Lower bounds on the $\ell$-rank of ideal class groups
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917894761218048 |
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| author | Martin, Daniel E. |
| author_facet | Martin, Daniel E. |
| contents | For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is supported on prime ideals in $F$ over which at least one (rather than each) prime in $K$ has ramification index divisible by $\ell$. This bound holds with a proviso on the Galois group of the normal closure of $K/F$, which is satisfied by towers of Galois extensions, intermediate fields in nilpotent extensions, and intermediate fields in dihedral extensions of degree $8n$, to name a few. We also use our lower bound to prove a new density result on number fields with infinite class field towers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lower bounds on the $\ell$-rank of ideal class groups Martin, Daniel E. Number Theory 11R29 For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is supported on prime ideals in $F$ over which at least one (rather than each) prime in $K$ has ramification index divisible by $\ell$. This bound holds with a proviso on the Galois group of the normal closure of $K/F$, which is satisfied by towers of Galois extensions, intermediate fields in nilpotent extensions, and intermediate fields in dihedral extensions of degree $8n$, to name a few. We also use our lower bound to prove a new density result on number fields with infinite class field towers. |
| title | Lower bounds on the $\ell$-rank of ideal class groups |
| topic | Number Theory 11R29 |
| url | https://arxiv.org/abs/2501.09865 |