Genus formulas for dormant modular curves and asymptotic behavior of their function fields

Abstract: Towers of algebraic function fields over finite fields play a fundamental role in arithmetic geometry and coding theory. Classical examples arising from modular and Drinfeld modular curves exhibit asymptotically good behavior. In this paper, we introduce an analogous construction derived from the moduli spaces of higher-level dormant $\mathrm{PGL}_2$-opers of prescribed radii on $4$-pointed stable curves of genus $0$. These spaces, which we refer to as dormant modular curves, form projective systems under level reduction. Building on previous results in the moduli theory of dormant opers, we establish an explicit formula for computing the genera of these curves. This formula allows us to study the asymptotic behavior of the corresponding towers of function fields and to compare them with the classical modular and Drinfeld modular cases.

• The paper derives explicit genus formulas for dormant modular curves of type (0,4) using balanced edge numberings, local finite-level D-module classifications, and Chern character computations.
• The formulas yield a uniform genus bound of at most (p−1)p^(2N−1)/4 and determine when level-reduced curves are nonempty through combinatorial radius data.
• The associated function field towers are shown to be asymptotically α-good under digit-separation conditions, with α = log(s)/(2 log(p)) and a proven asymptotic quality of at least 1/2 in key cases.

This paper by Aoyama, Morita, and Wakabayashi develops an explicit genus formula for a new class of curves over finite fields — the dormant modular curves of type $(0,4)$ — and uses it to analyze the asymptotic behavior of towers of function fields obtained from these curves under level reduction. The construction is an analogue of classical modular and Drinfeld modular towers, with level structures replaced by higher-level dormant $\mathrm{PGL}_2$-opers on $4$-pointed stable genus-$0$ curves (2605.17973).

Background and construction

The paper works over a field $k$ of odd characteristic $p$. Building on Berthelot–Montagnon's sheaves of differential operators of finite level, the authors consider dormant $\mathrm{PGL}_2^{(N)}$-opers: $\mathrm{PGL}_2$-opers equipped with an $(N-1)$-PD stratification whose $p^N$-curvature vanishes identically. For each quadruple of radii $\mathrm{PGL}_2$0, the moduli stack $\mathrm{PGL}_2$1 classifies pairs consisting of a $\mathrm{PGL}_2$2-pointed stable curve of genus $\mathrm{PGL}_2$3 and a dormant $\mathrm{PGL}_2$4-oper of radii $\mathrm{PGL}_2$5. By prior work of Wakabayashi (and Mochizuki in the level-$\mathrm{PGL}_2$6 case), this stack is a geometrically connected, smooth, proper Deligne–Mumford $\mathrm{PGL}_2$7-stack of dimension $\mathrm{PGL}_2$8, hence a smooth projective curve when $\mathrm{PGL}_2$9; this is the dormant modular curve. A compatible system of radii $4$0 yields a projective system of such curves via level reduction, whose transition morphisms are finite, faithfully flat, and generically étale.

The analogy with classical modular curves is structural rather than incidental: the forgetful projection $4$1 plays the role of the usual projection from modular curves, and the oper data replaces Igusa structures.

Combinatorial description of the boundary

A key input is that, over the boundary of $4$2, dormant opers correspond to balanced $4$3-edge numberings on trivalent graphs satisfying explicit triangle inequalities. The authors define sets $4$4 ($4$5) parametrizing these numberings, and prove a canonical bijection between the $4$6-points of the boundary divisor and $4$7. Two consequences follow directly:

• Nonemptiness criterion: $4$8 is nonempty if and only if $4$9, and when nonempty, $0$0 for each $0$1, with $0$2.
• Degree bound: $0$3.

The authors also show that raising the level within a fixed radius class does not change the curve: the level-reduction map $0$4 is an isomorphism whenever $0$5 lifts $0$6 compatibly. This fact is used crucially in deriving the simplified genus formula below.

Local $0$7-module analysis

The technical core concerns logarithmic $0$8-modules near nodes of a family of pointed stable curves. Around a formal node $0$9, the authors construct explicit $k$0-modules $k$1 indexed by $k$2, all having vanishing $k$3-curvature, and prove two structural results. First, any free $k$4-module with vanishing $k$5-curvature splits as a direct sum of the $k$6, and the multiset of exponents $k$7 is an isomorphism invariant. Second, they compute exactly the kernel and cokernel of the Frobenius-adjunction morphism $k$8: both vanish when $k$9, and both have $p$0-dimension $p$1 otherwise. These dimensions feed directly into the Chern character computations at the nodes.

The genus formula

Let $p$2 denote the canonical representative of a radius class in $p$3. The main theorem states that, for nonempty $p$4,

$p$5

Moreover, if all radii satisfy $p$6, the formula simplifies to

$p$7

The proof computes the Chern character of $p$8 for the adjoint dormant $p$9-oper associated to the universal family, using the Kodaira–Spencer isomorphism to identify this bundle with the log tangent bundle of the curve. Contributions arise from three sources: the marked points (psi-class terms computed via residue isomorphisms), the nodes (the node-local dimensions above), and the blow-up comparison against $\mathrm{PGL}_2^{(N)}$0, where the cokernel of the Frobenius pushforward of $\mathrm{PGL}_2^{(N)}$1 contributes terms supported on exceptional divisors. The simplified formula follows by comparing the general formula at levels $\mathrm{PGL}_2^{(N)}$2 and $\mathrm{PGL}_2^{(N)}$3, exploiting the isomorphism between consecutive levels.

Two corollaries quantify the result. The genus satisfies the uniform upper bound

$\mathrm{PGL}_2^{(N)}$4

and the number of critical points of $\mathrm{PGL}_2^{(N)}$5 admits an explicit upper bound derived from Riemann–Hurwitz. Worked examples illustrate the formulas: for $\mathrm{PGL}_2^{(N)}$6, $\mathrm{PGL}_2^{(N)}$7, and $\mathrm{PGL}_2^{(N)}$8, one obtains $\mathrm{PGL}_2^{(N)}$9 (a non-rational elliptic curve), while the simplified formula would incorrectly give $\mathrm{PGL}_2$0 — confirming that its hypothesis is genuinely necessary. For equal radii $\mathrm{PGL}_2$1 with $\mathrm{PGL}_2$2, the genus evaluates to $\mathrm{PGL}_2$3.

Asymptotic behavior of the function field towers

For a compatible system of radii $\mathrm{PGL}_2$4, the function fields $\mathrm{PGL}_2$5 form a tower (finite separable extensions, since $\mathrm{PGL}_2$6 is generically étale), provided the genera tend to infinity. The authors introduce the notion of an asymptotically $\mathrm{PGL}_2$7-good tower, requiring $\mathrm{PGL}_2$8; asymptotic goodness in the classical sense corresponds to $\mathrm{PGL}_2$9.

The main asymptotic theorem fixes an integer $(N-1)$0 with $(N-1)$1 and considers radii lying in $(N-1)$2, where $(N-1)$3 imposes a uniform separation condition on the base-$(N-1)$4 digits of the radii. For such $(N-1)$5, the tower satisfies

$(N-1)$6

hence is asymptotically $(N-1)$7-good. The proof combines the genus upper bound with the lower bound $(N-1)$8 coming from the digit-separation condition. Note that $(N-1)$9 always, so these towers are not shown to be asymptotically good in the classical sense.

When $p^N$0, taking the symmetric choice $p^N$1 yields $p^N$2 and genus $p^N$3, giving

$p^N$4

More generally, any radii with $p^N$5 for large $p^N$6 produce $p^N$7-good towers.

Relation to Heun equations

An appendix connects level-$p^N$8 dormant $p^N$9-opers on the $\mathrm{PGL}_2$00-pointed projective line to Heun's differential operators. The authors prove that every dormant $\mathrm{PGL}_2$01-oper on $\mathrm{PGL}_2$02 arises from a Heun operator with a full set of root functions, and classify the redundancy: two parameter vectors determine the same oper precisely when the squares of the four exponent differences agree and one additional linear combination involving $\mathrm{PGL}_2$03 agrees. This identifies the radii appearing in the genus formula with classical Heun exponent parameters.

Limitations and open questions

Several restrictions are stated plainly in the paper. The genus computation for the cokernel term is carried out only for $\mathrm{PGL}_2$04; the analogous calculation for $\mathrm{PGL}_2$05 involves more intricate intersection theory on stacks and is left incomplete. The asymptotic analysis depends on the digit-separation condition defining $\mathrm{PGL}_2$06, and which patterns of radii yield asymptotically good (i.e., $\mathrm{PGL}_2$07-good) towers has not been determined. The authors conjecture that $\mathrm{PGL}_2$08 for every odd prime $\mathrm{PGL}_2$09, but prove only the lower bound. Whether any dormant modular tower attains the Drinfeld–Vlăduţ bound, as classical modular towers do, remains open, as does the extension of the framework to higher-rank groups $\mathrm{PGL}_2$10 and the study of zeta functions of these towers.

Conclusion

The paper supplies the first explicit genus formulas for moduli spaces of higher-level dormant opers on $\mathrm{PGL}_2$11-pointed stable rational curves, together with quantitative bounds showing that the associated function field towers are asymptotically $\mathrm{PGL}_2$12-good with $\mathrm{PGL}_2$13 up to $\mathrm{PGL}_2$14. The methods — local classification of finite-level $\mathrm{PGL}_2$15-modules at nodes, Frobenius-pullback comparisons, and Chern character computations tied to the combinatorics of balanced edge numberings — provide a template for computing global invariants of dormant-oper moduli spaces, and connect the theory to the anticipated $\mathrm{PGL}_2$16d TQFT structure and to enumerative questions beyond the cases treated here.