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

• Journal title : Kyungpook mathematical journal
• Volume 49, Issue 3,  2009, pp.419-424
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2009.49.3.419
Title & Authors
On the Mordell-Weil Groups of Jacobians of Hyperelliptic Curves over Certain Elementary Abelian 2-extensions
Moon, Hyun-Suk;

Abstract
Let J be the Jacobian variety of a hyperelliptic curve over $\small{\mathbb{Q}}$. 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 $\small{\mathbb{Z}}$-module of infinite rank. In particular, J(M) is not a divisible group. On the other hand, if $\small{\widetilde{M}}$ is an extension of M which contains all the torsion points of J over $\small{\widetilde{\mathbb{Q}}}$, then $\small{J(\widetilde{M}^{sol})/J(\widetilde{M}^{sol})_{tors}}$ is a divisible group of infinite rank, where $\small{\widetilde{M}^{sol}}$ is the maximal solvable extension of $\small{\widetilde{M}}$.
Keywords
Mordell-Weil groups;hyperelliptic curves;
Language
English
Cited by
References
1.
G. Frey and M. Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc., 28(1974), 112-128.

2.
G. Faltings, Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern, Invent. Math., 73(3)(1983), 349-366.

3.
G. Faltings, Erratum: "Finiteness theorems for abelian varieties over number fields", Invent. Math., 75(2)(1984), 381.

4.
H. Imai, On the rational points of some Jacobian varieties over large algebraic number fields, Kodai Math. J., 3(1980), 56-58.

5.
S. Ohtani, On certain closed normal subgroups of free profinite groups of countably infinite rank, Comm. Algebra, 32(2004), 3257-3262.

6.
K. Ribet, Torsion points of abelian varieties in cyclotomic extensions, appendix to N. Katz and S. Lang, Finiteness theorems in geometric classfield theory, Enseign. Math., (2) 27(3-4)(1981), 285-319.

7.
S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.

8.
J. Top, A remark on the rank of Jacobians of hyperelliptic curves over \$\mathbb{Q}\$ over certain elementary Abelian 2-extension, Tohoku Math. J., 40(1988), 613-616.