Diagonal quartic surfaces with a Brauer-Manin obstruction

TL;DR

This paper studies diagonal quartic surfaces and quantifies how many exhibit a Brauer-Manin obstruction to the Hasse principle, providing an asymptotic count and analyzing the nature of their Brauer groups.

Contribution

It derives an asymptotic formula for the number of such surfaces with a Brauer-Manin obstruction, extending Heath-Brown's method and showing the non-existence of a uniform generator.

Findings

Asymptotic count of surfaces with Brauer-Manin obstruction

Generalization of Heath-Brown's sum method

No uniform generator exists for the family

Abstract

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer-Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.