Hirzebruch-Zagier cycles and twisted triple product Selmer groups

• 成果类型: Article
• 署名作者: Liu, Yifeng
• 署名单位: Northwestern University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-016-0645-9
• 发表日期: 2016
• 页码: 693-780
• 关键词: elliptic-curves; REPRESENTATIONS; points; VALUES; forms; DECOMPOSITION; intersection; COHOMOLOGY; etale

摘要

Let E be an elliptic curve over and A another elliptic curve over a real quadratic number field. We construct a -motive of rank 8, together with a distinguished class in the associated Bloch-Kato Selmer group, using Hirzebruch-Zagier cycles, that is, graphs of Hirzebruch-Zagier morphisms. We show that, under certain assumptions on E and A, the non-vanishing of the central critical value of the (twisted) triple product L-function attached to (E, A) implies that the dimension of the associated Bloch-Kato Selmer group of the motive is 0; and the non-vanishing of the distinguished class implies that the dimension of the associated Bloch-Kato Selmer group of the motive is 1. This can be viewed as the triple product version of Kolyvagin's work on bounding Selmer groups of a single elliptic curve using Heegner points.