$p$-adic Wan-Riemann Hypothesis for $\mathbb{Z}_p$-towers of curves

Abstract

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.