Remark on Kohnen-Zagier's paper concerning Fourier coefficients of modular forms of half intergral weight
Remark on Kohnen-Zagier's paper concerning Fourier coefficients of modular forms of half intergral weight
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关于 Kohnen-Zagier 关于半积分权模形式的傅里叶系数的论文的评论
DOI：
10.3792/pjaa.69.383
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发表时间：
1993
期刊：
影响因子：
--
通讯作者：
H. Kojima
中科院分区：
文献类型：
--
作者：
H. Kojima
Introduetion。在[5]中，Shimura建立了半积分权的模形式空间与积分权的模形式空间的对应关系。Waldspurger[8]和[9]利用元群的表示理论的方法和语言，证明了一个模形式为f(z) n的自由平方整数n的傅里叶系数的平方a(n) n—1的半积分权值a(n)e[nz]本质上正比于在模形式f上的某个整数上的ζ函数的特殊值，如果f对应于f by并且f是Hecke算子的特征函数。另一方面，Kohnen- zagier[1]和[2]明确地确定了f(z) a(n)e[nz]的自由平方整数n的平方所占的比例，该整数是Hecke算子的特征函数，属于Kohnen子空间S<k+I)/。（4N） {f(z) <_)h,, 0,l<4)a(n)e[nz] (4N)} S<。+)n（4N）通过与积分权值的模形式F (F)相关的函数的特殊值。柯南-扎吉尔[1]（音）Kohnen b[2])处理了N I (p。N是一个奇平方自由整数)。本文的目的是推导出在S<+)/(4N)中f是Hecke算子的特征函数的情况下[1]和[2]的类比。<。+)/(4N， ZN))和（f)是$2（2N）中的原始形式，其中S<k+I）/。(4 n)(分别地。S<+I)/(4N, z))表示由权值(2k- f1)/2和阶数为4N （resp. 1）的模尖形式组成的向量空间。用字符Z表示第4N级)。由于满足我们条件的模形式包含在Kohnen空间的正交补中，所以Kohnen- zagier的结果与我们的结果没有重叠。我们的证明方法与[1]的证明方法类似。为了证明我们的结果，我们需要修改他们的方法。1. 符号和开头。我们用Z、Q、R、C分别表示有理数环、有理数域、实数域和复数域。对于zc，我们用e[z] exp（2riz）我们定义v z使r/2 < arg(z1/) _
Introduetion. In [5], Shimura has established a correspondence between the space of modular forms of half integral weight and the space of those of integral weight. Using the methods and languages of representation theory of adeles of metaplectic groups, Waldspurger [8] and [9] showed that the square of Fourier coefficients a(n)for a square free integer n of the modular form f(z) n.--1 a(n)e[nz] of half integral weight is essentially proportional to the special value of the zeta function at a certain integer attached to the modular form F if f corresponds to F by and f is an eigen-function of Hecke operators. On the other hand, Kohnen-Zagier [1] and [2] determined explicitly the proportion of the square of a(n) for a square free integer n of f(z) a(n)e[nz] which is an eigen-function of Hecke operators and belongs to the Kohnen’s subspace S<k+I)/.(4N) {f(z) <_)h,,o,l<4)a(n)e[nz] (4N)} of S<.+)n(4N) by the special value of the zeta function associated with the modular form F (f)of integral weight. Kohnen-Zagier [1] (resp. Kohnen [2]) treated the case where N I (resp. N is an odd square free integer). The purpose of this note is to derive an analogy of [1] and [2] in the case where f is an eigen-function of Hecke operators in S<+)/(4N) (resp. S<.+)/(4N, ZN)) and (f) is a primitive form in $2(2N), where S<k+I)/.(4N) (resp. S<+I)/(4N, z)) means the vector space consisting of modular cusp forms of weight (2k-F 1)/2 and of level 4N (resp. of level 4N with the character Z). Since the modular form f satisfying our conditions is contained in orthogonal complement of Kohnen’s space, there is no overlap between Kohnen-Zagier’s results and ours. The method of our proof is similar to that of [1]. To prove our results, we need to modify their methods. 1. Notation and preliminaries. 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 z C, we put e[z]--exp(2riz) and we define v z so that r/2 < arg(z1/) _