Massey products in the étale cohomology of number fields
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918222942437376 |
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| author | Ahlqvist, Eric Carlson, Magnus |
| author_facet | Ahlqvist, Eric Carlson, Magnus |
| contents | We give formulas for 3-fold Massey products in the étale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an infinite $p$-class field tower, where $p$ is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of $p$-extensions, for the vanishing of 3-fold Massey products is given. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman's $(3,3)$-conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of $G_{\mathbb{Q},S}$ which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_06353 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Massey products in the étale cohomology of number fields Ahlqvist, Eric Carlson, Magnus Number Theory Algebraic Geometry 11R34, 14F20, 14G25 We give formulas for 3-fold Massey products in the étale cohomology of the ring of integers of a number field and use these to find the first known examples of imaginary quadratic fields with class group of $p$-rank two possessing an infinite $p$-class field tower, where $p$ is an odd prime. Furthermore, a necessary and sufficient condition, in terms of class groups of $p$-extensions, for the vanishing of 3-fold Massey products is given. As a consequence, we give an elementary and sufficient condition for the infinitude of class field towers of imaginary quadratic fields. We also disprove McLeman's $(3,3)$-conjecture. Lastly, we relate the vanishing of Massey products to the existence of Galois representations of $G_{\mathbb{Q},S}$ which realize an unexpectedly large class group for certain extensions of a quadratic imaginary number field. |
| title | Massey products in the étale cohomology of number fields |
| topic | Number Theory Algebraic Geometry 11R34, 14F20, 14G25 |
| url | https://arxiv.org/abs/2207.06353 |