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arXiv · 2201.04573

Diagonal quartic surfaces with a Brauer-Manin obstruction

Tim Santens
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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.

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Tim Santens. 2022-01-12. Diagonal quartic surfaces with a Brauer-Manin obstruction. https://arxiv.org/abs/2201.04573

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