Let E be an elliptic curve over Q and A another elliptic curve over a real quadratic number field. We construct a Q-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.
ASJC Scopus subject areas
- General Mathematics