On the Mordell-Weil Groups of Jacobians of Hyperelliptic Curves over Certain Elementary Abelian 2-extensions
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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 fM is an extension of M which contains all the torsion points of J over Q, then [수식] is a divisible group of infinite rank, where fMsol is the maximal solvable extension of fM.
Abstract
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 fM is an extension of M which contains all the torsion points of J over Q, then [수식] is a divisible group of infinite rank, where fMsol is the maximal solvable extension of fM.
참고문헌 (8)
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