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In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within the framework of the Gan–Gross–Prasad conjecture. We show that if the central critical value of the Rankin–Selberg L-function does not vanish, then the Bloch–Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch–Kato Selmer group constructed from a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin–Selberg L-function, then the Bloch–Kato Selmer group is of rank one. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over $${{\mathbb {Q}}}$$ and A another elliptic curve over a real quadratic number field. We construct a $${{\mathbb {Q}}}$$ -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. [ABSTRACT FROM AUTHOR]