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

• Moon, Hyun-Suk (Department of Mathematics, College of Natural Sciences, Kyungpook National University)
• Accepted : 2008.04.17
• Published : 2009.09.30
• 31 25

#### Abstract

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 $\widetilde{M}$.

#### Keywords

Mordell-Weil groups;hyperelliptic curves

#### References

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#### Cited by

1. The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture vol.55, pp.04, 2012, https://doi.org/10.4153/CMB-2011-140-5