Polynomiality of the Generalized Verschiebung Degree

Technical Summary: Polynomiality of the Generalized Verschiebung Degree

Problem Statement
Let $X$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of characteristic $p > 2$. The generalized Verschiebung map $V: M_2(X') \dashrightarrow M_2(X)$ is a generically finite rational map between moduli spaces of rank 2 vector bundles with trivial determinant, induced by the Frobenius pullback $E \mapsto F_X^* E$. A central question in the study of this map is determining its generic degree, $\deg(V)$.

Previous work by Kondo–Wakabayashi established that $\deg(V)$ can be expressed as the ratio of the cardinalities of two specific sets of edge-numberings on a trivalent graph $G$ of genus $g$: $\deg(V) = | \text{Ed}_{p,2,G} | / | \text{Ed}_{p,1,G} |$. Furthermore, they demonstrated that as a function of the characteristic $p$, this degree is a quasi-polynomial of degree $3g-3$. While they proved via direct computation that this quasi-polynomial is actually a polynomial for the specific case of genus $g=3$, the general case for arbitrary $g \ge 2$ remained open. This paper addresses the question of whether $\deg(V)$ is a polynomial in $p$ for all genera and, if so, provides an explicit formula.

Methodology
The paper employs a combination of combinatorial geometry and enumerative combinatorics to resolve the problem. The methodology proceeds in three main stages:

1. Combinatorial Reduction (Level-Reduction):
   The core technical innovation is a new combinatorial result concerning the sets $\text{Ed}_{P,N,G}$, which consist of edge-numberings on a trivalent graph $G$ constrained by local conditions at vertices defined by a set $\dagger C_N(P)$. The author proves a "Level-Reduction" theorem (Theorem 2), establishing a natural bijection:
   $$\text{Ed}_{P,N,G} \cong \text{Ed}_{P,1,G} \times \text{Ed}_{\frac{N-1}{2}P, 1, G}$$
   for odd $P \ge 3$ and $N \ge 1$. This result relies on a "local factorization lemma" (Lemma 7) that constructs a bijection between the local constraint sets $\dagger C_N(P)$ and the product $\dagger C_1(2P) \times \dagger C_{N-1}(P)$.

   Applying this to the specific case of the Verschiebung degree (where $P=p$ and $N=2$), the paper reduces the problem to calculating $|\text{Ed}_{2p,1,G}|$, effectively eliminating the denominator $|\text{Ed}_{p,1,G}|$ from the expression.

2. Transfer Matrix and Diagonalization:
   To compute $|\text{Ed}_{2p,1,G}|$, the author models the counting problem using transfer matrices. By constructing a graph $G$ from blocks and defining a transition matrix $T$ based on the number of valid labelings for local configurations, the total count is expressed as the trace of a matrix power: $|\text{Ed}_{2p,1,G}| = \text{Tr}(T^{g-1})$.

   The matrix $T$ is diagonalized using an orthogonal matrix $S$ constructed from trigonometric sums involving sine functions. This step transforms the combinatorial counting problem into a spectral problem, allowing the trace to be computed as a sum of eigenvalues.

3. Explicit Polynomial Derivation:
   The resulting trace formula involves a sum of cosecant powers. The author utilizes a known formula by Zagier regarding the sums of powers of cosecants to express this sum explicitly in terms of Bernoulli numbers and coefficients of Laurent expansions. This step confirms that the resulting expression is a polynomial in $p$ rather than a quasi-polynomial.

Key Contributions and Results

• Theorem 1 (Polynomiality): The paper proves that for a general curve of genus $g \ge 2$, the generic degree $\deg(V)$ is strictly a polynomial in $p$ (specifically, a polynomial in $p^2$ scaled by $p^{g-1}$). The explicit formula is given as:
  $$\deg(V) = p^{g-1} R_g(p^2)$$
  where $R_g(t) \in \mathbb{Q}[t]$ is defined by:
  $$R_g(t) = 1 - \frac{1}{2g} + \frac{1}{2g} \sum_{r=0}^{g-1} (-1)^{r-1} \frac{2^{4r} B_{2r}}{(2r)!} [z^{-2r}] \csc^{2g-2}(z) \, t^r$$
  Here, $B_{2r}$ denotes the $2r$-th Bernoulli number, and $[z^{-2r}]$ denotes the coefficient of $z^{-2r}$ in the Laurent expansion of $\csc^{2g-2}(z)$ at $z=0$.

• Theorem 2 (Level-Reduction): The paper establishes the bijection $\text{Ed}_{P,N,G} \cong \text{Ed}_{P,1,G} \times \text{Ed}_{\frac{N-1}{2}P, 1, G}$. This combinatorial result is of independent interest, as it reduces higher-level edge-numbering problems to level-1 counterparts, simplifying the enumeration of dormant opers.

• Explicit Formula for Even $P$: The paper provides an elementary derivation for the number of level-1 edge-numberings $|\text{Ed}_{P,1,G}|$ when $P$ is even, complementing existing results for odd $P$:
  $$|\text{Ed}_{P,1,G}| = \left(\frac{P}{2}\right)^{g-1} + \left(\frac{P}{4}\right)^{g-1} \sum_{\theta=1}^{P/2-1} \csc^{2g-2}\left(\frac{\pi\theta}{P}\right)$$

• Formalization: The combinatorial proofs (Lemma 7, Theorem 2, and Lemma 10) have been formalized and verified in Lean 4.

Significance and Claims
The paper claims to resolve the question of the polynomiality of the generalized Verschiebung degree for arbitrary genus, moving beyond the specific case of $g=3$ previously verified by Kondo–Wakabayashi. By providing an explicit polynomial formula, the work removes the ambiguity of the quasi-polynomial nature suggested by Ehrhart theory in this context.

The author notes a heuristic observation: the generic degree $\deg(V)$ in characteristic $p$ coincides with the number of dormant opers over a hypothetical "field of characteristic $2p$." This is derived from the relation $\deg(V) = |\text{Ed}_{2p,1,G}| / |\text{Ed}_{p,1,G}|$ combined with the level-reduction bijection, which implies $|\text{Ed}_{2p,1,G}| = |\text{Ed}_{p,1,G}| \cdot |\text{Ed}_{p,1,G}|$ is not the case, but rather the reduction leads to $\deg(V) = |\text{Ed}_{2p,1,G}|$ (after correcting the denominator logic in the specific context of the paper's derivation).

The paper does not propose new applications or future experimental directions but focuses on the rigorous derivation of the polynomial structure and the explicit coefficients, grounding the result in the enumerative geometry of dormant opers and the combinatorics of trivalent graphs.