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

Moon, Hyun-Suk

  • Received : 2008.02.08
  • Accepted : 2008.04.17
  • Published : 2009.09.30


Let J be the Jacobian variety of a hyperelliptic curve over $\mathbb{Q}$. 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 $\mathbb{Z}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $\widetilde{M}$ is an extension of M which contains all the torsion points of J over $\widetilde{\mathbb{Q}}$, then $J(\widetilde{M}^{sol})/J(\widetilde{M}^{sol})_{tors}$ is a divisible group of infinite rank, where $\widetilde{M}^{sol}$ is the maximal solvable extension of $\widetilde{M}$.


Mordell-Weil groups;hyperelliptic curves


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