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

Sairaiji, Fumio ; Yamauchi, Takuya

Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 842-849

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-140-5

Mots-clés : 11G05, 11D25, 14G25, 14K07, Mordell–Weil rank, Jacobian varieties, Frey-Jarden conjecture, abelian points

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