标题： 关于姜和Rallis的一个定理 显示英文标题

标题： On a Theorem of Jiang and Rallis

摘要： 蒋和拉利斯（1997）定义了一个与三次多项式相关的局部积分族，并证明了在非阿基米德局部域 $F$上它们的显式求值，当 $F$包含三个三次单位根，或者定义多项式是可约的。 对 $F$的限制使他们能够将形式为 $x^3-a$的不可约多项式的案例进行简化。 普莱索（2009）开始去除对$F$的限制，通过将积分表示为三次多项式$x^3 - b x - c$的$16$个积分之和，其中$b,c\in F$，并计算了其中的九个。 在本工作中，我们计算了普莱索积分的$15$，并将最后的结果简化为关于有限域上曲面点数的一个基本断言，在$F$是$p$有理数、$F=\mathbb{Q}_p$和$p$与$5$模$6$等价的特殊情况下。我们的计算在该特殊情况下基本上完成了普莱索的工作。在此期间，熊（2020）通过一种完全不同的方法计算了任意非阿基米德局部域的积分。我们的直接方法可能更容易扩展到使用五次多项式定义的高阶情形下的类似积分。 显示英文摘要

摘要： Jiang and Rallis (1997) defined a family of local integrals attached to a cubic polynomial and proved explicit evaluations of them over a non-archimedean local field $F$, when either $F$ contains three third roots of unity, or the defining polynomial is reducible. The restriction on $F$ allowed them, among other things, to reduce the case of irreducible polynomials of the form $x^3-a$. Pleso (2009) began the work of removing the restriction on $F$ by expressing the integral as a sum of $16$ integrals for the cubic polynomial $x^3 - b x - c$ with $b,c\in F$, and computing nine of them. In this work, we compute $15$ of Pleso's integrals, and reduce the last to an elementary assertion about the number of points on a surface over a finite field, in the special case when $F$ is the $p$-adic numbers, $F=\mathbb{Q}_p$, and $p$ is equivalent to $5$ mod $6$. Our computations essentially complete Pleso's work in that special case. In the interim, Xiong (2020) has computed the integrals for an arbitrary non-archimedean local field by a totally different approach. Our direct approach might be more extendable to analogous integrals defined using quintic polynomials, in a higher-rank setting.