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

Let J be the Jacobian variety of a hyperelliptic curve over Q. Let M be the field generated by all square roots of rational integers over a finite number field K. Then we prove that the Mordell-Weil group J(M) is the direct sum of a finite torsion group and a free Z-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if (M) over tilde is an extension of M which contains all the torsion points of J over (Q) over bar, then J((M) over tilde (sol))/J((M) over tilde (sol))(tors) is a divisible group of infinite rank, where (M) over tilde (sol) is the maximal solvable extension of (M) over tilde.