关于局部密度公式和Gross-Keating不变量及其附录“二元二次型的局部密度”。Ikeda和H.桂田
On the local density formula and the Gross-Keating invariant with an Appendix `The local density of a binary quadratic form' by T. Ikeda and H. Katsurada
摘要：
T. Ikeda和H. Katsurada在他们最近的论文[IK 1]和[IK 2]中发展了二次型的Gross-Keating不变量理论。特别地，他们证明了Siegel-Eisenstein级数的傅里叶系数的局部因子完全由Gross-Keating不变量确定，该不变量具有额外的数据，称为扩展的GK数据，在[IK 2]中。另一方面，这种局部因子是两个二次型的局部密度的一种特殊情况。因此，我们提出了一个一般性的问题，如果局部密度可以确定的某些系列的Gross-Keating不变量和扩展的GK基准。本文证明了对于定义在$\mathbb{Z}_2$的非分歧有限扩张上的单个二次型的局部稠密性，这个问题的答案是肯定的。在附录中，T. Ikeda和H. Katsurada计算了定义在$\mathbb{Z}_2$的任意有限扩张上的单个二元二次型的局部密度公式。
英文摘要：
T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariants and the extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of $\mathbb{Z}_2$.
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