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

Published:2011-08-03
Printed: Dec 2012
• Fumio Sairaiji,
Hiroshima International University, Hiro, Hiroshima 737-0112, Japan
• Takuya Yamauchi,
Faculty of Education, Kagoshima University, 1-20-6 Korimoto, Kagoshima, 890-0065, Japan
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## Abstract

Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell-Weil rank over the maximal abelian extension $K^{\operatorname{ab}}$. 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 $\sharp C(K^{\operatorname{ab}})=\infty$ and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.
 Keywords: Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points
 MSC Classifications: 11G05 - Elliptic curves over global fields [See also 14H52] 11D25 - Cubic and quartic equations 14G25 - Global ground fields 14K07 - unknown classification 14K07