On Greenberg's generalized conjecture for families of number fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909607717240832 |
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| author | Do, Thong Nguyen Quang |
| author_facet | Do, Thong Nguyen Quang |
| contents | For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{Λ}$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the maximal abelian unramified pro-$p$-extension of $\tilde{k}$. Greenberg's generalized conjecture (GGC for short) asserts that the $\tildeΛ$-module $X(\tilde{k})$ is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special ${\mathbb Z}^2_p$-extension of $k$, and these conditions are partially or entirely fullfilled by certain families of number fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07529 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Greenberg's generalized conjecture for families of number fields Do, Thong Nguyen Quang Number Theory 11R23 11R34 11R37 For a number field $k$ and an odd prime $p$, let $\tilde{k}$ be the compositum of all the ${\mathbb Z}_p$-extensions of $k$, $\tilde{Λ}$ the associated Iwasawa algebra, and $X(\tilde{k})$ the Galois group over $\tilde{k}$ of the maximal abelian unramified pro-$p$-extension of $\tilde{k}$. Greenberg's generalized conjecture (GGC for short) asserts that the $\tildeΛ$-module $X(\tilde{k})$ is pseudo-null. Very few theoritical results toward GGC are known. We show here that for an imaginary k, GGC is implied by certain pseudo-nullity conditions imposed on a special ${\mathbb Z}^2_p$-extension of $k$, and these conditions are partially or entirely fullfilled by certain families of number fields. |
| title | On Greenberg's generalized conjecture for families of number fields |
| topic | Number Theory 11R23 11R34 11R37 |
| url | https://arxiv.org/abs/2505.07529 |