A family of abelian varieties rationally isogenous to no Jacobian

Abstract:

Let ${E_d}$, for any $d \in {\mathbf {Q}}{(i)^*}$, be the curve ${x^3} - dx{z^2} = {y^2}z$, and let $g$ be any positive integer. It is shown that, if $d$ is not a square in ${\mathbf {Q}}(i)$ and $g > 1$, the abelian variety $E_d^g$ is not isogenous over ${\mathbf {Q}}(i)$ to the Jacobian of any genus- $g$ curve. The proof proceeds by showing that any curve whose Jacobian is isogenous to $E_d^g$ over ${\mathbf {Q}}(i)$ must be hyperelliptic, and then showing that no hyperelliptic curve can have Jacobian isogenous to $E_d^g$ over ${\mathbf {Q}}(i)$.