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

Kevin Chang, Dušan Dragutinović, Steven R. Groen, Yuxin Lin, Natalia Pacheco-Tallaj, Deepesh Singhal

Published 2025-02-13Version 1

In this paper 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.