On the Fourier coefficients of nonholomorphic Hilbert modular forms of half-integral weight
On the Fourier coefficients of nonholomorphic Hilbert modular forms of half-integral weight
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半积分权非全纯希尔伯特模形式的傅立叶系数
DOI：
10.1215/s0012-7094-96-08414-8
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发表时间：
1996
影响因子：
2.5
通讯作者：
Kamal Khuri
中科院分区：
文献类型：
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作者：
Kamal Khuri
设f(Z)=∑n≥1和2πinz是半整数权m+1/2的Hecke本征形，g(Z)=∑n≥1 bne 2πinz是相应的偶权形式，在[Sh 73]的意义下.特别地，g的权为2m，并且与f属于相同的Hecke算子的特征值。如果n=Qr且无平方r，则An可用Ar和{bj}表示。在[Sh 77]的末尾，Shimura提出Ar应该与与g相关的Dirichlet级数的特定值相关，这在[Wa 81]中得到了证明，其中Waldspurger证明了这样一个惊人的关系：对于无平方r，Ar本质上与∑n≥1φr(N)bnn−S||S=m成正比。这里我们用由f的特征标和二次特征标(r·)得到的特征标φr扭转了g的标准狄里克莱级数。本文的目的是在f和g是全实数域F上的非全纯Hilbert模形式的情况下，得到具有显式比例常数的Waldspurger关系的推广。如果F=Q，则这种形式也称为Maass形式。证明方法应推广到任意数域，但在[Sh 93A]中作了一些简化，它处理全纯Hilbert模形式的情况。以前对这个问题的研究包括Kohnen和Zagier([Ko-Za 81]和[Ko 85])在全纯情形下的工作，Katok和Sarnak([Ka-Sa 93])在非全纯情形下的工作。这两种处理都只处理上半平面上的形式(即F=Q)，但有一些额外的限制。M.Furusawa最近(尚未发表)的工作建议，在Sp(N)和O(2n+1)上的自同构形之间的对应的情况下，[Sh 93A]和本文中的方法应该推广以得到类似的公式。将Shimura在[Sh 93A]中的工作推广到非全纯情形涉及两个主要困难。首先，由于Maass形式的傅立叶展开涉及惠特克函数而不是指数，梅林变换和Rankin-Selberg卷积产生比通常更复杂的“Gamma因子”；为了得到Waldspurger关系的精确版本，必须显式地评估这些因子。其次，全纯形式的傅里叶展开式只由域F的全正元素来标引，而Maass形式的展开式则由任意签名的域元素来标引，这使得计算相当精细。本文第三节详细地说明了如何从任意整权的Hilbert模形式构造狄利克雷级数时克服这两个问题。第3节中的结果原则上从[Ma 53]、[J-L]和[W]中已知，但在文献中没有这种形式(见第3节开头的讨论)。这尤其适用于第三节的附录，在那里我们显式地计算了所有可能的惠特克函数在阿基米德地方的梅林变换。第1节到第4节没有什么新内容，而是建立了所有自同构形式、特殊函数和参数的精确定义和规范化
Let f(z) = ∑ n≥1 ane 2πinz be a Hecke eigenform of half-integral weight m+1/2, and let g(z) = ∑ n≥1 bne 2πinz be the corresponding even-weight form, in the sense of [Sh 73]. In particular, g has weight 2m, and belongs to the same eigenvalues of Hecke operators as f . If n = qr with squarefree r, then an is expressible in terms of ar and the {bj}. At the end of [Sh 77], Shimura suggested that ar should be related to special values of Dirichlet series associated to g. This was borne out in [Wa 81], where Waldspurger proved the striking relation that for squarefree r, ar is essentially proportional to ∑ n≥1 φr(n)bnn −s ∣∣ s=m . Here we have twisted the standard Dirichlet series for g by a character φr obtained from the character of f and the quadratic character ( r · ) . The purpose of this paper is to derive generalizations of Waldspurger’s relation, with an explicit proportionality constant, in the case where f and g are nonholomorphic Hilbert modular forms over a totally real number field F . If F = Q, such forms are also called Maass forms. The method of proof, which ought to generalize to arbitrary number fields, follows, with some simplifications, that in [Sh 93a], which treats the case of holomorphic Hilbert modular forms. Previous investigations into this topic have included work by Kohnen and Zagier ([Ko-Za 81] and [Ko 85]) in the holomorphic case, and Katok and Sarnak ([Ka-Sa 93]) in the nonholomorphic case. Both of these treatments deal only with forms on the upper half-plane (i.e. F = Q), with some additional restrictions. Recent (not yet published) work of M. Furusawa suggests that the method in [Sh 93a] and in this paper should generalize to yield a similar formula, in the case of the correspondence between automorphic forms on Sp(n) and on O(2n+ 1). Extending Shimura’s work in [Sh 93a] to the nonholomorphic case involves two main difficulties. First, as the Fourier expansions of Maass forms involve Whittaker functions instead of exponentials, Mellin transforms and Rankin-Selberg convolutions produce more complicated “Gamma-factors” than usual; these factors must be explicitly evaluated, in order to yield precise versions of Waldspurger’s relation. Second, whereas the Fourier expansions of holomorphic forms are indexed only by totally positive elements of the field F , the expansions of Maass forms are indexed by field elements of arbitrary signature; this makes the calculations rather more delicate. Section 3 of this paper explains in explicit detail how one overcomes both of these problems in constructing a Dirichlet series from Hilbert modular forms of arbitrary integral weight. The results in section 3 are in principle known from [Ma 53], [J-L], and [W], but are not found in this form in the literature (see the discussion at the beginning of section 3). This applies particularly to the appendix to section 3, where we explicitly compute the Mellin transforms of all possible Whittaker functions at the archimedean places. Sections 1 through 4 contain nothing new, but rather set up precise definitions and normalizations of all automorphic forms, special functions, and parameters