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

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).

💡 Research Summary

The paper resolves a long‑standing number‑theoretic conjecture that underlies the determination of spin parity for odd‑order k‑differentials on genus‑zero and genus‑one Riemann surfaces. A k‑differential ξ is a section of the k‑th power of the canonical bundle; its zero and pole orders are encoded by a tuple μ=(m₁,…,m_n) satisfying ∑m_i = k(2g−2). The moduli space Ω_k M_g(μ) can be disconnected, and the spin parity—defined via the theta‑characteristic of the associated flat surface—is the only known invariant that distinguishes its connected components when k is odd.

Earlier work (particularly the Appendix of