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

TL;DR

This paper fully determines the spin parity of k-differentials on genus zero and one Riemann surfaces, resolving a previously conditional result by proving a key number-theoretic hypothesis through combinatorial and algebraic methods.

Contribution

It proves a conjecture relating to spin parity of k-differentials by establishing a number-theoretic hypothesis using Jacobi symbols and formal proof systems.

Findings

Complete classification of spin parity for genus zero and one

Proof of the number-theoretic hypothesis via Jacobi symbols

Formal verification of the proof in Lean/Mathlib

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).