Abelian p-extensions with restricted p-ramification and the cyclotomic ℤ2-extension of ℚ (√q)
Abstract
We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted p-ramification. Let K be an algebraic number field, and p a prime number which splits into two distinct primes p, p' in K. Assume that K and p satisfies several (somewhat strict) conditions. Let K∞ /K be the cyclotomic ℤp-extension. In the present paper, we give a method to study the structure of the unramified Iwasawa module X (K∞) by using abelian p-extensions unramified outside p. We give a sufficient condition for |X (K∞)| to be finite in terms of such extensions. We also give a similar criterion for X (K∞) to be finitely generated over ℤp. In the latter part of the present paper, we consider the case where k = ℚ (√q) with an odd prime number q and apply our results to the cyclotomic ℤ2-extension k∞ /k. We give a necessary and sufficient condition for X (k∞) to be cyclic over ℤ2, which is different from the former results given by either Mouhib-Movahhedi or Mizusawa-Mouhib. We also give several sufficient conditions for the validity of Greenberg's conjecture.
Related
- On finiteness properties of the unramified Iwasawa module of a ℤp-extension with restricted ramification
- Multiplicator freeness for restricted-ramification Galois groups over global function fields
- Ramification invariants of pure extensions
- A lower bound for the Call-Silverman height in cyclotomic extensions
- A Global Generic Locus for Elementary Abelian p-covers of P^1 with two Ramification Points
BibTeX
@misc{itoh2026abelian,
title = {{Abelian $p$-extensions with restricted $p$-ramification and the cyclotomic $\mathbbZ_2$-extension of $\mathbbQ (\sqrtq)$}},
author = {Tsuyoshi Itoh and Naoki Kumakawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25866},
eprint = {2609.25866},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25866},
}
RIS
TY - UNPB
TI - Abelian p-extensions with restricted p-ramification and the cyclotomic ℤ_2-extension of ℚ (√q)
AU - Tsuyoshi Itoh
AU - Naoki Kumakawa
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.25866
UR - https://arxiv.org/abs/2609.25866
AB - We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted p-ramification. Let K be an algebraic number field, and p a prime number which splits into two distinct primes p, p' in K. Assume that K and p satisfies several (somewhat strict) conditions. Let K_∞ /K be the cyclotomic ℤ_p-extension. In the present paper, we give a method to study the structure of the unramified Iwasawa module X (K_∞) by using abelian p-extensions unramified outside p. We give a sufficient condition for |X (K_∞)| to be finite in terms of such extensions. We also give a similar criterion for X (K_∞) to be finitely generated over ℤ_p. In the latter part of the present paper, we consider the case where k = ℚ (√q) with an odd prime number q and apply our results to the cyclotomic ℤ_2-extension k_∞ /k. We give a necessary and sufficient condition for X (k_∞) to be cyclic over ℤ_2, which is different from the former results given by either Mouhib-Movahhedi or Mizusawa-Mouhib. We also give several sufficient conditions for the validity of Greenberg's conjecture.
ER -