Odd torsion Brauer elements and arithmetic of diagonal quartic surfaces over number fields

Abstract

We use recent advances in the local evaluation of Brauer elements to study the role played by odd torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over arbitrary number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, under which the odd torsion does not obstruct weak approximation. We also note a systematic way to produce K 3 K3 surfaces over Q 2 \mathbb {Q}_2 with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero.

An example of a Brauer–Manin obstruction to weak approximation at a prime with good reduction

“…Theorem 5.5). This theorem proves what whose already claimed by Ieronymous after some computational evidences, see [18,Remark 2.6].…”

“…Let p be a prime above 2 and O p be the valuation ring of k p . We have that 2 • α -1 ∈ O p if and only if α 2 ≡ 0 mod 8; we can define X α to be the O p -scheme defined by equation (18). If α 2 ≡ 0 mod 8, then X α is smooth and hence X α has good reduction at p. Theorem 7.2.…”