On the Prym map of degree 4 cyclic covers of hyperelliptic curves

In this paper, we study the Prym map associated to degree 4 étale cyclic covers of genus $g$ hyperelliptic curves restricted to the irreducible component ${\mathcal{R}\mathcal{H}}_g[4{]}^{hyp}$ of the moduli space of such covers where an intermediate cover is hyperelliptic. We show that for $g\ge 3$ the Prym map is injective on ${\mathcal{R}\mathcal{H}}_g[4{]}^{hyp}$. In the case $g=2$ (where ${\mathcal{R}\mathcal{H}}_2[4{]}^{hyp}={\mathcal{R}\mathcal{H}}_2[4]$) we prove that non-empty fibers of the Prym map, apart from two exceptional fibers, are isomorphic to the projective line without 8 points. Moreover, we obtain a new description of the space ${\mathcal{R}\mathcal{H}}_g[4{]}^{hyp}$ in terms of tuples of complex numbers and find equations of hyperelliptic curves arising from such covers.