p-converse to a theorem of Gross-Zagier, Kolyvagin and Rubin

• 成果类型: Article
• 署名作者: Burungale, Ashay A.; Tian, Ye
• 署名单位: California Institute of Technology; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS; Chinese Academy of Sciences
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00929-7
• 发表日期: 2020
• 页码: 211-253
• 关键词: adic l-functions; elliptic-curves; main conjectures; rational-points; iwasawa theory; selmer groups; FORMULA; characters

摘要

Let E be a CM elliptic curve over the rationals and p>3-Selmer group Selp infinity(E/Q) and the complex L-function L(s,E/Q). In particular, the Tate-Shafarevich group X(E/Q) is finite whenever corankZpSelp infinity(E/Q)=1. We also prove an analogous p-converse for CM abelian varieties arising from weight two elliptic CM modular forms with trivial central character. For non-CM elliptic curves over the rationals, first general results towards such a p-converse theorem are independently due to Skinner (A converse to a theorem of Gross, Zagier and Kolyvagin, , 2014) and Zhang (Camb J Math 2(2):191-253, 2014).