Decomposition of Some Jacobian Varieties of Dimension 3

Abstract

We study degree 2 and 4 elliptic subcovers of hyperelliptic curves of genus 3 defined over 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 $\wp \in{\mathcal H}_3$ 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, …, t ₆ as in [12]. 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 ≅ 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.