Hirzebruch-Zagier cycles and twisted triple product Selmer groups

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $A$ be 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.

71 pages

References in corpus (1)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

Cited by in corpus (7)

• Euler systems for Hilbert modular surfaces
• Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups
• Twisted triple product p-adic L-functions and Hirzebruch-Zagier cycles
• Tate cycles on some quaternionic Shimura varieties mod p
• Indivisibility of Heegner cycles over Shimura curves and Selmer groups
• Plectic Galois action on CM points and connected components of Hilbert modular varieties
• Bounding cubic-triple product Selmer groups of elliptic curves