LAUSR

Abstract Our goal in this paper is to investigate four conjectures, proposed by Daqing Wan, about the stable behavior of a geometric Z p -tower of curves X ∞ / X . Let h n 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 v p ( h n ) equals to for n sufficiently large. Let Q p , n be the splitting field (over Q p ) of the zeta-function of n-th layer in X ∞ / X . The p-adic Wan-Riemann Hypothesis conjectures that the extension degree [ Q p , n : 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 λ ( X ∞ / X ) is nonzero.