On the Mordell-Weil Groups of Jacobians of Hyperelliptic Curves over Certain Elementary Abelian 2-extensions

Abstract

Abstract. Let J be the Jacobian variety of a hyperelliptic curve over Q. Let M be thefield generated by all square roots of rational integers over a finite number field K. Thenwe prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion groupand a free Z-module of infinite rank. In particular, J(M) is not a divisible group. On theother hand, if Mf is an extension of M which contains all the torsion points of J over Q,then J(Mf sol )/J(Mf ) tors is a divisible group of infinite rank, where Mf sol is the maximalsolvable extension of Mf. 1. IntroductionLet K be a number field. Let A be a nonzero abelian variety defined over K.For an extension M over K, we denote the group of M-rational points by A(M)and its torsion subgroup by A(M) tors . We call A(M) is the Mordell-Weil group ofA over M. In [1], Frey and Jarden have asked whether the Mordell-Weil group ofevery nonzero abelian variety A defined over K has infinite Mordell-Weil rank overthe maximal abelian extension K ab of K. They proved that for elliptic curves Edefined over Q, the Mordell-Weil group E(Q