Tate-Shafarevich groups and $K3$ surfaces

Patrick Corn
2010 Mathematics of Computation  
This paper explores a topic taken up recently by Logan and van Luijk in [12]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
more » ... obtained here from explicitly constructed quaternion algebras, rather than elliptic fibrations. (3) where V = V × k k and Br 1 V = ker(Br V → Br V ) is the "algebraic part" of the Brauer group. Proof: This is a standard consequence of the Hochschild-Serre spectral sequence; see for instance [8], Proposition 1.3.7. It is, unfortunately, very difficult in general to compute the inverse of the isomorphism (3) explicitly. The crux of the computation is an explicit use of the fact (due to Tate) 2
doi:10.1090/s0025-5718-09-02264-9 fatcat:q5eokpksfnco3kdpwpwqosm7si