Parity of $k$-differentials in genus zero and one

Abstract: Here we completely determine the spin parity of $k$-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appendix).

• The paper determines spin parity of k-differentials unconditionally for genus zero and one using combinatorial and number-theoretic techniques.
• It reduces parity computation via cyclic canonical covers and connects the invariants to Jacobi symbols through explicit residue counts.
• The work validates its methodology with AI-enabled formal proof verification, offering new insights for moduli space analysis.

Parity of $k$-Differentials in Genus Zero and One

Context and Problem Formulation

The study of $k$-differentials—global sections of $K_X^k$ for a Riemann surface $X$ of genus $g$—is central in moduli theory, flat surface dynamics, and combinatorial geometry. The moduli spaces $\Omega^k\mathcal{M}_g(\mu)$, where $\mu$ specifies prescribed orders of zeros and poles summing to $k(2g-2)$, have a rich structure: in many cases, they are disconnected, and their connected components can be distinguished by discrete invariants, such as spin parity. For $k\geq 3$, especially for odd $k$, the classification of connected components is largely unresolved, even for low genus.

For holomorphic ($k=1$) and quadratic ($k=2$) differentials, spin invariants play a critical role in component stratification [KZ03, La04S]. This paper focuses on the spin parity of $k$-differentials in genus zero and one for arbitrary $k$, completing a program previously conditional on a number-theoretic conjecture [CG22]. The principal achievement is an unconditional determination of spin parity for all $k$ in these genera by resolving the underlying arithmetic disparity.

Theoretical Contributions

Canonical Covers and Spin Parity Reduction

For $k>1$, the canonical construction of the cyclic cover $\pi:\widehat{X}\to X$ of degree $k$ allows reduction of the spin parity computation to the parity of an associated differential one-form $\widehat{\omega}$ on $\widehat{X}$. The parity is captured via the dimension modulo 2 of sections of half-canonical divisors, generalizing classical theta characteristics [At71, Mu71]. For $k$ odd, parity type occurs precisely when all entries of $\mu$ are even.

Number-Theoretic Reformulation

Chen and Gendron reduced the parity calculation to a combinatorial count $N_k(n)$ for odd $k$ and coprime $n$: the number of pairs $(b_1, b_2)$ under explicit congruence and sum constraints. The conjecture stated that

$$N_k(n) \equiv \left\lfloor\frac{k+1}{4}\right\rfloor \pmod{2}.$$

This paper proves the conjecture by recognizing a connection to Jacobi symbols and established number-theoretic identities, specifically Eisenstein's Lemma and the Gauss–Schering formula for counting residue sets [Tan00, Jenkins1867]. The floor-sum function $F_k(a)$, parameterized by an integer $a$, is shown to encode the parity via

$$F_k(a) \equiv \begin{cases} 0 \pmod{2}, & a \text{ odd} \ \left\lfloor \frac{k+1}{4} \right\rfloor \pmod{2}, & a \text{ even} \end{cases}$$

for all $a$ coprime to $k$.

The parity function $N_k(n)$ is further linked to Jacobi symbols $\left(\frac{2}{k}\right)$ by explicit combinatorial decomposition and summation arguments. The proof employs both analytic and formal verification techniques, including machine-verified formalizations in Lean.

Main Results

The core unconditional theorems formalized in the paper are:

• For genus zero and odd $k$, spin parity of $\Omega^k\mathcal{M}_0(2\mu)$ is given by $n_k(\mu)\pmod{2}$, where $n_k(\mu)$ counts components indexed by certain $q$-adic valuations arising in the prime factorization of $k$.
• For genus one and odd $k$, spin parity of the connected component of rotation number $d$, $\Omega^k\mathcal{M}_1(2\mu)^d$, is determined by $n_k(\mu) + d + 1 \pmod{2}$.

Moreover, the number-theoretic function $n_k(\mu)$ admits a reformulation in terms of Jacobi symbols: $$n_k(\mu) = \#\left\{ i : \left(\frac{2}{d_i}\right) \neq \left(\frac{2}{k}\right) \right\}$$ where $d_i = \gcd(k, m_i)$.

Strong Claims and Numerical Verification

The paper establishes the parity count $N_k(n)$ for all odd $k$ and suitable $n$ and $n+1$ unconditionally, eliminating the reliance on numerical evidence. The construction is validated both via analytic number theory and automated formal proof using the AxiomProver system in Lean, with all combinatorial core lemmas verified by symbolic computation.

Practical and Theoretical Implications

These results remove longstanding conditionality from the stratification program for moduli spaces of $k$-differentials in low genus, enabling practitioners to compute spin invariants for genus zero and one in complete generality. For $k\geq 3$, this method provides the only known invariant besides hyperelliptic and rotation numbers capable of distinguishing moduli space components, substantially advancing the analysis in open problems on strata structure [BCGGMk, CG22].

On the theoretical side, the connection to Jacobi symbols and parity periodicity modulo 8 suggests possible approaches to parity problems for $k$-differentials in higher genus, and raises prospects for further arithmetic-geometric correspondences in moduli theory. Furthermore, formalization via AI proof verification introduces new rigor and opens the possibility of systematic exploration of conjectures in algebraic geometry using computational methods.

Practically, these techniques may generalize to automated classification of moduli space components in more intricate settings, including holomorphic and meromorphic $k$-differentials in arbitrary genus.

Future Directions in AI-Driven Mathematical Research

The success of AxiomProver in autonomously formulating, refactoring, and verifying a nontrivial combinatorial proof exemplifies the evolving role of AI in mathematical research. While standard number-theoretic facts were invoked from classical sources rather than rederived, the modular workflow and capacity for Lean-based validation indicate potential for fully autonomous reasoning and formalization in future conjecture-driven programs. Scaling such systems could impact research workflows across algebraic geometry, combinatorics, and dynamics, providing new methods of rigor and discovery.

Conclusion

This paper provides a complete, unconditional determination of spin parity for $k$-differentials in genus zero and one, utilizing number-theoretic identities and formal proof verification techniques. By establishing the explicit parity count and formulating the accompanying combinatorial framework, it advances both the mathematical theory of moduli space stratification and proves the efficacy of AI-assisted research workflows in pure mathematics.

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References

Relevant source literature includes (2602.03722), [CG22], [KZ03], [La04S], [BCGGMk], [Tan00], [Jenkins1867], [At71], [Mu71], [Jo80], and computational formalization platforms [Lean, Mathlib2020].