Counting integral points on indefinite ternary quadratic equations over number fields

TL;DR

This paper develops an asymptotic formula for counting integral solutions to indefinite ternary quadratic equations over number fields, linking local densities and prime factors.

Contribution

It provides a new asymptotic formula for integral points on indefinite ternary quadratic forms over number fields, incorporating local densities and prime factors.

Findings

Finite part of the asymptotic formula expressed as product of local densities and (1 - p^{-1}) over primes.

Established conditions under which the formula applies, involving the form's determinant and representation of integers.

Extended classical results to the setting of number fields with indefinite quadratic forms.

Abstract

We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number field. In particular, we prove that the finite part of this asymptotic formula is given by the product of local density times $1-p^{-1}$ over all finite primes $p$ over $\Bbb Z$.