Solubility for families of norm equations coming from abelian number fields

TL;DR

This paper estimates how often values of certain binary quadratic forms are norms from abelian number fields with class number one, using sieve theory and geometry of numbers.

Contribution

It provides an order of magnitude estimate for the frequency of norm values from abelian number fields for specific quadratic forms.

Findings

Derived the order of magnitude for norm values of quadratic forms from abelian fields.

Applied sieve theory and geometry of numbers in the analysis.

Focused on forms irreducible over $\\mathbb{Q}$ and fields with class number 1.

Abstract

For $F \in \mathbb{Z}[s,t]$ a binary quadratic form which is irreducible over $\mathbb{Q}$, and $L$ an abelian number field with class number $1$, we obtain the order of magnitude for the number of values $F(s,t)$ which are a norm from $L$. Our result relies on the fundamental lemma of sieve theory and on geometry of numbers.