  Our goal in this paper is to investigate four conjectures proposed by Daqing
Wan about the stable behavior of a geometric $\mathbb{Z}_p$-tower of curves
$X_{\infty}/X$. Let $h_n$ be the class number of the $n$-th layer in
$X_{\infty}/X$. It is known from Iwasawa theory that there are integers
$\mu(X_{\infty}/X), \lambda(X_{\infty}/X)$ and $ \nu(X_{\infty}/X)$ such that
the $p$-adic valuation $v_p(h_n)$ equals to $\mu(X_{\infty}/X) p^n +
\lambda(X_{\infty}/X) n+ \nu(X_{\infty}/X)$ for $n$ sufficiently large. Let
$\mathbb{Q}_{p,n}$ be the splitting field (over $\mathbb{Q}_p$) of the
zeta-function of $n$-th layer in $X_{\infty}/X$. The $p$-adic Wan-Riemann
Hypothesis conjectures that the extension degree
$[\mathbb{Q}_{p,n}:\mathbb{Q}_p]$ goes to infinity as $n$ goes to infinity.
After motivating and introducing the conjectures, we prove the $p$-adic
Wan-Riemann Hypothesis when $\lambda(X_{\infty}/X)$ is nonzero.
