特殊函数的复乘值  

CM values of special functions

作　　者：杨同海 叶东曦 于鹏[3] Tonghai Yang;Dongxi Ye;Peng Yu

机构地区：[1]Department of Mathematics,University of Wisconsin Madison,Madison,WI 53706,USA [2]中山大学数学学院(珠海),珠海519082 [3]中国人民大学数学学院,北京100872

出　　处：《中国科学：数学》2024年第9期1421-1454,共34页Scientia Sinica：Mathematica

基　　金：Dorothy Gollmar主任基金、美国威斯康星大学麦迪逊分校Kellet职业中期研究成就奖、广东基础与应用基础研究基金(批准号:2023A1515010298);国家自然科学基金(批准号:12101605)资助项目。

摘　　要：本文综述来自正则化theta提升的特殊函数复乘值的研究结果.本文关注那些作为Borcherds提升的模函数和作为正则化theta提升的高次Green函数,并阐述这些漂亮公式背后的基本证明思路.这些函数是Gross和Zagier在奇异模数(singular moduli)上的知名分解公式和高次Green函数复乘值代数性猜想的推广.尽管最著名的Gross-Zagier公式的推广也适合本文的思路,但本文略去这些细节.本文所述的证明方法与Gross和Zagier原文中的并不相同.In this paper,we survey interesting results on CM values of special functions that come from regularized theta liftings.We focus on modular functions which are Borcherds liftings and higher green functions which are regularized theta liftings.We also describe the basic ideas behind these beautiful formulas.They are the generalization of the well-known Gross-Zagier formula on singular moduli and their algebraicity conjecture on the CM value of higher Green functions.We skip the generalization of their most famous Gross-Zagier formula in this survey although the generalization also fits the idea of this paper.The method described in this paper is different from Gross and Zagier’s original idea.

关 键 词：复乘理论 Gross-Zagier公式 模函数 theta提升 Shimura簇 

分 类 号：O156[理学—数学]