Ferraro's multiplicity conjecture for relative zeta functions

Let $R:=A$, $B:=B_A$, and $H:=H_A$. For a positive degree $d$, let $\mathcal{I}(d)$ be the set of ideals in $B$ whose norm has degree $d$, and define the analytic function $\zeta_{H,A}$ as the limit of the partial sums $\zeta_{H,A}(\leq d)$ in $\mathbb{T}_A$. Let $h$ denote the degree of the unramified extension $H/K$, and let $\Xi$ be the canonical point on the relevant curve. Ferraro's multiplicity conjecture. The analytic function $\zeta_{H,A}$ has a zero of multiplicity $h-1$ at $\Xi$. This conjecture is supported by the computations described in the paper and is presented as an analogue of the classical fact that the Dedekind zeta function of a number field has order of vanishing at zero equal to the rank of its unit group.