On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-8p})$ for $p\equiv -1\mathrm{mod}4$

Abstract

We use a variant of Vinogradov's method to show that the density of the set of prime numbers $p\equiv -1\mathrm{mod}4$ for which the class group of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-8p})$ has an element of order $16$ is equal to $1/16$, as predicted by the Cohen-Lenstra heuristics.

arXiv comment39 pages, 1 figure; small changes to the introduction and throughout; changed certain arguments from previous version to be shorter and clearer, mainly in Section 6; to appear in GAFA

Cited by in corpus (1)

• $2^{\infty }$-Selmer groups, $2^{\infty }$-class groups, and Goldfeld's conjecture