Improving the trivial bound for $\ell$-torsion in class groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917914081230848 |
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| author | Oliver, Robert J. Lemke Zaman, Asif |
| author_facet | Oliver, Robert J. Lemke Zaman, Asif |
| contents | For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. In fact, we obtain an explicit log-power saving. This is the first general unconditional saving over the trivial bound that holds for all $K$ and all $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improving the trivial bound for $\ell$-torsion in class groups Oliver, Robert J. Lemke Zaman, Asif Number Theory 11R29, 11R42 For any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. In fact, we obtain an explicit log-power saving. This is the first general unconditional saving over the trivial bound that holds for all $K$ and all $\ell$. |
| title | Improving the trivial bound for $\ell$-torsion in class groups |
| topic | Number Theory 11R29, 11R42 |
| url | https://arxiv.org/abs/2502.03464 |