2019/12/11 by Alvarenga, Roberto
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1912.05065
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 μ(X∞/X), λ(X∞/X) and ν(X∞/X) such that the p-adic valuation vp(hn) equals to μ(X∞/X) pn + λ(X∞/X) n+ ν(X∞/X) 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.