Critical values of the twisted tensor $L$-function in the imaginary quadratic case
Critical values of the twisted tensor $L$-function in the imaginary quadratic case
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虚数二次情况下扭曲张量 $L$ 函数的临界值
DOI：
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发表时间：
1999
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影响因子：
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通讯作者：
Eknath Ghate
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文献类型：
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作者：
Eknath Ghate
f 的扭曲张量 L 函数（我们用 G(s, f) 表示）是与数域 K/F 的二次扩展相关联的某个狄利克雷级数，以及 K 上的尖点自守函数 f 。在 Shimura 之前的工作之后，Asai 在 [1] 中引入了它，在 f 是 Q 的实二次扩展 K 上的希尔伯特模尖点形式的情况下。 在过去的二十多年里，这个L函数被更普遍地考虑：例如[11]和[12]处理完全实数域的二次扩展，[17]处理Q的虚数二次扩展，[3]、[4]和[14]处理数域的一般二次扩展。所有这些论文主要关注建立类似于[1]中的G(s, f)的解析性质，例如整个复平面的亚纯延拓、极点数量的位置和有限性以及函数方程。本文的目的是证明 G(s, f) 在虚数二次设置中的合理性结果。如果 K 是一个虚数二次场，并且 f 是与 K 相关的尖点形式，我们可以确定存在一个“周期”Ωj(f)，使得
The twisted tensor L-function of f , which we denote by G(s, f), is a certain Dirichlet series associated to a quadratic extension of number fields K/F , and a cuspidal automorphic function f over K. It was introduced in [1] by Asai, following previous work of Shimura, in the case when f is a Hilbert modular cusp form over a real quadratic extension K of Q. In the past twenty odd years, this L-function has been considered more generally: for instance [11] and [12] deal with quadratic extensions of totally real fields, [17] with imaginary quadratic extensions of Q, and [3], [4] and [14] with general quadratic extensions of number fields. All these papers have been primarily concerned with establishing analytic properties of G(s, f) analogous to those in [1], such as meromorphic continuation to the entire complex plane, location and finiteness of the number of poles, and functional equation. The aim of this paper is to prove a rationality result for G(s, f) in the imaginary quadratic setting. If K is an imaginary quadratic field, and f a cusp form associated to K, we establish that there is a ‘period’ Ωj(f) such that