Decomposition of Some Jacobian Varieties of Dimension 3

摘要

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 H_3. We determine this subvariety explicitly. For any given moduli point p ∈ H_3 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, E_1, and E_2 in terms of the absolute invariants t_1,...,t_6 as in. 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 X and E starting form the j-invariant of E_1.