Bounding cubic-triple product Selmer groups of elliptic curves

Abstract

Let $E$ be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over $\mathbb{Q}$ constructed from $E$ through multiplicative induction; it is of rank 8. We show that, under certain assumptions on $E$, the nonvanishing of the central critical value of the $L$-function attached to the motive implies that the dimension of the associated Bloch-Kato Selmer group is 0.