CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

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
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   LaTeX   MathJax   PDF  

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 Mordell-Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points
MSC Classifications: 11G05, 11D25, 14G25, 14K07 show english descriptions Elliptic curves over global fields [See also 14H52]
Cubic and quartic equations
Global ground fields
unknown classification 14K07
11G05 - Elliptic curves over global fields [See also 14H52]
11D25 - Cubic and quartic equations
14G25 - Global ground fields
14K07 - unknown classification 14K07
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/