Special value formulae of Rankin -Selberg L-functions
Special value formulae of Rankin -Selberg L-functions
复制标题
Rankin-Selberg L-函数的特殊值公式
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Haiping Yuan
中科院分区：
文献类型：
--
作者：
Haiping Yuan
我想感谢张守武，感谢他使用爱森斯坦级数技巧的绝妙想法以及他的基本论文[21]和[22]，我从这些论文中学习了他计算傅立叶系数的新语言。我还要感谢我的顾问Ted Chinburg，他一直以来的支持和鼓励。最后，我想表达我最衷心的感谢，叶天告诉我这个有趣的话题，并不断鼓励我解决问题，并与我分享他的想法总是。事实上，这是一个联合项目的一部分，叶天旨在建立一个p-adic模拟的特殊价值公式的希达家庭的希尔伯特新形式。本文证明了与全真实的域的全虚二次扩张的高权Hilbert模形式和环类特征标相关联的Rankin-Selberg L-函数的一个特殊的N级取值公式.该公式将Rankin-Selberg L-函数在s = 12处的特殊值与某个检验形式在一个与一个四元代数相关联的0维Shimura簇上的某个CM-点处的值联系起来。该公式推广了张守武所证明的公式，张守武所证明的公式是经典Gross-Zagier公式的广泛推广。证明是基于惠学的一个公式（水平N D），并结合Eisenstein级数的技巧来计算在比较水平N和水平N D两个公式时出现的普适常数。
ii Acknowledgement I would like to express my indebtness to Shou-Wu Zhang for his wonderful idea of using the trick of Eisenstein series as well as his fundamental papers [21] and [22] from which I learned his new language to compute Fourier coefficients. I also would like to thank my advisor Ted Chinburg for his constant support and encouragement. At last but not the least, I wish to express my heartiest thanks to Ye Tian who told me this interesting topic and constantly encourage me to work out the problem and shared his idea with me always. In fact this is the a part of a joint project with Ye Tian aiming to establishing a p-adic analogue of the special value formula of a Hida family of Hilbert newforms. In this paper, we prove a special value formula of level N of Rankin-Selberg L-function associated to a Hilbert modular form of higher weight and a ring class character of an totally imaginary quadratic extension of a totally real field. The formula relates the special value of the Rankin-Selberg L-functions at s = 1 2 to the value of certain test form at some CM-point on a 0-dimensional Shimura Variety associated to a quaterion algebra. The formula generalizes the formula proved by Shou-Wu Zhang which is a vast generalization of classical Gross-Zagier formula. The proof is based on a formula (level N D) of Hui Xue combined with a technique of Eisenstein Series to compute the universal constants which arise in the comparison of both formulae of level N and N D.