A note on the Siegel-Eisenstein series of weight 2 onSp2(Z)
A note on the Siegel-Eisenstein series of weight 2 onSp2(Z)
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关于权重 2 onSp2(Z) 的 Siegel-Eisenstein 级数的注释
DOI：
10.1007/bf02567045
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发表时间：
1992
影响因子：
0.6
通讯作者：
S. Nagaoka
中科院分区：
文献类型：
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作者：
S. Nagaoka
L在文[3]中研究了重量为2的油SPZ(Z)的Siegel-Eisenstein系列E~‘~(Z，S)的解析性质。FIE证明了E~‘)(Z，S)在S=0时是有限的，并给出了E~’~(Z，O)的傅里叶展开式。他的方法是以雅可比形式理论为基础的。在本文中，我们用不同的方法给出了他的公式的一个简短证明。我们方法的主要工具是Siegel级数和广义超几何函数。一般而言，Siegel-Eisenstein的傅里叶系数本质上由它们(Of.定理1......G.Shimura[5]对管域上的广义超几何函数进行了充分的研究。另一方面，研究了Siegel级数与二次型的局部密度的关系。对于二元二次型，G.L[2]给出了显式公式，它在我们的证明中起着重要的作用
In [3], W. l (ohnen investigated the analytic properties of tile Siegel-Eisenstein series E~'~(Z, s) of weight 2 oil SPz (Z). Fie showed that E~')(Z, s) is finite at s= 0 and gave the Fourier expansion of E~'~(Z, O) explicitly. His method is based on the theory of Jacobi forms. In the present note we give a short proof of his formula by a different method. Tile main tools in our method are the Siegel series and the generalized hypergeometric functions. In general, the Fourier coefficients of the Siegel-Eisenstein are essentially expressed by them (of. Theorem 1...... The generalized hypergeometric functions on tube domains were fully studied by G. Shimura [5]. On the other hand, the Siegel series was studied in connection with the local density of quadratic forms. In the case of binary quadratic forms, the explicit formula was given by G. l (aufhold [2], which plays an important role in our proof