Diagonal quartic surfaces and transcendental elements of the Brauer group

Abstract

We exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

Effective computation of Picard groups and Brauer-Manin obstructions of degree two $$K3$$ surfaces over number fields

“…Examples of computations of Brauer-Manin obstructions on K3 surfaces can be found in [Bri06], [HVV11], [Ie10], [SS05], [Wit04]. The results here, combined with results in [CS], imply as well an effective bound for the order of Br(Xk) Gal(k/k) .…”

Odd order Brauer–Manin obstruction on diagonal quartic surfaces

“…Let The first example of a diagonal quartic surface D with a non-constant 3-torsion element in Br(D) was found by Thomas Preu [13]. Preu's surface D = [1,3,4,9] has an obvious rational point, and he shows that for any rational point on his surface one has 3|yw while there are Q 3 -points not satisfying this condition. This failure of weak approximation is explained by a 3-torsion element in Br(D).…”

“…For everywhere locally soluble diagonal quartic surfaces D over Q whose coefficients a, b, c, d are general enough, Bright [4] has shown using [9,10] Conditional results on the existence of rational points on diagonal quartic surfaces can be found in [19] and [20, Thm. 1.51].…”

Rational points and zero-cycles on rationally connected varieties over number fields

“…Another example of a rationally connected variety X over a number field k such that X(A k ) Br(X) = X(A k ) Br 1 (X) is given in [DLAN17]. Transcendental elements and their influence on the Brauer-Manin set have received a lot of attention for other classes of varieties as well (see [Wit04], [Ier10], [HVAV11], [Pre13], [HVA13], [IS15], [New16], [CV15], [MSTVA16] for examples of K3 and Enriques surfaces for which transcendental elements play a role in the Brauer-Manin set).…”

“…The analogous problem for X(A Q ) Br 1 (X) was solved by Bright [Bri06]. It is known that transcendental elements cannot be ignored in this context: there exist diagonal quartic surfaces X over Q such that X(A Q ) Br(X) = X(A Q ) Br 1 (X) (see [Ier10], [Pre13], [IS15]). There also exist K3 surfaces X over Q such that…”