Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms
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| Format: | Preprint |
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2022
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| _version_ | 1866913881918537728 |
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| author | O'Dorney, Evan M. |
| author_facet | O'Dorney, Evan M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_10924 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms O'Dorney, Evan M. Number Theory 11R16, 11A15, 11E12, 11E08, 11R54 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. |
| title | Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms |
| topic | Number Theory 11R16, 11A15, 11E12, 11E08, 11R54 |
| url | https://arxiv.org/abs/2204.10924 |