Ohno-Nakagawa型的复合簇与反射定理
Composed varieties and reflection theorems of Ohno-Nakagawa type
AI总结：
本文借鉴Tate论文的傅里叶分析，利用adelic上同调提出新方法，证明并推广Ohno-Nakagawa反射定理，得到数域上三次、四次型及二次型的新O-N型反射定理，兼具理论与分析应用价值。
摘要：
Ohno-Nakagawa（O-N）反射定理是一个出人意料简洁的恒等式，它关联了两个不同判别式$D$、$-27D$的二元三次型（等价于三次环）的$\text{GL}_2(\boldsymbol{Z})$类的数量，该定理推广了三次互反律和Scholz反射定理。本文中，我们借鉴Tate论文中著名的$\boldsymbol{A}_K$上的傅里叶分析，利用有限伽罗瓦模的adelic上同调$H^1(\boldsymbol{A}_K, M)$上的傅里叶分析，提出了证明和推广该定理的新方法。该方法通过局部域上格计数的恒等式产生O-N型反射定理，得到了任意数域上三次型与三次环、四次型与四次环，以及按特殊不变量——超判别式$a(b^2 - 4ac)$计数的二次型的新O-N型反射定理。O-N型定理兼具内在美感与分析应用价值，其应用包括约化判别式恒等式、函数方程的对称化以及Shintani ζ函数的次凸性。
英文摘要：
The Ohno-Nakagawa (O-N) reflection theorem is an unexpectedly simple identity relating the number of $\mathrm{GL}_2 \mathbb{Z}$-classes of binary cubic forms (equivalently, cubic rings) of two different discriminants $D$, $-27D$; it generalizes cubic reciprocity and the Scholz reflection theorem. In this paper, we present a new approach to proving and generalizing this theorem using Fourier analysis on the adelic cohomology $H^1(\mathbb{A}_K, M)$ of a finite Galois module, modeled after the celebrated Fourier analysis on $\mathbb{A}_K$ used in Tate's thesis. This method produces reflection theorems of O-N type from identities of lattice counts over local fields and yields new reflection theorems of O-N type for cubic forms and rings over arbitrary number fields, quartic forms and rings, and also quadratic forms counting by a peculiar invariant, the \emph{superdiscriminant} $a(b^2 - 4ac)$. O-N-type theorems are valuable both for their intrinsic beauty and for analytic applications, which include discriminant-reducing identities, symmetrization of functional equations, and subconvexity for Shintani zeta functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
