The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves

TL;DR

This paper investigates the 2-part of the Bloch-Kato conjecture for specific elliptic curves, providing explicit descriptions, constructing elements with particular divisibility properties, and numerically testing the conjecture.

Contribution

It offers explicit descriptions of 2-torsion in Selmer groups, constructs special elements in $K_2$, and proves their indivisibility by 2, advancing understanding of the conjecture for certain elliptic curves.

Findings

Explicit description of 2-torsion in Selmer groups.

Construction of elements in $K_2$ with non-vanishing regulators.

Numerical evidence supporting the 2-part of the Bloch-Kato conjecture.

Abstract

For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture.