Critical groups in harmonic abelian quotients

A harmonic cover of graphs $p:\tilde{X}\to X$ induces a surjective pushforward morphism $p_{\ast }:\mathrm{Jac}(\tilde{X})\to \mathrm{Jac}(X)$ on the critical groups. In the case when $p$ is Galois with abelian Galois group, we compute the order of the kernel of $p_{\ast }$, and hence the relationship between the numbers of spanning trees of $\tilde{X}$ and $X$, in terms of Zaslavsky’s bias matroid associated to the cover $p:\tilde{X}\to X$.