Expand all Collapse all | Results 1 - 5 of 5 |
1. CMB 2011 (vol 55 pp. 38)
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)
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)
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)
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)
Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms |
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 |