Capitulation in the absolutely abelian extensions of some fields $\mathbb{Q}(\sqrt{p_1p_2q}, \sqrt{-1})$

TL;DR

This paper investigates the capitulation of 2-ideal classes in certain imaginary bicyclic biquadratic fields, showing that strongly ambiguous classes capitulate in the absolute genus field, which is smaller than the relative genus field.

Contribution

It provides explicit computation of capitulation kernels for three quadratic extensions within the absolute genus field of these specific fields, revealing new capitulation phenomena.

Findings

Strongly ambiguous classes capitulate in the absolute genus field

Capitulation kernels are explicitly computed for three quadratic extensions

The absolute genus field is smaller than the relative genus field in this context

Abstract

We study the capitulation of $2$-ideal classes of an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $\mathbf{k} =\mathbb{Q}(\sqrt{p_1p_2q}, i)$, where $i=\sqrt{-1}$ and $p_1\equiv p_2\equiv-q\equiv1 \pmod 4$ are different primes. For each of the three quadratic extensions $\mathbf{K}/\mathbf{k}$ inside the absolute genus field $\mathbf{k}^{(*)}$ of $\mathbf{k}$, we compute the capitulation kernel of $\mathbf{K}/\mathbf{k}$. Then we deduce that each strongly ambiguous class of $\mathbf{k}/\mathbb{Q}(i)$ capitulates already in $\mathbf{k}^{(*)}$, which is smaller than the relative genus field $(\mathbf{k}/\mathbb{Q}(i))^*$.