Greenberg’s conjecture for real quadratic fields and the cyclotomic $\mathbb {Z}_2$-extensions
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Abstract:
Let $\mathcal {A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $\mathbb {Z}_2$-extension of a real quadratic number field $F$. The cardinality of $\mathcal {A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $\mathcal {A}_n$’s stabilizes for the real fields $F=\mathbb {Q}(\sqrt {f})$ for any integer $0<f<10000$. Equivalently Greenberg’s conjecture holds for those fields.References
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Additional Information
- Lorenzo Pagani
- Affiliation: Dipartimento di matematica, università di Roma “La Sapienza”, piazzale Aldo Moro 5, 00185 Roma, Italy
- Email: pagani@mat.uniroma1.it
- Received by editor(s): March 23, 2021
- Received by editor(s) in revised form: August 8, 2021, September 28, 2021, and October 17, 2021
- Published electronically: December 30, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 91 (2022), 1437-1467
- MSC (2020): Primary 11R29, 11Y40; Secondary 11R23
- DOI: https://doi.org/10.1090/mcom/3712
- MathSciNet review: 4405501