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Search: MSC category 14G25 ( Global ground fields )

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1. CMB 2011 (vol 56 pp. 225)

Agashe, Amod
On the Notion of Visibility of Torsors
Let $J$ be an abelian variety and $A$ be an abelian subvariety of $J$, both defined over $\mathbf{Q}$. Let $x$ be an element of $H^1(\mathbf{Q},A)$. Then there are at least two definitions of $x$ being visible in $J$: one asks that the torsor corresponding to $x$ be isomorphic over $\mathbf{Q}$ to a subvariety of $J$, and the other asks that $x$ be in the kernel of the natural map $H^1(\mathbf{Q},A) \to H^1(\mathbf{Q},J)$. In this article, we clarify the relation between the two definitions.

Keywords:torsors, principal homogeneous spaces, visibility, Shafarevich-Tate group
Categories:11G35, 14G25

2. CMB 2011 (vol 55 pp. 842)

Sairaiji, Fumio; Yamauchi, Takuya
The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey-Jarden Conjecture
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
Categories:11G05, 11D25, 14G25, 14K07

3. CMB 2009 (vol 52 pp. 117)

Poulakis, Dimitrios
On the Rational Points of the Curve $f(X,Y)^q = h(X)g(X,Y)$
Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points.

Categories:11G30, 14G05, 14G25

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