Main stability conjecture for zeta functions of algebraic-geometric $\mathbb{Z}_p$-towers

Let $(C_n)_{n\geq 0}$ be a $\mathbb{Z}_p$-tower of curves from algebraic geometry. Let $g_n$ be the genus of $C_n$, let $d_\alpha(n)$ be the slope-counting quantity defined in the paper, and write the reciprocal roots of $P(C_n,s)$ as $\alpha_i(n)$, with $v_q(q)=1$. Main stability conjecture. The following hold:

1. There are constants $a,b,c$ with $a>0$ such that $g_n=ap^{2n}+bp^n+c$ for all sufficiently large $n$.
2. For every fixed $\alpha\in[0,\infty)$, there are constants $\mu_1(\alpha),\mu_2(\alpha)$ such that $d_\alpha(n)=p^n\mu_1(\alpha)+\mu_2(\alpha)$ for all sufficiently large $n$.
3. The multiset of $q$-slopes $\{v_q(\alpha_i(n))\}$ is equidistributed in $[0,1]$ as $n\to\infty$.
4. There is a positive integer $n_0$, depending on the tower, such that for all $n>n_0$, the rescaled slopes $\{p^nv_q(\alpha_i(n))\}$ are explicitly determined by their values for $0\leq n\leq n_0$ using finitely many arithmetic progressions.

This is the paper’s main geometric stability prediction and combines genus growth, fixed-slope asymptotics, global slope distribution, and finite slope-pattern behavior; the supplied text gives no resolution status.