The Maaß space and Hecke operators
The Maaß space and Hecke operators
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Maaß 空间和 Hecke 算子
DOI：
10.1007/bf02570891
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发表时间：
1990
影响因子：
0.8
通讯作者：
A. Krieg
中科院分区：
文献类型：
--
作者：
A. Krieg
H.Maag[20]在1979年提出了所谓的Spezialschar。后来，他研究了关于非平凡乘子系统的2次Siegel模形式的Spezialschar的模拟。对于与高斯数域相关的厄米特模形式，相应的Maab空间由Kojima[11]描述。Andrianov[1]、Maab[21]分别证明了这些Maab空间在Hecke算子下的不变性。Gritsenko[7]在[-13]中引入了二次四元数半空间上Spezialschar的类比。本文证明了Maab空间在所有Hecke算子下都是不变的。因此，我们可以得出结论，Siegel-Eisenstein系列总是属于Maab空间，正如在[-13]中已经宣布的那样。我们可以在Maal3空间与El3中的椭圆模形式空间的某个子空间同构的情况下计算Siegel-Eisenstein级数的像。因此，我们可以显式地计算Siegel-Eisenstein级数的傅里叶系数。令人惊讶的是，这些傅里叶系数的结构比Spezialschar中的Siegel-Eisenstein级数的傅里叶系数的结构简单得多(参见[19])。关于Hermitian半空间上的Siegel-Eisenstein级数(参看[22])。傅里叶系数的显式描述允许我们验证Resnikoff-Saldafia[26]关于Siegel-Eisenstein级数E~的傅里叶系数增长的猜想的一个特殊情况。傅立叶展开的知识也使我们能够更仔细地观察从四元数半空间到二次Siegel半空间的限制映射。我们已经知道这个限制映射到theta级数(参见[12]，IV，w1)。但这一限制一般不能将H(2；IH)上的Siegel-Eisenstein-级数映射到H(2；11)上的Siegel-Eisenstein-级数上。此时会出现一个附加的尖点形状。此外，我们还在Maag空间中引入了两类与模形式相关的Dirichlet级数。第一个是由傅里叶-雅可比展开中的第一个雅可比形式产生的。第二个是Andrianov Zeta函数。
H. Maag [20] introduced the so-called Spezialschar in 1979. Later on [21] he investigated the analogue of the Spezialschar for Siegel modular forms of degree 2 with respect to the non-trivial multiplier system. For Hermitian modular forms associated with the Gaussian number field the corresponding MaaB space was described by Kojima [11]. The invariance of these MaaB spaces under Hecke operators was demonstrated by Andrianov [1], MaaB [21], resp. Gritsenko [7].The analogue of the Spezialschar on the half-space of quaternions of degree two was introduced in [-13]. In this paper we show that the MaaB space is invariant under all Hecke operators. As a consequence we can conclude that the Siegel-Eisenstein-series always belongs to the MaaB space, as it was already announced in [-13]. We can compute the image of the Siegel-Eisenstein-series under the isomorphism between the Maal3 space and a certain subspace of the space of elliptic modular forms in El3]. As a consequence we can calculate the Fourier coefficients of the Siegel-Eisenstein-series explicitly. Surprisingly the structure of these Fourier coefficients is much simpler than the structure of the Fourier coefficients of the Siegel-Eisenstein-series in the Spezialschar (cf.[19]) resp. of the Siegel-Eisenstein-series on the Hermitian half-space (cf.[22]). The explicit description of the Fourier coefficients allows us to verify a particular case of a conjecture of Resnikoff-Saldafia [26] on the growth of the Fourier coefficients of the Siegel-Eisenstein-series E~. The knowledge of the Fourier expansion also enables us to have a closer look at the restriction map from the half-space of quaternions to the Siegel half-space of degree 2. We already know that this restriction maps theta series onto theta series (cf.[12], IV, w 1). But the restriction does not map the Siegel-Eisenstein-series on H (2; IH) onto the Siegel-Eisenstein-series on H (2; ll) in general. Here an additional cusp form appears. Moreover we introduce two types of Dirichlet series associated with modular forms in the Maag space. The first one arises from the first Jacobi form in the Fourier-Jacobi-expansion. The second one is the Andrianov zeta function.