Odd order obstructions to the Hasse principle on general K3 surfaces

Abstract

We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface Y of degree 2 over Q together with a three torsion Brauer class α that is unramified at all primes except for 3, but ramifies at all 3-adic points of Y . Motivated by Hodge theory, the pair (Y, α) is constructed from a cubic fourfold X of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for α. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold X (and hence the fibers) over Q 3 and local solubility at all other primes. 1 Brauer-Manin obstructions to the Hasse principle (see §6 for details):the constant classes Br 0 (X) := im (Br(k) → Br(X)) , and the algebraic classes Br 1 (X) := ker Br(X) → Br(X) . Elements ofBr(X) that are not algebraic are called transcendental. Brauer classes of odd order, whether algebraic or transcendental, had previously only been known to obstruct weak approximation on K3 surfaces [Pre13, IS15]. Ieronymou and Skorobogatov [IS15] showed that elements of odd order can never obstruct the Hasse principle on smooth diagonal quartics in P 3 Q . Their work prompted them to ask if there exists a locally soluble K3 surface over a number field with an odd-order Brauer-Manin obstruction to the Hasse principle [IS15, p. 183]. Since then, Skorobogatov and Zarhin [SZ16] have shown that such classes cannot obstruct the Hasse principle on Kummer surfaces. Recently, Corn and Nakahara [CN17] gave a positive answer to Ieronymou and Skorobogatov's question, by showing that a 3-torsion algebraic class can obstruct the Hasse principle on a degree 2 K3 surface over Q. Our main result shows that 3-torsion transcendental Brauer classes can obstruct the Hasse principle on K3 surfaces. Theorem 1.1. There exists a K3 surface Y over Q of degree 2, together with a class α in Br Y [3] such that Y (A Q ) = ∅ and Y (A Q ) {α} = ∅.Moreover, Pic Y ∼ = Z, and hence Br 1 (Y )/ Br 0 (Y ) = 0. In particular, there is no algebraic Brauer-Manin obstruction to the Hasse principle on Y .

Quantitative arithmetic of diagonal degree $2$ K3 surfaces

“…Nonetheless our next result will show that for such surfaces a Brauer-Manin obstruction to the Hasse principle is extremely rare. The second family (2) is much more common than (3), but our next theorem shows that between these two families a Brauer-Manin obstruction occurs for a comparable number surfaces asymptotically (differing by a small power of log T ), and that the surfaces in family (3) in fact dominate asymptotically. We denote by…”

“…We obtain the power savings over Theorem 1.1 by showing that if there is a Brauer-Manin obstruction then this imposes valuative conditions on the primes dividing the coefficients, so that up to small primes A i is essentially a square and A j /A k is essentially a cube, in both cases (2) and (3). To obtain the powers of log T saving, we show that the prime divisors of the coefficients must have special splitting types in certain number fields.…”

“…thus (1.2) cannot happen for Kummer surfaces. In [5] and [2] examples of K3 surfaces X are given with X(A Q ) = ∅, X(A Q ) Br 2 ⊥ = ∅, however these works left open the possibility of such counter-examples being explained by an even order element, i.e. whether X(A Q ) Br 2 = ∅.…”

Rational points and derived equivalence

“…The computations in § 4 were done in Magma [BCP97], and drew heavily on the program accompanying Berg and Várilly-Alvarado's paper [BV20]; we thank them for expert advice, S. Elsenhans for a crucial Magma tip, and B. Young for computer time.…”

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

“…Furthermore, there are examples of transcendental elements of order 3 on K3 surfaces that obstruct weak approximation (for example, see [Pre13], [New16] and [BVA20]). In all these cases, the evaluation map at the place at infinity has to be trivial, since Br(R) does not contain elements of order 3, and the obstruction to weak approximation comes from the evaluation map at the prime 3, which in every example is a prime of bad reduction for the K3 surface taken into account.…”