论文标题
ohno-nakagawa类型的反射定理，用于四分之一的戒指和$ n $ y-二次形式的对
Reflection theorems of Ohno-Nakagawa type for quartic rings and pairs of $n$-ary quadratic forms
论文作者
论文摘要
我们证明了由Nakagawa和Ohno猜想的反射定理，对于四分之一的环或成对的三元二次形式，并具有给定的立方分辨率。超过$ \ mathbb {z} $，我们的结果是无条件的；我们还允许基础是一般数字场的整数环，条件是在蒙特卡洛验证的某些代数身份上。我们还为四分之一的$ 11111 $ - forms和$ 48441 $ - forms建立了一个反射定理，将它们与给定特征多项式的$ 3 \ times 3 $对称矩阵相关联。一路上，我们发现了圆锥体的Igusa Zeta函数的优雅新结果，以及在本地字段中盒子上二次字符的平均值。 我们推测，反射定理适用于任何奇数$ n $的$ n $ - 二次二次表格对，我们证明了这是奇数cubefree Indientant的。对于科恩，迪亚兹·迪亚兹（Diaz Y Diaz）和奥利维尔（Olivier）提出的一个问题，这给了一个更令人满意的答案，即是否存在一个无限的反思定理家族。
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.
