The symbol length for elementary type pro-$p$ groups and Massey products
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913303031185408 |
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| author | Efrat, Ido |
| author_facet | Efrat, Ido |
| contents | For a prime number $p$ and an integer $m\geq2$, we prove that the symbol length of all elements of $m$-fold Massey products in $H^2(G,\mathbb{F}_p)$, for pro-$p$ groups $G$ of elementary type, is bounded by $(m^2/4)+m$. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro-$p$ Galois groups $G=G_F(p)$ of fields $F$ which contain a root of unity of order $p$. More generally, we provide such a uniform bound for the symbol length of all pullbacks $ρ^*(\barω)$ of a given cohomology element $\barω\in H^n(\bar G,\mathbb{F}_p)$, where $\bar G$ is a finite $p$-group, $n\geq2$, and $ρ\colon G\to \bar G$ is a pro-$p$ group homomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_02249 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The symbol length for elementary type pro-$p$ groups and Massey products Efrat, Ido Number Theory Primary 12G05, Secondary 20J06, 55S30, 12F10, 12E30 For a prime number $p$ and an integer $m\geq2$, we prove that the symbol length of all elements of $m$-fold Massey products in $H^2(G,\mathbb{F}_p)$, for pro-$p$ groups $G$ of elementary type, is bounded by $(m^2/4)+m$. Assuming the Elementary Type Conjecture, this applies to all finitely generated maximal pro-$p$ Galois groups $G=G_F(p)$ of fields $F$ which contain a root of unity of order $p$. More generally, we provide such a uniform bound for the symbol length of all pullbacks $ρ^*(\barω)$ of a given cohomology element $\barω\in H^n(\bar G,\mathbb{F}_p)$, where $\bar G$ is a finite $p$-group, $n\geq2$, and $ρ\colon G\to \bar G$ is a pro-$p$ group homomorphism. |
| title | The symbol length for elementary type pro-$p$ groups and Massey products |
| topic | Number Theory Primary 12G05, Secondary 20J06, 55S30, 12F10, 12E30 |
| url | https://arxiv.org/abs/2212.02249 |