On ℤ₂-extensions of real quadratic fields with class group of $2$-rank three

@misc{saito2026ℤ2extensions,
  title = {{On $\mathbbZ_2$-extensions of real quadratic fields with class group of $2$-rank three}},
  author = {Yuito Saito},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23063},
  eprint = {2609.23063},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23063},
}

TY  - UNPB
TI  - On ℤ_2-extensions of real quadratic fields with class group of $2$-rank three
AU  - Yuito Saito
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23063
UR  - https://arxiv.org/abs/2609.23063
AB  - In this paper, we study the unramified Iwasawa module over the cyclotomic ℤ_2-extension of the real quadratic field ℚ(√(p_1p_2p_3p_4)), where p_1, p_2, p_3, and p_4 are distinct odd prime numbers. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The criterion is based on the nonexistence of a $2$-group with a certain prescribed quotient. Using this criterion, we construct an infinite family of such real quadratic fields with unramified Iwasawa module of type ℤ/4ℤ ⊕ ℤ/2ℤ ⊕ ℤ/2ℤ and $2$-class group of type ℤ/2ℤ ⊕ ℤ/2ℤ ⊕ ℤ/2ℤ. This gives the first example of an infinite family of real quadratic fields whose ideal class group has $2$-rank three and whose cyclotomic ℤ_2-extension is totally ramified and satisfies Greenberg's conjecture.
ER  -