$p$-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.

9 pages

References in corpus (1)

• Zeta functions of Z_p-towers of curves