Genus 3 hyperelliptic curves with (2, 4, 4)-split Jacobians

Abstract

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over $\mathbb{C}$. The family of genus 3 hyperelliptic curves which have a degree 2 cover to an elliptic curve $E$ and degree 4 covers to elliptic curves $E_1$ and $E_2$ is a 2-dimensional subvariety of the hyperelliptic moduli ${\mathcal{H}}_3$. We determine this subvariety explicitly. For any given moduli point $\mathfrak{p}\in {\mathcal{H}}_3$ we determine explicitly if the corresponding genus 3 curve $\mathcal{X}$ belongs or not to such family. When it does, we can determine elliptic subcovers $E$, $E_1$, and $E_2$ in terms of the absolute invariants $t_1,\dots ,t_6$ as in \cite{hyp_mod_3}. This variety provides a new family of hyperelliptic curves of genus 3 for which the Jacobians completely split. The sublocus of such family when $E_1$ is isomorphic to $E_2$ is a 1-dimensional variety which we determine explicitly. We can also determine $\mathcal{X}$ and $E$ starting form the $j$-invariant of $E_1$.

References in corpus (5)

• Determining equations of families of cyclic curves
• Some special families of hyperelliptic curves
• Bielliptic curves of genus 3 in the hyperelliptic moduli
• On Jacobians of curves with superelliptic components
• Families of genus two curves with many elliptic subcovers

Cited by in corpus (2)

• On Jacobians of curves with superelliptic components
• Heights on algebraic curves