On the $p$-ranks of class groups of certain Galois extensions
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arXiv
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| Format: | Preprint |
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2024
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| author | Asarhasa, Ufuoma Gambheera, Rusiru Kundu, Debanjana Lon-wo, Enrique Nunez Sheth, Arshay |
| author_facet | Asarhasa, Ufuoma Gambheera, Rusiru Kundu, Debanjana Lon-wo, Enrique Nunez Sheth, Arshay |
| contents | Let $p$ be an odd prime, let $N$ be a prime with $N \equiv 1 \pmod{p}$, and let $ζ_p$ be a primitive $p$-th root of unity. We study the $p$-rank of the class group of $\mathbb{Q}(ζ_p, N^{1/p})$ using Galois cohomological methods and obtain an exact formula for the $p$-rank in terms of the dimensions of certain Selmer groups. Using our formula, we provide a numerical criterion to establish upper and lower bounds for the $p$-rank, analogous to the numerical criteria provided by F.~Calegari--M.~Emerton and K.~Schaefer--E.~Stubley for the $p$-ranks of the class group of $\mathbb{Q}(N^{1/p})$. In the case $p=3$, we use Redei matrices to provide a numerical criterion to exactly calculate the $3$-rank, and also study the distribution of the $3$-ranks as $N$ varies through primes which are $4,7 \pmod{9}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $p$-ranks of class groups of certain Galois extensions Asarhasa, Ufuoma Gambheera, Rusiru Kundu, Debanjana Lon-wo, Enrique Nunez Sheth, Arshay Number Theory 11R29, 11R34 Let $p$ be an odd prime, let $N$ be a prime with $N \equiv 1 \pmod{p}$, and let $ζ_p$ be a primitive $p$-th root of unity. We study the $p$-rank of the class group of $\mathbb{Q}(ζ_p, N^{1/p})$ using Galois cohomological methods and obtain an exact formula for the $p$-rank in terms of the dimensions of certain Selmer groups. Using our formula, we provide a numerical criterion to establish upper and lower bounds for the $p$-rank, analogous to the numerical criteria provided by F.~Calegari--M.~Emerton and K.~Schaefer--E.~Stubley for the $p$-ranks of the class group of $\mathbb{Q}(N^{1/p})$. In the case $p=3$, we use Redei matrices to provide a numerical criterion to exactly calculate the $3$-rank, and also study the distribution of the $3$-ranks as $N$ varies through primes which are $4,7 \pmod{9}$. |
| title | On the $p$-ranks of class groups of certain Galois extensions |
| topic | Number Theory 11R29, 11R34 |
| url | https://arxiv.org/abs/2408.04481 |