Elliptic curves related to cyclic cubic extensions

2007/11/01 by Kozuma, Rintaro
#11G05 #11G07 #11R16 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.0711.0083

Abstract

The aim of this paper is to study certain family of elliptic curves \\mathscrX_H\_H defined over a number field F arising from hyperplane sections of some cubic surface \mathscrX/F associated to a cyclic cubic extension K/F. We show that each \mathscrX_H admits a 3-isogeny ϕ over F and the dual Selmer group S^(ϕ)(\mathscrX_H/F) is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve \mathscrX_H with nice property. We also prove that the Shafarevich-Tate group \cyr X (\mathscrX_H/\rat)[ϕ] coincides with a class group of K as a special case.

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