Horizontal non-vanishing of Heegner points and toric periods

doi:10.1016/j.aim.2019.106938

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	Horizontal non-vanishing of Heegner points and toric periods
	Burungale, Ashay A.1,2; Tian, Ye 3,4
	2020-03-04
发表期刊	ADVANCES IN MATHEMATICS
ISSN	0001-8708
卷号	362 页码:35
摘要	Let F be a totally real number field and A a modular GL(2)-type abelia.n variety over F. Let K/F be a CM quadratic extension. Let x be a class group character over K such that the Rankin-Selberg convolution L(s, A, chi) is self-dual with root number -1. We show that the number of class group characters chi with bounded ramification such that L'(1, A, chi) not equal 0 increases with the absolute value of the discriminant of K. We also consider a rather general rank zero situation. Let pi be a cuspidal cohomological automorphic representation over GL(2)(A(F)). Let chi be a Hecke character over K such that the Rankin-Selberg convolution L(s, pi, chi) is self-dual with root number 1. We show that the number of Hecke characters chi with fixed infinity-type and bounded ramification such that L(1/2, pi, chi) not equal 0 increases with the absolute value of the discriminant of K. The Gross-Zagier formula and the Waldspurger formula relate the question to horizontal non-vanishing of Heegner points and tonic periods, respectively. For both situations, the strategy is geometric relying on the Zariski density of CM points on self-products of a quaternionic Shimura variety. The recent result [26,31,1] on the Andre-Oort conjecture is accordingly fundamental to the approach. (C) 2019 Elsevier Inc. All rights reserved.
关键词	Heegner points Tonic periods Gross-Zagier formula Waldspurger formula
DOI	10.1016/j.aim.2019.106938
收录类别	SCI
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000510113300005
出版者	ACADEMIC PRESS INC ELSEVIER SCIENCE
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/50808
专题	中国科学院数学与系统科学研究院
通讯作者	Burungale, Ashay A.
作者单位	1.Inst Adv Study, Einstein Dr, Princeton, NJ 08540 USA 2.CALTECH, 1200 E Calif Blvd, Pasadena, CA 91125 USA 3.Chinese Acad Sci, Acad Math & Syst Sci, HLM, MCM, Beijing 100190, Peoples R China 4.Univ Chinese Acad Sci, Sch Math Sci, Beijing 10049, Peoples R China
推荐引用方式 GB/T 7714	Burungale, Ashay A.,Tian, Ye. Horizontal non-vanishing of Heegner points and toric periods[J]. ADVANCES IN MATHEMATICS,2020,362:35.
APA	Burungale, Ashay A.,&Tian, Ye.(2020).Horizontal non-vanishing of Heegner points and toric periods.ADVANCES IN MATHEMATICS,362,35.
MLA	Burungale, Ashay A.,et al."Horizontal non-vanishing of Heegner points and toric periods".ADVANCES IN MATHEMATICS 362(2020):35.