Mordell-Weil Groups of Elliptic Curves Over Cyclotomic Fields

Abstract

Let E be an elliptic curve defined over Q. For any field F containing Q let E(F) denote the group of points of E rational over F. The Mordel1-Weil theorem asserts that if F is a finite extension of Q then E(F) is finitely generated. Our main result shows that for a large class of elliptic curves over Q, and for certain infinite abelian extensions F of Q, the group E(F) remains finitely generated. it should be noted that the torsion subgroup of E(F) over such fields has been studied by Ribet. As a special case of a much more general theorem [16] he has proved that, for any elliptic curve E over Q and any abelian extension F of Q, the group E(F)tors is finite.KeywordsExact SequenceElliptic CurfClass NumberDirichlet CharacterAbelian ExtensionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.