On the $p$-ranks of class groups of certain Galois extensions

TL;DR

This paper derives an exact formula for the $p$-rank of class groups in certain Galois extensions, providing numerical criteria and analyzing the distribution of $3$-ranks for primes in specific residue classes.

Contribution

It introduces a Galois cohomological method to precisely compute the $p$-rank and establishes numerical criteria for bounds, extending previous work on class group ranks.

Findings

Exact formula for $p$-rank in terms of Selmer groups

Numerical criteria for upper and lower bounds of $p$-rank

Distribution analysis of $3$-ranks for primes in specific residue classes

Abstract

Let $p$ be an odd prime, let $N$ be a prime with $N \equiv 1 \pmod{p}$, and let $\zeta_p$ be a primitive $p$-th root of unity. We study the $p$-rank of the class group of $\mathbb{Q}(\zeta_p, N^{1/p})$ 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 $4,7 \pmod{9}$.