On the 16-rank of class groups of $\mathbb{Q}(\sqrt{-3p})$ for primes $p$ congruent to 1 modulo 4
Volume 202 / 2022
Acta Arithmetica 202 (2022), 1-20
MSC: Primary 11N45, 11R29; Secondary 11R44, 11N36.
DOI: 10.4064/aa200422-9-6
Published online: 3 December 2021
Abstract
For fixed $q\in \{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb {Q}(\sqrt {-qp})$ has order divisible by $16$. We show that this density is equal to $1/8$, in line with a more general conjecture of Gerth. Vinogradov’s method is the key analytic tool for our work.
