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Diagonal quartic surfaces with a Brauer–Manin obstruction
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 4 / April 2023
- Published online by Cambridge University Press:
- 17 March 2023, pp. 659-710
- Print publication:
- April 2023
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- Article
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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.
