Parity of $k$-differentials in genus zero and one
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910010539245568 |
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| author | Chen, Dawei Chen, Evan Lau, Kenny Ono, Ken Zhang, Jujian |
| author_facet | Chen, Dawei Chen, Evan Lau, Kenny Ono, Ken Zhang, Jujian |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03722 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Parity of $k$-differentials in genus zero and one Chen, Dawei Chen, Evan Lau, Kenny Ono, Ken Zhang, Jujian Number Theory Algebraic Geometry Geometric Topology 11A07 (Primary) 14H10, 32G15 (Secondary) 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). |
| title | Parity of $k$-differentials in genus zero and one |
| topic | Number Theory Algebraic Geometry Geometric Topology 11A07 (Primary) 14H10, 32G15 (Secondary) |
| url | https://arxiv.org/abs/2602.03722 |