Hirzebruch-Zagier cycles and twisted triple product Selmer groups

TL;DR

This paper constructs a motive related to elliptic curves over $Q$ and real quadratic fields, linking special values of triple product L-functions to the structure of associated Selmer groups, extending Kolyvagin's methods.

Contribution

It introduces a new motive using Hirzebruch-Zagier cycles and establishes a connection between L-function values and Selmer group dimensions for elliptic curves.

Findings

Non-vanishing of the central L-value implies Selmer group is trivial.

Non-vanishing of a distinguished class implies Selmer group has dimension one.

Provides a triple product analogue of Kolyvagin's results on elliptic curves.

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.