Class numbers and p-ranks in [formula omitted]-towers

Abstract

To extend Iwasawa's classical theorem from Zp-towers to Zpd-towers, Greenberg conjectured that the exponent of p in the n-th class number in a Zpd-tower of a global field K ramified at finitely many primes is given by a polynomial in pn 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.