Representations of binary quadratic forms by quaternary quadratic forms

TL;DR

This paper establishes a local-global principle for representing binary quadratic forms with quaternary forms, combining ergodic theory, measure classification, and counting techniques.

Contribution

It introduces a novel approach that integrates ergodic methods, measure classification, and the determinant method to solve the density problem for quadratic form representations.

Findings

Proves a local-global principle for primitive representations.

Reduces the density problem to a counting problem on an affine variety.

Successfully applies the determinant method to solve the counting problem.

Abstract

We prove a local-global principle for primitive representations of binary quadratic forms by quaternary quadratic forms. Our method is a variant of Linnik's ergodic method showing density for certain homogenous toral sets. The central ingredient is a measure classification result of Einsiedler and Lindenstrauss for actions of rank two diagonalizable groups on quotients of products of $\mathrm{SL}_2$. This rigidity result together with an application of the Siegel mass formula reduces the density problem to a counting problem on a certain affine variety. We solve that counting problem using the determinant method of Bombieri-Pila and Heath-Brown.