Greenberg’s conjecture for real quadratic fields and the cyclotomic ℤ₂-extensions

Pagani, Lorenzo

doi:10.1090/mcom/3712

Greenberg’s conjecture for real quadratic fields and the cyclotomic $\mathbb {Z}_2$-extensions
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Math. Comp. 91 (2022), 1437-1467 Request permission

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.

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