Tate-Shafarevich groups and $K3$ surfaces

Abstract

Abstract. This paper explores a topic taken up recently by Logan and van Luijk, finding nontrivial 2-torsion elements of the Tate-Shafarevich group of the Jacobian of a genus-2 curve by exhibiting Brauer-Manin obstructions to rational points on certain quotients of principal homogeneous spaces of the Jacobian, whose desingularizations are explicit K3 surfaces. The main difference between the methods used in this paper and those of Logan and van Luijk is that the obstructions are obtained here from explicitly constructed quaternion algebras, rather than elliptic fibrations.

Transcendental obstructions to weak approximation on general K3 surfaces

“…In contrast, for K3 surfaces the group Br(X)/ Br 1 (X) can be nontrivial, albeit finite [SZ08, Theorem 1.2]. For these surfaces, the arithmetically interesting groups Br(X)/ Br 0 (X), Br(X)/ Br 1 (X) and Br 1 (X)/ Br 0 (X) are the object of much recent research [Bri06, SZ08, ISZ09, KT09, LvL09, SZ09,Cor10]. Explicit transcendental elements of Br(X), or the lack thereof, play a central role in [Wit04,SSD05,HS05,Ier09].…”

Homogeneous spaces and degree 4 del Pezzo surfaces

“…Our main aim will be to show how, given any degree 4 del Pezzo surface V , to find C, δ such that V is the same as V δ (up to linear change in variable); the algorithm in the next section will not require anything of the geometry of H δ or J. For a description of the underlying geometry, see [12], [20] and Lemma 6.1 of [8].…”

Brauer–Manin obstructions on degree 2 K3 surfaces

“…The idea is to make the key isomorphism Br 1 (X)/Br 0 (X) → H 1 (k, P) explicit. As in [11], our program proceeds by first analyzing H 1 (H, P) for all the subgroups H of G, then identifying those H which will give rise to nonconstant cyclic algebras in Br 1 (X). Since |G| = 864 = 2 5 · 3 3 , only 6-primary elements appear in Br 1 (X)/Br 0 (X).…”