Class groups of Kummer extensions via cup products in Galois cohomology

Abstract

We use Galois cohomology to study the p p -rank of the class group of Q ( N 1 / p ) \mathbf {Q}(N^{1/p}) , where N ≡ 1 mod p N \equiv 1 \bmod {p} is prime. We prove a partial converse to a theorem of Calegari–Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case p = 5 p = 5 , we prove a complete characterization of the 5 5 -rank of the class group of Q ( N 1 / 5 ) \mathbf {Q}(N^{1/5}) in terms of whether or not ∏ k = 1 ( N − 1 ) / 2 k k \prod _{k=1}^{(N-1)/2} k^{k} and 5 − 1 2 \frac {\sqrt {5} - 1}{2} are 5 5 th powers mod N N .