Asymptotics For Local Solubility Of Diagonal Quadrics Over A Split Quadric Surface

TL;DR

This paper establishes asymptotic formulas for the density of certain diagonal quadric surfaces that are locally soluble and parametrized by rational points on a split quadric surface, excluding some special cases.

Contribution

It provides new asymptotic results for the distribution of locally soluble diagonal quadrics over a split quadric surface, extending understanding of their density.

Findings

Asymptotic formulas for the density of locally soluble diagonal quadrics.

Identification of conditions excluding certain special cases.

Extension of previous results on quadratic forms and local solubility.

Abstract

We prove asymptotics for the density of everywhere locally soluble diagonal quadric surfaces parameterised by rational points on the split quadric surface $y_0y_1=y_2y_3$ which do not satisfy $-y_0y_1=\square$ nor $-y_0y_3=\square$.