The p -rank stratification of the moduli space of double covers of a fixed elliptic curve

In this article, we investigate the p -rank stratification of the moduli space of curves of genus g that admit a double cover to a fixed elliptic curve E in characteristic $p>2$ . We show that the closed p -rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of E of all the possible values of the p -rank on this moduli space.