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

and

such that the p-adic valuation $v_p(h_n)$ equals to

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.

中文翻译：

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的p -adic Wan-Riemann假设${\mathbb{ž}}_p$-曲线塔

我们的目标是研究大庆万提出的关于几何的稳定行为的四个猜想 ${\mathbb{ž}}_p$-曲线塔 $X_{\infty }/X$。让$H_{ñ}$是第n层的类编号$X_{\infty }/X$。从岩泽学说知道整数

和

这样p- adic估值$v_p（H_{ñ}）$ 等于

对于n足够大。让${\mathbb{问}}_{p，ñ}$ 是分割字段（在 ${\mathbb{问}}_p$）中第n层的zeta函数$X_{\infty }/X$。该p进制万黎曼假设臆想扩展度$[{\mathbb{问}}_{p，ñ}：{\mathbb{问}}_p]$变为无穷大，因为n变为无穷大。在激发和介绍了这些猜想之后，我们证明了p -adic Wan-Riemann假说在$\lambda （X_{\infty }/X）$ 不为零。