Zeta functions in several variables associated with prehomogeneous vector spaces, II. A convergence criterion
Zeta functions in several variables associated with prehomogeneous vector spaces, II. A convergence criterion
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与预齐次向量空间相关的几个变量中的 Zeta 函数，II。
DOI：
10.3792/pjaa.57.126
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
Fumihiro Sato
中科院分区：
文献类型：
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作者：
Fumihiro Sato
在文献[14]中，我们引入了与预齐次向量空间相联系的zeta函数，并证明了它们关于Q-正则子空间的函数方程。为了应用文[14]中的结果，需要找到zeta函数收敛的实用判据。本文的目的是给出zeta函数绝对收敛的一个充分条件，它是Suzuki [22]方法的推广.在§ 1中，我们回顾了与预齐次向量空间相关的zeta函数的定义，并给出了主要结果（定理1）。定理1的证明在§ 2中给出。我们的论证是基于小野[10]、[12]和[13]发展的阿黛尔几何中的技巧。我们将给出§ 3和即将发表的论文[15]中定理1的一些应用。作者感谢T。铃木为许多刺激的讨论。在下文中，我们分别用Z、Q、R和C表示有理数整数环、有理数域、真实的数域和复数域。对于Q的素数v（有限或无限），Qv是Q关于v的完备化。对于有限素数p，Zp是p-adic整数环，Fp是p元有限域。我们使用伽罗瓦上同调和阿黛尔几何中的标准符号。特别是对于定义在Q上的任何仿射代数集X，XQv（resp. XZp）是（^-有理数（resp. X在Q上的代数化记为XA。对于X上的Q-有理规范形式ω和Q的素数v，|α>| v是ω诱导的XQu上的测度。我们用^（VA）表示Q-向量空间V的抽象化VA上的Schwartz-Bruhat空间。集合X的基数用#（X）表示。对于线性代数群G，我们分别用ε ^（G）和RU（G）表示它的导群和幂幺根。
In the previous paper [14], we introduced zeta functions associated with prehomogeneous vector spaces and proved their functional equations with respect to a Q-regular subspace. For application of the results in [14], it is desirable to find a practical criterion for convergence of zeta functions. The purpose of the present paper is to give a certain sufficient condition for absolute convergence of zeta functions, which is a generalization of the method used by Suzuki [22]. In § 1, we recall the definition of zeta functions associated with prehomogeneous vector spaces and formulate the main result (Theorem 1). The proof of Theorem 1 is given in § 2. Our argument is based upon the techniques in adele geometry developed by Ono [10], [12] and [13]. We shall give some applications of Theorem 1 in § 3 and the forthcoming paper [15]. The author would like to thank T. Suzuki for many stimulating discussions. In what follows, we denote by Z, Q, R and C the ring of rational integers, the rational number field, the real number field and the complex number field, respectively. For a prime v (finite or infinite) of Q, Qv is the completion of Q with respect to v. For a finite prime p, Zp is the ring of p-adic integers and Fp is the finite field with p elements. We use the standard notation in Galois cohomology and adele geometry. In particular for any affine algebraic set X defined over Q, XQv (resp. XZp) are the set of (^-rational (resp. Zp-integral) points of X The adelization of X over Q is denoted by XA. For a Q-rational gauge form ω on X and a prime v of Q, |α>|v is the measure on XQu induced by ω. We denote by ^{VA) the Schwartz-Bruhat space on the adelization VA of a Q-vector space V. The cardinality of a set X is denoted by #(X). For a linear algbraic group G, we denote by £^(G) and RU{G) its derived group and its unipotent radical, respectively.