On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-3p})$ for primes $p$ congruent to $1$ modulo $4$

Margherita Piccolo

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arXiv · 2106.04309

On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-3p})$ for primes $p$ congruent to $1$ modulo $4$

Margherita Piccolo

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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.

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BibTeX RIS

Margherita Piccolo. 2021-06-08. On the $16$-rank of class groups of $\mathbb{Q}(\sqrt{-3p})$ for primes $p$ congruent to $1$ modulo $4$. https://arxiv.org/abs/2106.04309

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