Abstract
Our goal in this paper is to investigate four conjectures, proposed by Daqing Wan, about the stable behavior of a geometric Zp-tower of curves X∞/X. Let hn be the class number of the n-th layer in X∞/X. It is known from Iwasawa theory that there are integers and such that the p-adic valuation vp(hn) equals to for n sufficiently large. Let Qp,n be the splitting field (over Qp) of the zeta-function of n-th layer in X∞/X. The p-adic Wan-Riemann Hypothesis conjectures that the extension degree [Qp,n:Qp] goes to infinity as n goes to infinity. After motivating and introducing the conjectures, we prove the p-adic Wan-Riemann Hypothesis when λ(X∞/X) is nonzero.
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