Properties of Eisenstein series formed with modular symbols
Properties of Eisenstein series formed with modular symbols
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用模符号构成的爱森斯坦级数的性质
DOI：
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发表时间：
2000
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通讯作者：
C. O’Sullivan
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文献类型：
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作者：
C. O’Sullivan
本文研究了由Goldfeld首次提出的一类新的非全纯爱森斯坦级数。对于任意一类Fuchsian群，我们固定了一个全纯尖形，并考虑了用与此尖形相关的模符号构造的爱森斯坦级数。我们从其亚纯延拓到整个复平面开始，与通常的爱森斯坦级数相似地发展了该理论。函数方程然后获得相关的值(s 1−s。介绍让H = {z∈C: Im z > 0}是上半平面,让Γ⊂SL2 (R)是一个固定的非co-compact富克斯的第一种,(例如Γ(N),Γ0 (N)),作用于H .简单假设Γ有独特的尖端与稳定集团在无穷远处Γ∞={±(1 m 0 1) m∈z}。对于Γ中的每个γ，我们将标记它的矩阵元素（γa γb γc γd）。设f(z)是S2（Γ）的一个元素，S2（Γ）是Γ的权值2的空间全纯顶点形式。在[Go]之后，我们定义了一个改进的爱森斯坦级数（0.1）E∗(z, s) =∑γ∈Γ∞Γ < γ, f > Im(γz)s, z∈H，其中对于γ∈Γ，模符号为< γ, f > = - 2πi∫γw w f(τ) dτ，该定义与w∈H无关。注意，由于< γ1γ2, f > = < γ1, f > + < γ2, f >，该级数不是自同态的。对于所有γ∈Γ，变换规则为E∗(γz, s) = E∗(z, s)−< γ, f > E(z, s)，其中E（z, s）是Γ的通常爱森斯坦级数。Goldfeld为了研究模符号< γ, f >在群Γ上γ范围内的分布性质，引入了这种新型的非全纯Eisenstein级数。级数（0.1）收敛于Re(s) >2， Goldfeld假设它应该具有解析延拓和泛函方程。在本文中，对Selberg的方法（在[Iw]， [He]中描述）进行了推广，得到了以下结果。AMS-TEX排版
In this work a new kind of non-holomorphic Eisenstein series, first introduced by Goldfeld, is studied. For an arbitrary Fuchsian group of the first kind we fix a holomorphic cusp form and consider Eisenstein series constructed with the modular symbol associated with this cusp form. We develop the theory analogously with that of the usual Eisenstein series starting with its meromorphic continuation to the entire complex plane. A functional equation is then obtained relating the values at s to those at 1− s. Introduction Let H = {z ∈ C : Im z > 0} be the upper half plane and let Γ ⊂ SL2(R) be a fixed non co-compact Fuchsian group of the first kind, (for example Γ(N), Γ0(N)), acting on H. For simplicity assume that Γ has a unique cusp at infinity with stability group Γ∞ = { ± ( 1 m 0 1 ) ,m ∈ Z } . For each γ in Γ we shall label its matrix elements ( γa γb γc γd ) . Let f(z) be an element of S2(Γ), the space holomorphic cusp forms of weight 2 for Γ. Following [Go] we define a modified Eisenstein series (0.1) E∗(z, s) = ∑ γ∈Γ∞Γ 〈 γ, f 〉Im(γz)s, z ∈ H, where for γ ∈ Γ the modular symbol is given by 〈 γ, f 〉 = −2πi ∫ γw w f(τ) dτ, the definition being independent of w ∈ H. Note that since 〈 γ1γ2, f 〉 = 〈 γ1, f 〉+ 〈 γ2, f 〉 the series is not automorphic. The transformation rule is E∗(γz, s) = E∗(z, s)− 〈 γ, f 〉E(z, s), for all γ ∈ Γ where E(z, s) is the usual Eisenstein series for Γ. This new type of non-holomorphic Eisenstein series was introduced by Goldfeld in order to study the distribution properties of modular symbols 〈 γ, f 〉 as γ ranges over the group Γ. The series (0.1) converges for Re(s) > 2 and Goldfeld hypothesised that it should have an analytic continuation and a functional equation. In this paper Selberg’s method, (described in [Iw], [He]), is extended to establish the following results. Typeset by AMS-TEX 1