Hirzebruch--Zagier cycles and twisted triple product Selmer groups

Yifeng Liu Northwestern University

arXiv subject: Number Theory (math.NT) mathscidoc:2105.60001

Invent. math., 205, 693-780, 2016.12
Let $E$ be an elliptic curve over $\dQ$ and $A$ another elliptic curve over a real quadratic number field. We construct a $\dQ$-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.
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@inproceedings{yifeng2016hirzebruch--zagier,
  title={Hirzebruch--Zagier cycles and twisted triple product Selmer groups},
  author={Yifeng Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210523180258002929827},
  booktitle={Invent. math.},
  volume={205},
  pages={693-780},
  year={2016},
}
Yifeng Liu. Hirzebruch--Zagier cycles and twisted triple product Selmer groups. 2016. Vol. 205. In Invent. math.. pp.693-780. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210523180258002929827.
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