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The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture

Published online by Cambridge University Press:  20 November 2018

Fumio Sairaiji  and
Takuya Yamauchi
Fumio Sairaiji
Affiliation:
Hiroshima International University, Hiro, Hiroshima 737-0112, Japane-mail: sairaiji@it.hirokoku-u.ac.jp
Takuya Yamauchi
Affiliation:
Faculty of Education, Kagoshima University, 1-20-6 Korimoto, Kagoshima, 890-0065, Japane-mail: yamauchi@edu.kagoshima-u.ac.jp
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Abstract

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Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell–Weil rank over the maximal abelian extension ${{K}^{\text{ab}}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C\left( {{K}^{\text{ab}}} \right)\,=\,\infty $ and for any abelian variety of $\text{G}{{\text{L}}_{2}}$-type with trivial character.

Keywords

11G0511D2514G2514K07Mordell–Weil rankJacobian varietiesFrey-Jarden conjectureabelian points

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 842 - 849
Copyright
Copyright © Canadian Mathematical Society 2012

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