On the Brauer group of diagonal quartic surfaces

Abstract

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Masumi Mizukami.

Appendix "The Fermat quartic as a Kummer surface (after Mizukami)", by Sir Peter Swinnerton-Dyer

References in corpus (2)

• Point counting on reductions of CM elliptic curves
• A finiteness theorem for the Brauer group of abelian varieties and K3 surfaces

Cited by in corpus (16)

• Transcendental obstructions to weak approximation on general K3 surfaces
• Finiteness theorems for K3 surfaces and abelian varieties of CM type
• Transcendental Brauer groups of products of CM elliptic curves
• Effective bounds for Brauer groups of Kummer surfaces over number fields
• Bad reduction of the Brauer-Manin obstruction
• Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces
• Degree and the Brauer-Manin obstruction
• Failure of the Hasse principle on general K3 surfaces
• A uniform bound on the Brauer groups of certain log K3 surfaces
• An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction
• Effective computation of Picard groups and Brauer-Manin obstruction of degree two K3 surfaces over number fields
• Effectivity of Brauer-Manin obstructions on surfaces
• Brauer-Manin obstructions on degree 2 K3 surfaces
• Arithmetic of rational points and zero-cycles on Kummer varieties
• Good reduction of the Brauer-Manin obstruction
• Transcendental Brauer-Manin obstructions on singular K3 surfaces