Class numbers and p-ranks in Zpd-towers

Abstract

Abstract To extend Iwasawa s classical theorem from Z p -towers to Z p d -towers, Greenberg conjectured that the exponent of p in the n-th class number in a Z p d -tower of a global field K ramified at finitely many primes is given by a polynomial in p n and n of total degree at most d for sufficiently large n. This conjecture remains open for d ≥ 2 . In this paper, we prove that this conjecture is true in the function field case. Further, we propose a series of general conjectures on p-adic stability of zeta functions in a p-adic Lie tower of function fields.