On the Derivative of an Eisenstein Series of Weight One
On the Derivative of an Eisenstein Series of Weight One
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关于权重一的爱森斯坦级数的导数
DOI：
10.1155/s1073792899000185
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发表时间：
1999
影响因子：
1
通讯作者：
Tonghai Yang
中科院分区：
文献类型：
--
作者：
S. Kudla;M. Rapoport;Tonghai Yang
在文[17]中，引入了一族权为(g+1)/2的Siegel Eisenstein级数。它们有一个奇怪的函数方程，因此在它们的对称中心有一个自然零(S=0)。作者认为，这种级数在S=0处的导数，我们称之为非相干艾森斯坦级数，应该与算术代数几何有关。给出了亏格为2且权为3/2的情形的一些证据。在这种情况下，证明了中心导数的某些傅立叶系数涉及Shimura曲线上Heegner点的高度对。另外的证据出现在格罗斯和基廷早期的工作中[12]，其中隐含地出现了权为2的Siegel Eisenstein级数在SP3上的导数。高维情形在[19](SP4，权重5/2)和[21](SP3，权重2)中被研究。在本文中，我们考虑非相干Eisenstein级数及其中心导数的最简单可能的例子。更确切地说，设q>；3是3的模4的素同余。有两类权为1的艾森斯坦级数与虚二次域k=q(√−q)有关。第一个是连贯的爱森斯坦系列。对于上半平面的τ=u+iv和具有Re(S)>；1的S∈C，这个级数的形式为
In [17], a certain family of Siegel Eisenstein series of genus g and weight (g + 1)/2 was introduced. They have an odd functional equation and hence have a natural zero at their center of symmetry (s = 0). It was suggested that the derivatives at s = 0 of such series, which we will refer to as incoherent Eisenstein series, should have some connection with arithmetical algebraic geometry. Some evidence was provided in the case of genus 2 and weight 3/2. In that case, certain of the Fourier coefficients of the central derivative were shown to involve (parts of) the height pairing of Heegner points on Shimura curves. Additional evidence occurred earlier in the work of Gross and Keating [12], where, implicitly, derivatives of Siegel Eisenstein series on Sp3 of weight 2 arise. Higher dimensional cases are studied in [19] (Sp4, weight 5/2) and [21] (Sp3, weight 2). In the present paper,we consider the simplest possible example of an incoherent Eisenstein series and its central derivative. More precisely, let q > 3 be a prime congruent to 3 modulo 4. There are two types of Eisenstein series of weight 1 associated to the imaginary quadratic field k = Q(√−q). The first is a coherent Eisenstein series. For τ = u+ iv in the upper half-plane and s ∈ C with Re(s) > 1, this series has the form