论文标题
二次空间三元的傅立叶变换
The Fourier transform for triples of quadratic spaces
论文作者
论文摘要
令$ v_1，v_2，v_3 $为一个数字矢量空间的三重数字$ f $，配备了非等级二次二次形式$ \ mathcal {q} _1，\ mathcal {q} _2，_2，_2，\ mathcal {q} _3 $ _3 $。令$ y \ subset \ prod_ {i = 1}^3 v_i $为封闭的子顺序，由$（v_1，v_2，v_3）$组成，这样$ \ mathcal {q} _1（v_1）= \ \ m nathcal {q}一个人在适当的假设上对所涉及功能具有泊松求和公式，但相关的傅立叶变换以前仅被定义为对应关系。在当前的论文中，我们采用了一个新颖的全球到本地论点来证明，这种傅立叶变换在$ y（\ mathbb {a} _f）的Schwartz空间上得到了很好的定义。$ $ $以执行全球到本地的论点，我们引入边界术语，从而将Poisson求发公式扩展到更广泛的测试功能。这是第一次证明具有边界项的总和公式，而不是Braverman-Kazhdan空间的球形品种。
Let $V_1,V_2,V_3$ be a triple of even dimensional vector spaces over a number field $F$ equipped with nondegenerate quadratic forms $\mathcal{Q}_1,\mathcal{Q}_2,\mathcal{Q}_3$, respectively. Let $Y \subset \prod_{i=1}^3 V_i$ be the closed subscheme consisting of $(v_1,v_2,v_3)$ such that $\mathcal{Q}_1(v_1)=\mathcal{Q}_2(v_2)=\mathcal{Q}_3(v_3)$. One has a Poisson summation formula for this scheme under suitable assumptions on the functions involved, but the relevant Fourier transform was previously only defined as a correspondence. In the current paper we employ a novel global-to-local argument to prove that this Fourier transform is well-defined on the Schwartz space of $Y(\mathbb{A}_F).$ To execute the global-to-local argument, we introduce boundary terms and thereby extend the Poisson summation formula to a broader class of test functions. This is the first time a summation formula with boundary terms has been proven for a spherical variety that is not a Braverman-Kazhdan space.
