Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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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)
  • Received : 2008.02.08
  • Accepted : 2008.04.17
  • Published : 2009.09.30
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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 $\widetilde{M}$.

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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. https://doi.org/10.1112/plms/s3-28.1.112
  2. G. Faltings, Endlichkeitssatze fur abelsche Varietaten uber Zahlkorpern, Invent. Math., 73(3)(1983), 349-366. https://doi.org/10.1007/BF01388432
  3. G. Faltings, Erratum: "Finiteness theorems for abelian varieties over number fields", Invent. Math., 75(2)(1984), 381. https://doi.org/10.1007/BF01388572
  4. H. Imai, On the rational points of some Jacobian varieties over large algebraic number fields, Kodai Math. J., 3(1980), 56-58. https://doi.org/10.2996/kmj/1138036119
  5. S. Ohtani, On certain closed normal subgroups of free profinite groups of countably infinite rank, Comm. Algebra, 32(2004), 3257-3262. https://doi.org/10.1081/AGB-120039290
  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. https://doi.org/10.2748/tmj/1178227925

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