Bounds for moments of $\ell$-torsion in class groups
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912152418254848 |
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| author | Koymans, Peter Thorner, Jesse |
| author_facet | Koymans, Peter Thorner, Jesse |
| contents | Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all $r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of $\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of degree $n$ $S_n$-extensions of $k$, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large $r$, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of $\mathbb{Q}$. When $r=1$, our results are new even for the family of all quadratic extensions of $\mathbb{Q}$, leading to an improved upper bound for the count of degree $p$ $D_p$-extensions over $\mathbb{Q}$ (where $D_p$ is the dihedral group of order $2p$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07204 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bounds for moments of $\ell$-torsion in class groups Koymans, Peter Thorner, Jesse Number Theory Fix a number field $k$, integers $\ell, n \geq 2$, and a prime $p$. For all $r \geq 1$, we prove strong unconditional upper bounds on the $r$-th moment of $\ell$-torsion in the ideal class groups of degree $p$ extensions of $k$ and of degree $n$ $S_n$-extensions of $k$, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large $r$, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of $\mathbb{Q}$. When $r=1$, our results are new even for the family of all quadratic extensions of $\mathbb{Q}$, leading to an improved upper bound for the count of degree $p$ $D_p$-extensions over $\mathbb{Q}$ (where $D_p$ is the dihedral group of order $2p$). |
| title | Bounds for moments of $\ell$-torsion in class groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2309.07204 |