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Search: MSC category 11G10 ( Abelian varieties of dimension $> 1$ [See also 14Kxx] )

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1. CMB 2011 (vol 55 pp. 38)

Butske, William
Endomorphisms of Two Dimensional Jacobians and Related Finite Algebras
Zarhin proves that if $C$ is the curve $y^2=f(x)$ where $\textrm{Gal}_{\mathbb{Q}}(f(x))=S_n$ or $A_n$, then ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)=\mathbb{Z}$. In seeking to examine his result in the genus $g=2$ case supposing other Galois groups, we calculate $\textrm{End}_{\overline{\mathbb{Q}}}(J)\otimes_{\mathbb{Z}} \mathbb{F}_2$ for a genus $2$ curve where $f(x)$ is irreducible. In particular, we show that unless the Galois group is $S_5$ or $A_5$, the Galois group does not determine ${\textrm{End}}_{\overline{\mathbb{Q}}}(J)$.

Categories:11G10, 20C20

2. CMB 2009 (vol 53 pp. 58)

Dąbrowski, Andrzej; Jędrzejak, Tomasz
Ranks in Families of Jacobian Varieties of Twisted Fermat Curves
In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

Keywords:Fermat curve, Jacobian variety, elliptic curve, canonical height
Categories:11G10, 11G05, 11G50, 14G05, 11G30, 14H45, 14K15

3. CMB 2009 (vol 53 pp. 95)

Ghioca, Dragos
Towards the Full Mordell-Lang Conjecture for Drinfeld Modules
Let $\phi$ be a Drinfeld module of generic characteristic, and let X be a sufficiently generic affine subvariety of $\mathbb{G_a^g}$. We show that the intersection of X with a finite rank $\phi$-submodule of $\mathbb{G_a^g}$ is finite.

Keywords:Drinfeld module, Mordell-Lang conjecture
Categories:11G09, 11G10

4. CMB 2009 (vol 52 pp. 237)

Ghioca, Dragos
Points of Small Height on Varieties Defined over a Function Field
We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree $1$ over a finite field.

Keywords:heights, Bogomolov conjecture
Categories:11G50, 11G25, 11G10

5. CMB 2004 (vol 47 pp. 271)

Naumann, Niko
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms
We study the interplay between canonical heights and endomorphisms of an abelian variety $A$ over a number field $k$. In particular we show that whenever the ring of endomorphisms defined over $k$ is strictly larger than $\Z$ there will be $\Q$-linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of $A(k)$.

Categories:11G10, 14K15

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