Polynomiality of the Generalized Verschiebung Degree

Abstract: For a general curve in positive characteristic, taking the Frobenius pullback induces a generically finite rational map V on the moduli space of rank 2 vector bundles with trivial determinant. Recently, Kondo--Wakabayashi show that the generic degree of V, considered as a function on the characteristic of the base field, is a quasi-polynomial. In this paper, we show that this quasi-polynomial is indeed a polynomial, and we write out this polynomial explicitly.

• The paper proves that the degree of the generalized Verschiebung map on the moduli space of rank 2 vector bundles with trivial determinant is a polynomial in the characteristic, valid for any genus g ≥ 2.
• The derivations count dormant opers and provide an explicit polynomial ρ-g(p2) describing the degree for various genera.
• The method involves a level-reduction theorem and a combinatorial bijection.
• Followup questions to facilitate further research and understanding.

Overview

This paper, by Siqing Zhang, resolves a question left open by Kondo–Wakabayashi concerning the generic degree of the generalized Verschiebung map on the moduli space of rank 2 vector bundles with trivial determinant. For a smooth curve $X$ of genus $g \geq 2$ over an algebraically closed field of characteristic $p > 2$, Frobenius pullback $E \mapsto F_X^*E$ induces a generically finite rational map $V: M_2(X') \dashrightarrow M_2(X)$ between moduli spaces on $X'$ and $X$. Kondo–Wakabayashi had shown that $\deg(V)$, as a function of $p$, is a quasi-polynomial of degree $3g-3$, and verified polynomiality only for $g \geq 2$0 by direct computation. The main result here establishes that the quasi-polynomial is in fact a genuine polynomial for all $g \geq 2$1, and gives it explicitly:

$g \geq 2$2

where $g \geq 2$3 is a rational-coefficient polynomial built from Bernoulli numbers and Laurent coefficients of $g \geq 2$4 at $g \geq 2$5.

Main results

The paper contains two theorems. Theorem (Polynomiality) states that for a general curve $g \geq 2$6 of genus $g \geq 2$7,

$g \geq 2$8

with $g \geq 2$9 the Bernoulli numbers. This recovers Osserman's and Lange–Pauly's genus 2 value $p > 2$0 as a special case, and extends Kondo–Wakabayashi's $p > 2$1 computation to arbitrary genus without case-by-case verification.

Theorem (Level-reduction) is the combinatorial engine behind this formula. Kondo–Wakabayashi express the degree as a ratio of cardinalities of edge-numbering sets,

$p > 2$2

for any connected trivalent graph $p > 2$3 of genus $p > 2$4, where $p > 2$5 consists of labelings of edges by triples satisfying certain inequalities modulo powers of $p > 2$6. Zhang proves a natural bijection

$p > 2$7

for odd $p > 2$8. Applied to the degree formula, this yields the striking simplification $p > 2$9: the generic degree in characteristic $E \mapsto F_X^*E$0 equals a level-1 count at characteristic $E \mapsto F_X^*E$1. The author notes the amusing heuristic that $E \mapsto F_X^*E$2 agrees with the number of dormant opers over a "field of characteristic $E \mapsto F_X^*E$3", which does not exist.

Method of proof

The level-reduction theorem rests on a local factorization lemma: for odd $E \mapsto F_X^*E$4 and $E \mapsto F_X^*E$5, an explicit map $E \mapsto F_X^*E$6 built coordinatewise from a function $E \mapsto F_X^*E$7 (distinguishing parity of $E \mapsto F_X^*E$8) restricts to a bijection $E \mapsto F_X^*E$9. The proof analyzes four linear inequalities characterizing membership via quantities $V: M_2(X') \dashrightarrow M_2(X)$0, splitting according to the parity pattern of the triple; surjectivity uses the fact that coordinates of elements of $V: M_2(X') \dashrightarrow M_2(X)$1 are bounded by $V: M_2(X') \dashrightarrow M_2(X)$2, forcing uniqueness of the inverse. Gluing these bijections edge-wise over $V: M_2(X') \dashrightarrow M_2(X)$3 gives the global result.

For polynomiality, the argument proceeds by choosing $V: M_2(X') \dashrightarrow M_2(X)$4 assembled from two-vertex blocks and computing $V: M_2(X') \dashrightarrow M_2(X)$5 as $V: M_2(X') \dashrightarrow M_2(X)$6 for a transfer matrix $V: M_2(X') \dashrightarrow M_2(X)$7, where $V: M_2(X') \dashrightarrow M_2(X)$8 counts admissible interior labels of a block. The matrix $V: M_2(X') \dashrightarrow M_2(X)$9 recording whether $X'$0 resembles the even part of the $X'$1 fusion ring, and admits a finite analogue of the Verlinde formula: with an explicit orthogonal matrix $X'$2 of sine entries,

$X'$3

Diagonalizing then gives $X'$4 and hence

$X'$5

Zagier's evaluation of power sums of cosecants converts this trigonometric sum into the stated polynomial. Notably, the paper supplies an elementary derivation of the even-$X'$6 counting formula for $X'$7, complementing Wakabayashi's earlier derivation of the odd-$X'$8 formula via Quot schemes and Gromov–Witten theory; the present proof assumes only basic enumerative combinatorics.

Consequences for dormant opers

Recall that a dormant oper is a $X'$9-oper with vanishing $X$0-curvature, equivalently one that is étale locally isomorphic to the trivial $X$1-bundle with its canonical connection. Wakabayashi showed that for a general curve $X$2 with a lift $X$3 over the Witt ring $X$4, the number of dormant opers over $X$5 equals $X$6. Combining this with level-reduction yields a clean multiplicative structure: if $X$7 and $X$8 denote the sets of isomorphism classes of dormant opers over $X$9 and $\deg(V)$0 respectively, then

$\deg(V)$1

Thus the count of higher-level dormant opers over a lift is governed entirely by the Verschiebung degree, which is now known to be polynomial in $\deg(V)$2.

Verification and limitations

A notable feature is formalization: the proofs of the local factorization lemma, the level-reduction theorem, and the trigonometric lemma have been verified in Lean 4 in a public repository, providing machine-checked assurance for the combinatorial core.

Several qualifications apply. The results concern general curves; behavior for special curves is not addressed. The characteristic restriction $\deg(V)$3 is essential throughout, and the level-reduction bijection requires $\deg(V)$4 odd — the case $\deg(V)$5 with $\deg(V)$6 odd is what makes the application to $\deg(V)$7 work. The identification $\deg(V)$8 depends on Kondo–Wakabayashi's graph-theoretic expression for the degree, which itself relies on their prior work rather than being reproved here. Finally, while the paper notes the resemblance of $\deg(V)$9 to the even fusion ring, no categorical or TQFT interpretation of the finite Verlinde-type formula is developed; establishing such a structural explanation remains open, as does any geometric account of why the "characteristic $p$0" heuristic should be more than accidental.

Conclusion

The paper establishes that the generic degree of the generalized Verschiebung map is given by an explicit polynomial $p$1 in the characteristic, upgrading the quasi-polynomiality of Kondo–Wakabayashi to full polynomiality for all genera. The key input is a level-reduction bijection reducing $p$2-edge-numberings to products of level-1 counts, proved by an elementary but delicate parity analysis, together with a finite Verlinde-style diagonalization of the resulting transfer matrix. As a corollary, counts of dormant opers over lifts to Witt vectors factor as $p$3, tying the arithmetic geometry of opers directly to the Verschiebung degree.