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

TL;DR

The paper settles the problem of spin parity for odd $k$-differentials on genus $0$ and $1$ by proving a number-theoretic conjecture that reexpresses the parity condition through Jacobi symbols. It reduces the core argument to a combinatorial floor-sum identity, culminating in the key relation $N_k(n)=F_k(n+1)-F_k(n)$ and the parity evaluation $F_k(a)\equiv 0$ or $\left\lfloor\frac{k+1}{4}\right\rfloor \pmod{2}$ depending on the parity of $a$, which together yield $N_k(n)\equiv \left\lfloor\frac{k+1}{4}\right\rfloor \pmod{2}$ when $\gcd(n,k)=\gcd(n+1,k)=1$. This unconditionally determines the spin parity for genus $0$ and $1$ via $n_k(\mu)$ (and, in genus one, via $n_k(\mu)+d+1$ for rotation number $d$). The authors also relate $n_k(\mu)$ to Jacobi-symbol data and include an AI-assisted formal verification in Lean via AxiomProver, underscoring a novel workflow for mathematical proofs and their verification. This completes the conditional results from CG22 and provides a fully explicit spin-parity classification for the targeted genera and odd $k$.

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).

Theorems & Definitions (16)

• Conjecture 1.1: CG22
• Theorem 1.2
• Theorem 1.3: CG22
• Theorem 1.4: CG22
• Lemma 2.1: Eisenstein Tan00
• Remark 1
• Lemma 2.2: Gauss--Schering Jenkins1867
• Lemma 2.3: Supplementary law for $(\frac{2}{k})$ Tan00
• Lemma 2.4
• proof