Bounding cubic-triple product Selmer groups of elliptic curves

TL;DR

This paper proves that, for certain elliptic curves over totally real cubic fields, the non-vanishing of a specific L-function value ensures the associated Selmer group has dimension zero, advancing understanding of elliptic curve arithmetic.

Contribution

It establishes a new link between the non-vanishing of a cubic-triple product L-value and the triviality of the corresponding Selmer group for elliptic curves over cubic fields.

Findings

Non-vanishing of the L-value implies Selmer group dimension zero

Constructs a cubic-triple product motive from elliptic curves

Provides conditions under which the main result holds

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 non-vanishing 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 $0$.