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The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture |
Canad. Math. Bull. 55(2012), 842-849
Published:2011-08-03
Printed: Dec 2012
Printed: Dec 2012
- Fumio Sairaiji,
- Takuya Yamauchi,
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 |