Localized quantitative Boston–Markin conjectures over function fields

Let $G$ be a nontrivial finite group, let $d_\lhd(G)$ be the least number of conjugacy classes in $G$ that generate $G$, and fix a prime power $q$ coprime to $|G|$. For prescribed generating conjugacy classes $C_1,\dots,C_{d_\lhd(G)}$, let $\mathcal E_q^C(G;n_1,\dots,n_{d_\lhd(G)})$ be the family of regular $G$-extensions of $\mathbb F_q(T)$ split completely at infinity, with inertia generators in $C=\bigcup_jC_j$ and with the corresponding branch-polynomial degrees equal to $n_j$. Let $\zeta_q(2)$ denote the zeta function of $\mathbb F_q[T]$ evaluated at $2$.

Localized function-field Boston–Markin conjectures. There exists a positive real number $\delta_q^{G,C}$ such that, as $n_1,\dots,n_{d_\lhd(G)}\to\infty$,

$$\sum_{K\in\mathcal E_q^C(G;n_1,\dots,n_{d_\lhd(G)})}\mathbf{1}_{|\operatorname{ram}(K)|=d_\lhd(G)}\sim\delta_q^{G,C}\prod_{j=1}^{d_\lhd(G)}\frac{\zeta_q(2)}{n_j}\,\left|\mathcal E_q^C(G;n_1,\dots,n_{d_\lhd(G)})\right|.$$

Moreover, as soon as at least one of the $n_j$ tends to infinity,

$$\sum_{K\in\mathcal E_q^C(G;n_1,\dots,n_{d_\lhd(G)})}(-1)^{|\operatorname{ram}(K)|}=o\left(\left|\mathcal E_q^C(G;n_1,\dots,n_{d_\lhd(G)})\right|\right).$$

These are localized analogues over $\mathbb F_q(T)$ of the quantitative and Möbius conjectures for number fields. The source motivates them using available upper and lower bounds for the relevant extension families; their asserted asymptotics remain conjectural.