Class Groups of Kummer Extensions via Cup Products in Galois Cohomology

In this work, we study the class group of the number field $\Q(N^{1/p})$ where $p$ is an odd prime number and $N > 1$ is coprime to $p$ and $p$th-power-free. We generalize theorems of Calegari--Emerton and Lecouturier and prove a result analogous to the Herbrand--Ribet Theorem for $\Q(\zeta_p)$. Our main tool is Galois cohomology. We relate the $p$-cotorsion in the class group of $K = \Q(N^{1/p})$ to Selmer subgroups in the cohomology of a $(p-1)$-dimensional Galois representation. The Selmer conditions we used are designed to detect the vanishing of a global cup product using only local information. We then bound the size of the cohomology of that representation through a detailed study of the cohomology of powers of the cyclotomic character, making use of several duality theorems. This allows us to bound the rank $r_K$ of the $p$-cotorsion of the class group of $K$ above and below in terms of the numbers of prime factors of $N$ satisfying certain congruence conditions modulo $p$. When $p = 3$ we completely determine the rank $r_K$ in terms of a matrix of cubic residue symbols, and when $p = 5$ and $N \equiv 1 \bmod p$ is prime, we completely determine this rank in terms of whether or not $\prod_{k=1}^{(N-1)/2} k^k$ and $\frac{\sqrt{5}-1}{2}$ are $5$th powers modulo $N$. For $p = 5$ and $p = 7$ we provide some data on the distribution of the rank $r_K$ and use this to formulate a conjecture on this distribution.