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 E1 and E2 is a 2-dimensional subvariety of the hyperelliptic moduli H3. We determine this subvariety explicitly. For any given moduli point p ∈ H3 we determine explicitly if the corresponding genus 3 curve X belongs or not to such family. When it does, we can determine elliptic subcovers E, E1, and E2 in terms of the absolute invariants t1,…, t6 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 E1≅E2 is a 1-dimensional variety which we determine explicitly. We can also determine X and E starting form the j-invariant of E1.