Class numbers and p-ranks in \mathbb Z_p^d-towers

2017/12/08 by Daqing Wan, Wan, Daqing
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1712.02906

openalex publication_date 2017/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To extend Iwasawa's classical theorem from \mathbb Z_p-towers to \mathbb Z_p^d-towers, Greenberg conjectured that the exponent of p in the n-th class number in a \mathbb Z_p^d-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.

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