{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/reflection-theorems-of-ohno-nakagawa-type-for","title":"Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms","arxiv_id":"2204.10924","date":"2022-04-22","proceeding":null,"authors":["Evan M. O'Dorney"],"abstract":"We prove a reflection theorem, conjectured by Nakagawa and Ohno, for the number of quartic rings, or pairs of ternary quadratic forms, with a given cubic resolvent. Over $\\mathbb{Z}$, our results are unconditional; we also allow the base to be the ring of integers of a general number field, conditional on some algebraic identities that are Monte Carlo verified. We also establish a reflection theorem for quartic $11111$-forms and $48441$-forms that relates them to the number of $3\\times 3$ symmetric matrices with given characteristic polynomial. Along the way, we find elegant new results on Igusa zeta functions of conics and the average value of a quadratic character over a box in a local field. We conjecture that a reflection theorem holds for pairs of $n$-ary quadratic forms for any odd $n$, and we prove this for odd cubefree discriminant. This furnishes a more satisfactory answer for a question raised by Cohen, Diaz y Diaz, and Olivier, namely whether there exist an infinite family of reflection theorems of Ohno-Nakagawa type.","url_abs":"https://arxiv.org/abs/2204.10924v2","url_pdf":"https://arxiv.org/pdf/2204.10924v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"reflection-theorems-of-ohno-nakagawa-type-for","repo_url":"https://github.com/emo916math/Quartic-Ohno-Nakagawa","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}