the moduli of pointed stable curves. In the $p$11 case there is a canonical isomorphism

$p$12

identifying pre-Tango structures with dormant generic Miura $p$13-opers of matching exponent. When nonempty, these stacks are smooth, proper, and finite over $p$14, with dimension

$p$15

This links dormant opers to classical positive-characteristic pathologies such as failures of Kodaira vanishing (Wakabayashi, 2017).

The relation to Tango structures also appears through the Gaudin model modulo $p$16. Dormant generic Miura $p$17-opers correspond bijectively to Tango structures, and mod-$p$18 Bethe ansatz solutions yield explicit Tango curves. In the unramified case, the Bethe equations reduce to $p$19 for $p$20, and the desingularization of

$p$21

is a Tango curve under the hypotheses stated in the paper (Wakabayashi, 2019).

Parabolic dormant opers introduce weighted flags and logarithmic poles. A generalization of Cartier descent gives an equivalence between parabolic bundles on the Frobenius twist $p$22 with weights $p$23 and parabolic $p$24-flat bundles on $p$25 with weights $p$26. Under this correspondence, maximally Frobenius-destabilized parabolic bundles correspond bijectively to dormant parabolic opers with prescribed exponents. In rank $p$27, the number of such bundles for a sufficiently general pointed curve is given by an explicit trigonometric formula under parity and inequality hypotheses (Wakabayashi, 2024).

The higher-level theory further associates dormant $p$28-opers with arithmetic liftings. For $p$29, generic étaleness yields a canonical diagonal lifting of a dormant $p$30-oper on a general curve to characteristic $p$31, and the degrees of the associated moduli spaces can be computed combinatorially (Wakabayashi, 2022).

Elliptic curves form a special test case. For an ordinary elliptic curve $p$32, dormant generic Miura $p$33-opers are described by

$p$34

while dormant $p$35-opers are described by

$p$36

The Miura transformation is a finite étale Galois covering over the ordinary locus with group $p$37 (Karube et al., 1 Apr 2025).

6. Combinatorics, TQFT, Gauss maps, and arithmetic applications

A major development is the emergence of $p$38d TQFT from the enumerative geometry of dormant opers. For semisimple $p$39, Wakabayashi introduced the compact moduli stack of dormant faithful twisted $p$40-opers, or $p$41-do’pers, together with a perfect obstruction theory and virtual fundamental class. The resulting correlators define a semisimple $p$42d TQFT whose state space is indexed by radii, and the generic numbers of $p$43-do’pers are the corresponding structure constants (Wakabayashi, 2017).

In rank $p$44 and higher level, the same factorization principle becomes explicitly combinatorial. For totally degenerate curves with trivalent dual graph $p$45, the number of dormant $p$46-opers is the number of balanced $p$47-edge numberings on $p$48. These counting functions are quasi-polynomials in $p$49 arising from lattice points in generalized rational polytopes, and they also compute the number of second-order differential equations in characteristic $p$50 with a full set of solutions (Wakabayashi, 2022).

This combinatorics controls other arithmetic invariants. In rank $p$51, the generic degree of the generalized Verschiebung is governed by the same balanced edge numberings: $p$52 For $p$53,

$p$54

and for $p$55,

$p$56

These formulas resolve the rank-$p$57 case of a conjectural relation between higher-level dormant $p$58-opers and generalized Verschiebung degrees (Kondo et al., 4 Sep 2025).

The Miura transformation also has a graph-theoretic description. On a totally degenerate curve, dormant generic Miura $p$59-opers correspond to strict $p$60-branch numberings of the associated $p$61-regular graph, while dormant $p$62-opers correspond to balanced $p$63-edge numberings. The combinatorial Miura transformation maps the former to the latter, and there are no dormant generic Miura $p$64-opers on a totally degenerate stable curve of genus $p$65 (Wakabayashi, 2019).

Dormant opers also enter projective geometry in characteristic $p$66. For smooth projective varieties, Wakabayashi established a correspondence between dormant $p$67-opers and closed immersions with purely inseparable Gauss map, and for $p$68 this correspondence is bijective. For curves, this identifies precisely the subfields arising from Gauss maps: $p$69 The same mechanism yields an $p$70-projective structure on the Fermat hypersurface

$p$71

whose Gauss map is the $p$72-th Frobenius morphism (Wakabayashi, 2022).

Recent work has pushed the explicit theory further in low characteristic and in one-dimensional towers. Dormant $p$73-opers arising from generalized hypergeometric equations in characteristic $p$74 are rigid within the class of dormant opers, and this rigidity determines explicit $p$75d TQFTs for counting them (Mori et al., 4 Sep 2025). For $p$76-pointed stable curves of genus $p$77, higher-level dormant $p$78-opers with prescribed radii form projective systems of “dormant modular curves,” and explicit genus formulas make it possible to study asymptotic behavior of the associated towers of function fields (Aoyama et al., 18 May 2026).

Taken together, these developments show that dormant opers are not merely a special locus inside the oper moduli problem. They organize a broad positive-characteristic theory encompassing explicit degree formulas, canonical dualities, Frobenius descent, combinatorial recursion, TQFT, and geometric realizations through Gauss maps and projective structures, with especially strong structural results in rank $p$79 and on generic curves (Wakabayashi, 2014, Wakabayashi, 2022).